Magnetic excitations in SrCu2O3: a Raman scattering study

0.6 MB·

by  A. Gössling

Magnetic excitations in SrCu O : a Raman scattering study
2 3
A. G¨oßling, U. Kuhlmann, and C. Thomsen
Institut fu¨r Festk¨orperphysik, Technische Universit¨at Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
A. Lo¨ffert, C. Gross, and W. Assmus
Physikalisches Institut, J.W. Goethe-Universit¨at, Robert-Mayer-Str. 2-4, D-60054 Frankfurt a. M., Germany
(Dated: 6 February 2003)
We investigated temperature dependentRaman spectra of the one-dimensional spin-laddercom-
pound SrCu2O3. At low temperatures a two-magnon peak is identified at 3160±10 cm−1 and
its temperature dependence analyzed in terms of a thermal expansion model. We find that the
8 two-magnon peak position must include a cyclic ring exchange of J /J = 0.09−0.25 with a
0 couplingconstantalongtherungsofJ ≈1215cm−1 (1750K)inordceyrcltob⊥econsistent withother
0 ⊥
experimentsand theoretical results.
2

n PACSnumbers: 78.30.-j,75.50.Ee
a
J
8 Antiferromagneticcopperoxidespin-laddershavebeen metrical considerations and Jk/J⊥ ∼ 1.7 − 2.0.15,16,17
investigated intensively from a theoretical and exper- This ring exchange can be understood as a superpo-

