Collective spin excitations in a quantum spin ladder probed by high-resolution Resonant Inelastic X-ray Scattering

0.19 MB·English

by  J. Schlappa

Collective spin excitations in a quantum spin ladder probed by high-resolution
Resonant Inelastic X-ray Scattering
J. Schlappa,1 T. Schmitt,1,∗ F. Vernay,1 V. N. Strocov,1 V. Ilakovac,2,3 B. Thielemann,4 H. M. Rønnow,5 Vanishri
S.,6 A.Piazzalunga,7 X.Wang,5 L.Braicovich,7G.Ghiringhelli,7 C.Marin,6 J.Mesot,4,5 B.Delley,1 andL.Patthey1
1Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
2Universit´e Pierre et Marie Curie - CNRS UMR 7614, LCP-MR, Paris, France
3Universit´e de Cergy-Pontoise, D´epartement de Physique, F-95000 Cergy-Pontoise, France
4Laboratory for Neutron Scattering, ETH Zurich and Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
5Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
9 6INAC/SPSMS/DRFMC, CEA-Grenoble, 17, rue des Martyrs, 38054 Grenoble Cedex 9, France
0 7CNR/INFM Coherentia/Soft - Dip. Fisica, Politecnico di Milano, p. Leonardo da Vinci 32, 20133 Milano, Italy
0 (Dated: January 31, 2009)
2
We investigate magnetic excitations in the spin-ladder compound Sr14Cu24O41 using high-
n
resolution Cu L3-edge Resonant Inelastic X-ray Scattering (RIXS).Our findings demonstrate that
a
RIXScouplestocollectivespinexcitationsfrom aquantumspin-liquidgroundstate. Incontrastto
J
InelasticNeutronScattering(INS),theRIXScross section changesonlymoderately overtheentire
1
BrillouinZone(BZ),revealingahighsensitivityalsoatsmallmomentumtransfers. Thetwo-triplon
3
energygapisfoundtobe100±30meV.Ourresultsaresupportedbycalculationswithinaneffective

Hubbardmodel for a finite-sizecluster.
]
l
e PACSnumbers: 78.70.En,75.25.+z,71.10.Pm,75.30.Ds
-
r
t
s Collective excitations in strongly correlated electron netic excitations fromsucha quantumgroundstate. We
.
t materials remain a pivotal challenge in contemporary demonstrate unambigously that this technique couples
a
m solid state physics. Magnetic excitations are heavily to purely quantum mechanicalexcitations from a singlet
debated to provide the pairing interaction in the high- ground state. While the INS cross section is inherently
-
d temperature and unconventional superconductors [1, 2]. lowaroundtheBrillouinzone(BZ)center(smallmomen-
n
Fromthatperspectivequantumspinsystemsattractcon- tum transfers) in a similar compound [20], the observed
o
siderable interest. While most such materials, e.g., the RIXS signal is found to be intense all over the BZ. A
c
[ cuprate superconductors, exhibit enormous complexity, numerical investigation of a Hubbard model as well as

the two-leg spin ladder is easier to tract theoretically the opticaltransitionselectionrules leads us to conclude
2
[3, 4, 5, 6]. It consists of two parallel chains (legs) with that the response is due to two-triplon excitations. We
v
1 a transverse (rung) exchange coupling. This system fea- demonstrate that in the case of a gapped spin liquid,
3 turesa singletgroundstate anddispersivetripletexcita- RIXS is particularly sensitive to these excitations. We
3 tions, that both have quantum mechanical origin with- thus are in position to directly evaluate the two-triplon
1
out any classical counterpart. To date, mainly two tech- energy gap in Sr Cu O at zero momentum transfer
14 24 41
.
1 niques havebeen establishedasmomentum- andenergy- as 100±30 meV.
0 resolvedprobesofthedispersionofcollectiveexcitations:
9 angle-resolvedphotoelectronspectroscopy(ARPES)and RIXS experiments were performed at the Advanced
0
inelasticneutronscattering(INS)forchargeandspinde- Resonant Spectroscopies (ADRESS) beamline [10] at
:
v greesoffreedom,respectively[7,8]. Duetothelatestin- the Swiss Light Source (SLS), Paul Scherrer Insti-
i
X strumentalimprovements[9,10],theenergyscaleofmag- tut, Switzerland, using the Super-Advanced X-ray
neticexchangeisbecomingreadilyaccessibleforresonant Emission Spectrometer (SAXES) [9]. A flux of
r
a inelasticx-rayscattering(RIXS)[11,12,13,14],whichis 1013 photons/sec/0.01% bandwidth was focused to a
promising to give information on both, spin and charge spot size below 8×100 µm (V×H). RIXS spectra were
degreesoffreedom,andinadditionisanelement-specific recordedintypically1houracquisitiontime,achievinga
technique. Furthermore, RIXS requires only small sam- statistics of 50-200 photons in peak maxima. The com-
ple volumes (< 0.1 mm3). Recent RIXS studies of mag- bined energy resolution was 120 meV at the Cu L3 edge
netic systems focussed on spin excitations in long-range (∼930 eV). The Sr14Cu24O41 single crystal was grown
ordered magnets [15, 16, 17]. with the traveling-solvent floating zone method. Sam-
pleswerecleavedex-situ,producing amirror-likesurface
In this letter, we report our study of the two-leg with b-orientation. The sample was mounted with b-
quantum spin ladder Sr Cu O [18, 19] by means andc-direction(leg-direction)inthe scatteringplane, as
14 24 41
of momentum-resolved high-resolution RIXS at the Cu depicted in the sketch of the experimental geometry in
L edge. One important question is how RIXS, which Fig.1. Twogeometrieswereusedwiththeanglebetween
3
also couples to charge, can provide information on mag- the incident (k) and scattered (k′) light being 90◦ (up-
2
90° b
nt. (a. u.) Cu Lgin3eco. m20. °90°XAS, TFY RIX S1 R5T K θ c qk´ 0.2
I Ph9o3to0n e9n3e2rgy9 (3e4V) 130° k cb a. u.) 0.1
-0.8 -0.6 -0.4 -0.2 0.0 0.2 θ k´ sity ( 0.0 cq
Energy loss (eV) k q en (2
nt -0.1 π
F(sIoGli.d1l:inLee)fta:nCduRTL3(dRaIsXhSedofliSnre1)4:C2u02◦4Ogr4a1zimngeaisnucrieddenacte,1590K◦ ring i -0.2 L/c)
e
scatteringangle, σ-polarized light (E ka). Excitation energy att
isindicatedintheXASdata(inset)intotalfluorescenceyield c -0.3
S
(TFY) mode, acquiredat 15 K in thesame geometry. Right:
◦ ◦
sketchoftheexperimentalset-up: 90 -(top)and130 -(bot- -0.4
tom) scattering geometries, k (k′) denotes the wavevector of
the incident (scattered) light and q =k′−k the transferred
-0.8 -0.4 0.0
momentum.
Energy loss (eV)
per part) and 130◦ (lower part). With this setup one V) 0.6 Int. (a.u.)
e
can cover at the Cu L3-edge up to 90% of the BZ in r ( 3.0
Sr Cu O (the lattice constant of the ladder system e 0.4 2.5
14 24 41 sf
is c = 3.93 ˚A). Incident light was linearly polarized ei- n 2.0
L a
tphlaenreou(πto-pfotlhaeriszcaattitoenr)i.ngplane(σ-polarization)orinthe gy tr 0.2 11..50
er 0.5
TheleftpanelinFig.1displaysCuL3RIXSspectraof En 0.0
Sr Cu O measuredatroomtemperature(RT)andat
14 24 41
15K.Spectrawereobtainedwithσ-polarizedlightat20◦ -0.4 -0.2 0.0 0.2
grazingincidencein90◦ geometry. Theexcitationenergy q (2π/c )
c L
was detuned by ∼ 0.2 eV from the resonance maximum
to reduce the elastic contribution, indicated in the x-ray
FIG. 2: (color online) Dispersion of magnetic excitations in
absorption data (XAS) in the inset (acquired in TFY Cu L3 RIXS from Sr14Cu24O41. Upper panel: experimental
modewithaphotodiode). BothRIXSspectrarevealtwo spectraobtainedat15Kwithσ-polarizedlight. Lower: RIXS
intense well-separated structures. One peak at zero en- data as intensity map vs. momentum and energy transfer
ergy loss consists of the elastic signal with unresolved after subtraction of theelastic signal.