] imental point of view.1 Superconducting under high sition of clockwise and counter clockwise permutations
l
e pressure2 they form a bridge between 1D Heisenberg of four spins around a plaquette (positions ABCD in
r- chains and a 2D Heisenberg square-lattice, which is also Fig.1). AtermHcycl =JcyclPiKAiBCD withKABCD =
t a model for high-T superconductors. The magnetic (S S )(S S )+(S S )(S S )−(S S )(S S ) has
s C A B C D A D B C A C B D
. ground state of a spin ladder is surprisingly not a long to be added to the conventional Heisenberg Hamilto-
t
a ranged N´eel state but a short ranged resonating valence nian H.18 In order to achieve the isotropic limit of
m bond state.3 The first excited state is separated from J /J ∼ 1 a ring exchange of J /J ∼ 0.18− 0.30
k ⊥ cycl ⊥
- the ground state by a finite energy ∆. The spin gap was introduced.16,17
d was first predicted theoretically4,5 and later confirmed InthispaperwestudiedtheRamanspectraofthetwo
n
o experimentally.6 leg S = 21 spin-ladder compound SrCu2O3. In addition
c to phonons we investigatedthe two-magnonpeak in this
[ Typical realizations of a two-leg spin-ladder are the compound at temperatures between 25 K and 300 K.
1 compoundsSrCu2O3(Ref.7)and(Sr,Ca,La)14Cu24O41.8 We find that the inclusion of a ring exchange is neces-
v The first compound is a prototype of weakly coupled sary for the understanding of the magnetic properties of
3 Cu2O3 spin-ladders while the latter consists of a lad- SrCu2O3.
0 der and a Cu2O edge sharing chain part. In contrast
3 to SrCu O , (Sr,Ca,La) Cu O is intrinsically doped
2 3 14 24 41
1 with holes. A schematic view of the compound SrCu O
2 3
1. is shown in Fig. 1. Cu atoms, represented by d-orbitals,
0 are antiferromagnetically coupled via an intermediate
8 oxygen p-orbital by superexchange9. The Sr atoms are
0 located in between the planes containing the Cu O
2 3
:
v atoms. The coupling constants along the legs and the
A D
i rungs are denoted with J and J . The interladder cou-
X k ⊥ J
pling is negligible to first approximation because the su- (cid:1) J
r perexchange via a Cu-O-Cu path with a 90◦ bond an- cycl
a
gle has a smaller orbital overlap than with a bond an- B C
J
gle of 180◦.10 Thus, a ladder can be considered an iso- (cid:0)
a
lated quasi one-dimensional object with a Heisenberg
Hamiltonian H = J P S ·S +J P S ·S . In
⊥ rung i j k leg i j b
the literature coupling ratios J /J ∼ 1.7 − 2.0 have O(p ) Cu (d ) Sr
k ⊥ x x²-y²
been reported11,12 taking the Hamiltonian mentioned
above into account for analyzing the data. On the other
FIG. 1: Schematic view of the two-leg ladder SrCu2O3 pro-
hand with almost identical Cu-Cu distances in leg and
jected on the ab-plane. The Sr atoms are located in between
rung direction7,8,13 one would expect an isotropic ra- the planes containing the Cu2O3 atoms. The magnetic cou-
tio Jk/J⊥ ∼ 1. This picture was also confirmed by plingconstantsalong therungsandthelegs areindicated by
an analysis of optical conductivity data.14 Recently the J⊥ and Jk and the cyclic ring exchange by Jcycl. The cou-
inclusion of a cyclic ring exchange J was suggested pling between two Cu d-orbitals is caused by superexchange
cycl
in order to resolve the discrepancy between the geo- interaction via an O p-orbital.
2
Polycrystalline SrCu O was grown under high pres-
2 3
sure asdescribed by Lo¨ffertet al.19 The crystallographic
structurewasverifiedbyx-raydiffraction. Inadditionto
SrCu O small amounts of Sr Cu O , CuO and Cu O
2 3 2 3 5 2
were detected. Measurements were performed using a
LabRam spectrometer (Dilor) including a grating with
600 and 1800 grooves/mm in backscattering geometry
and a multichannel CCD detector. An Ar+ laser spot
(488nm)wasfocusedbya 80×microscopeobjectiveon
smallindividualcrystallitesonthe samplesurfacewitha
diameter of 1-2 µm. The spectra in Fig. 2 were mea-
sured at room temperature. The crystallite was cho-
sen by using a polarization microscope :20 The sample
surface was illuminated normally with linear polarized
white light. The reflected light was detected through
an analyzer crossed to the polarizer with a CCD cam-
era. Being an anisotropic material the reflected light is
in general elliptically polarized. As for the ladder com-
pound LaCuO ,20 we assume the direction along the
2.5
ladder (a-axis) to be the one with the largestanisotropy
in SrCu O , whereas the b and c axes are approximately
2 3
optically isotropic. Thus, there should be a difference in
brightness along and perpendicular to the a-axis when
rotating the sample. If the incident wave vector k is
i
parallel to the a-axis the image stays dark while rotat-
ing the sample. For k ⊥ a the crystallite appears dark
i FIG. 2: Raman spectra in parallel and crossed polarization
and bright for four times when turning the sample 360◦.
measured at T=295 K.Theinsets showthelow energy parts
The spectra in Fig. 2 were measured on a crystallite for of the spectra measured in parallel polarization.
k ⊥a. For our measurements down to 25 K (spectra in
i
Fig. 3) the sample was mounted on the cold finger of an
Oxford microcryostat. lowed A modes. In Sr Cu O modes at 316 cm−1
g 14 24 41
The Raman spectra of SrCu2O3 can be divided into and 549 cm−1 were observed.23 The latter is believed to
a low (100-1700 cm−1) and a high-energy (2000-4600
be the breathing mode of the O-ladder atoms locatedon