low-energy contributions from phonons and presumably
magnetic excitations of the chains. The second peak at
final energy loss represents a low-energy excitation. The tainedin90◦geometry(black)andforhighermomentum
position at around 270 meV corresponds to the energy transfer in 130◦ geometry (gray). All spectra reveal the
range of intra-ladder exchange coupling. Previous Cu same pronouncedmagnetic mode as in Fig.1, dispersing
K RIXS investigations reported on charge excitations in strongly across the BZ [24].
the energyrangeof2-6eV[21, 22]. The spectrumatRT The lower panel of Fig. 2 displays an intensity map of
is slightly broader than at 15 K, which is the expected the RIXS data plotted vs. momentumand energytrans-
temperature dependence of spin excitations. fer. The elastic contribution has been subtracted, after
Tounderstandthe localvs. collectivecharacterofthis fitting each experimental spectrum with two Gaussians.
excitation we studied its dispersion upon q , momentum The magnetic excitation is seen here to disperse around
c
transfer along the leg-direction. Since Sr14Cu24O41 is a the BZ center (qc = 0), where it also reaches its mini-
low-dimensional system, where we expect no dispersion mum in energy loss. With larger |qc| it moves first to-
along b, we could map out q by simply rotating the wardshigher energy losses andis then folding back close
c
sample around a. Momentum transfer dispersion was to qc = 0.3×2π/cL towards the BZ edge. The width
measuredat15K usingthe samephotonenergyandpo- increases slightly towards BZ edge.
larization as in Fig. 1. RIXS data for different q trans- This dispersing behavior reveals the collective charac-
c
ferarepresentedinFig.2. Theupperpaneldisplaysraw ter of the observed magnetic mode. Similar dispersion
spectranormalizedwithacquisitiontimeandageometry- hasbeenpartiallyobservedinINSfromLa Sr Cu O
4 10 24 41
dependent factor accounting for variations in scattering [20]. Our dispersion curve revealed by the Cu L RIXS
3
volume [23]. Spectra for |q | < 0.21×2π/c were ob- data matches well with the two-triplon mode measured
c L
3
-0.4 -0.2 0.0 0.2 0.4
3 L and S remain good quantum numbers and in the elec-
0.4
tricdipoleapproximationonlytransitionswith∆L=±1
V) C
s (e 0.3 2 ente aelnedm∆enSta=ry0mwagilnlebtiecaelxlocwiteadti.onIn(ttrhipelopnr)esceonntssisytssteinmptrhoe-
of mas 0.2 r of ma mtoo∆tinSg=a1spainnd-sinnogtlettoinatdoipaotleri-pallelotw, ewdhticrhancslietaiornly. lHeaadvs-
Center 0.1 1 ⊥ss (J) einvgenan∆umSb=er0ofetxhceitsaettiorinplnoencsestsoagreitlhyemr,etahnesleeaxdciitnigngpraon-
cess being thus a two-triplon excitation. We simulate
0.0 0
which kind of magnetic excitations will occur in the lad-
-0.4 -0.2 0.0 0.2 0.4
der system in the Cu L RIXS process by confining our
q (2π/c ) 3
c L considerations to the optical selection rules. As a mini-
nt. (a. u.) σπ -- ppoollaarriizzaattiioonn 1/2 mfroamlmaomdeullatin-beaffnedctHivuebHbuabrdbamrdodHealmisilutosendia[n26d,o2w7n]:folded
catt. i Residual H= X tij(cid:16)d†i,σdj,σ + h. c.(cid:17)+UXni,↑ni,↓ (1)
S hi,ji,σ i
-0.6 -0.4 -0.2 0.0
Energy loss (eV) with n = d† d . The hopping parameters in (1) are
i,σ i,σ i,σ
taken as t = 0.35 eV, t = 0.3 eV while the on-site
⊥ k
FIG. 3: (color online) Upper panel: dispersion curve of the
Coulomb repulsion is U = 3.5 eV. According to results
collective spinexcitations across thefirstBZ. Theopen sym-
◦ of XAS, the concentration of holes in the ladder system
bolsrepresentmirroreddatapointsfrom130 geometry. The
is smaller than 10% [28]. We consider therefore that we
right axis is scaled in units of J along the rungs (J⊥), ex-
tractedfromthetheoreticalmodelbelow. Lower: RIXSspec- are at half-filling (1 hole per Cu-site). From the above
tra measured close to the Γ point for qc = −0.04×2π/cL, parameters we extract J⊥ ∼ 140 meV, which is close
using σ- and π-polarized light. The π-polarized spectrum is to the experimental value measured with INS or Raman
fitted by two Gaussians, the residual is represented by the scattering [20, 29, 30]. In this picture the experiment
thin blue line.
can be considered as a coherent process of two optical
transitions: promoting a Cu-2p electron to the 3d-band
and the recombination of the 2p-hole with an electron
with INS, however, the observed intensity vs. momen- from the 3d band. In essence, this can be rationalized
tum dependence is different. While INS intensities are as having a non-magnetic impurity in the 3d-band for
high in the region qc > 0.25 and low for small momen- the intermediate state. In the lower energy-loss region,
tum transfer,ourRIXS data revealsuniformintensity of this will naturally lead to magnetic rearrangements and
the excitation over the BZ. The magnetic excitation can finite overlaps with final excited states in different sym-
be detected close to the BZ center, where it approaches metry sectors than the ground-state. The Hamiltonian
a finite energy loss value. in Eq. (1) was fully diagonalized for an 8-site cluster
Using the procedure described above, we extracted and eigenvalues and eigenvectors were obtained for the
from our data the center of mass of the magnetic ex- ground and final states at half-filling (8 particles) and
citation for the different points in the BZ. The resulting for the intermediate states (7 particles). Spectral inten-
dispersion curve is presented in upper graph of Fig. 3. sities were calculated using the Kramers-Heisenberg for-
Data points corresponding to spectra measured in 130◦ mula [11] with the optical transition operator expressed
scatteringgeometryhavebeenmirroredupontheBZcen- intheholerepresentation: Ok =Pj,σp†j,σdj,σeik·rj,dre-
ter. Values close to qc = 0 (qc < 0.05×2π/cL) were moving a hole in the 3d-band and p† creating one in the
obtained from π-polarized data to suppress contribution Cu-2p shell. Presence of a Cu-2p core-hole in the inter-
from the elastic channel. We can follow for the observed mediate states was accounted for by an on-site Coulomb
energy gap the energy loss of 100±30 meV and for the interaction[31]. The calculatedRIXS profiles forthe ac-
maximum loss value 320 meV. The lower panel of Fig. 3 cessible k-points are displayed in Fig. 4. These spectra
presents fit of a π-polarized spectrum (thick solid line). show a dispersive low-energy excitation of energy loss
Interestingly, the residual reveals weak intensity in the ≤ 400 meV. Comparison of the energy position in the
energy range between -0.2 and -0.6 eV. simulated and the experimental data reveals an offset
To assignthe magnetic excitationwhichwe observein of ∼ 100 meV between them, which can be ascribed to
our RIXS data, we take a look at the electrical dipole finite-sizeeffects. Nevertheless,despite thefinite cluster-
transition selection rules. We confine our considerations size, the excitation disperses in qualitatively the same
to collective excitations and neglect in a first approxi- way as in the experiment. We therefore conclude that
mation the spin-orbit coupling [25]. As a consequence the observed mode in our RIXS data is in the ∆S = 0
4
SAXES spectrometer and C. Quitman for his critical
0.6
reading of the manuscript. Work at the EPFL is sup-
0.5 ported by the Swiss NSF and work at CEA-Grenoble by
V)
s (e0.4 the Indo-French project 3408-4.
s
o
y l0.3
g
er
En0.2
∗
0.1 Electronic address: [email protected]
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3
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