cm−1) region as shown in Figs. 2 and 3. We verified the ladder legs23, which is in accordance with our as-
the chemical composition of our polycrystalline sample signment. The structure at about 1150 cm−1 was also
by an analysis of the phonons in the low energy part of
observed in Sr Cu O , its origin attributed to two-
14 24 41
thespectra. Westartwithafactorgroupanalysis(FGA) phononprocesses.23,27 Knowingonly few phonons atthe
of the vibrational modes according to Rousseau et al.21
Γ point the details of the broad structure around 1150
The unit cell of SrCu2O3 consists of two formula units. cm−1 in SrCu O remainunclear. In summary,we iden-
Structure analysis7,22 attributes SrCu O to the space 2 3
2 3 tified the two allowedA ladder phonons at 310 and 550
group Cmmm (D21h9). The FGA results in cm−1 in SrCu2O3 at roogm temperature, the B1g modes
were too weak to be observed, and the B not allowed
Γ =2A +4B +2B +4B +4B +2B (1) 3g
SrCu2O3 g 1u 1g 2u 3u 3g for our geometry.
Having a center of inversion only even modes show Ra- In Fig. 2 in addition to the phonons a peak around
manactivity,resultinginsixRamanactivephonons. We 3000 cm−1 was observed in (aa) and (bb) polarizations
expectthetwoA modesinthespectrameasuredinpar- while it was absent in the crossed polarizations (ab) and
g
allel polarizationwhile the four evenB modes should be (ba). It can be shown from the two-magnon Raman
observed in crossed polarization. Hamiltonian28 that for a two-leg Heisenberg ladder the
Figure2showsfourspectrameasuredatT=295Ktwo two-magnon peak is forbidden in all crossed polariza-
in parallel and two in crossed polarization. The axes tions. The observed spectra follow the selection rules
were identified using the described analysis with polar- of a two-magnonpeak.
izedlight;comparingFig.2tothespectraoftheverysim- In Fig. 3 the Raman spectra of the high-energy re-
ilar ladder compound Sr Cu O ,12,23,24,25 we identify gion are presented for different temperatures. The spec-
14 24 41
the a (leg)andb(rung)axesasindicatedin the figure.26 tra were measured on another crystallite between 25 K
In (bb) polarization only a single phonon at 550 cm−1 and 300 K. Comparing them to the aligned spectra in
isobserved,whilein(aa)twophononsat310,550cm−1, Fig. 2 we assign the upper spectrum as (aa) and the
and in addition several broad peaks around 1150 cm−1 lower one as (bb) polarized. In (aa) polarization a sharp
are visible. We assign these two phonons as the two al- peakisvisiblewhichbroadensandshiftstolowerenergies
3
resentingthestrengthoftheexpansion,andω isanaver-
agedphononenergy(Einsteinmodel). Within the three-
bandHubbard-modelJ equals4t4 /ε2 (U −1+ε−1)with
pd pd d pd
t being the overlapintegralbetween Cu-O sites, U the
pd
Coulombrepulsion,andε thechargetransferenergy.29
pd
The overlap integral t ∝ d−4 and the Coulomb repul-
pd
sionU ∝d−1areafunctionsofdasshownbyHarrison.31
(p. 643) Using the parameters C , J can be written as
1,2
J =C d−15+C d−16 (2)
1 2
The temperature dependence of the two-magnonenergy,
whichisproportionaltoJ,arisesfromaTaylorexpansion
around d .
0
Bω
E =E − (3)
2−mag 0 exp(ω/T)−1
E is the two-magnon energy at T=0 K, B is a dimen-
0
sionless constant proportional to A, and ω is the aver-
aged phonon frequency. Other power-law exponents in
Eq. 2 have been used in similar systems in the litera-
ture (e.g., Kawada et al.30); the further analysis does
notdependmuchonthischoiceaslongB issimplyapa-
rameter. The temperature dependence of the peak/edge
FIG.3: Ramanspectrameasured withthesamepolarization
maxima of both spectra in the insets of Fig. 3 were fit-
asinFig.2withλex =488nmat25K(solid),150K(dashed)
ted simultaneously using Eq. (3) with the parameters
and 250 K (dotted). Insets: peak positions (top) and edge
B =0.94±0.05,ω =160±10cm−1, E (aa)=3160±10
position (bottom) of the two-magnon peak as a function of 0
temperature. cm−1 andE0(bb)=3130±10cm−1. Thefitshowsexcel-
lentagreementwiththeexperimentaldata. Theaveraged
phonon frequency ω is comparable to those of high-T
C
withincreasingtemperature. In(bb)polarizationamuch cuprate materials.32 The different values for E0 in the
broaderpeak ismeasured;itremainsbroadalsodownto (aa) and (bb) spectra in Fig. 3 are caused by the differ-
the lowest temperatures. We identify these features to ence between peak edge and peak maximum. For the
be the two-magnon peaks which were also observed in further analysis we take only the energy E0(aa)=3160
Sr Cu O at a lower energy (maximum for T=8-20 cm−1 into account. The observed temperature depen-
14 24 41
K at 2900-3000 cm−1).12,23,24,25 The energy of the peak dence is inaccordancewiththe assumedmagneticorigin
maxima as a function of temperature is shown in the of the peak. The common picture of superexchange is
upper inset of Fig. 3. In (bb) polarization the left edge confirmed by our measurements.
position is plotted as a function of temperature in the We now show that in order to compare the energy
lower insets of Fig. 3 due to difficulties in determining E0 with existing experiments and theoretical results in
the precise peak maxima. The two-magnon peak shifts a consistent way the cyclic ring exchange must be in-
for T >75 K almost linearly with temperature and satu- cluded. Note, that it is not possible to determine the
rates at low temperatures at 3160 cm−1 (peak position) magnetic constants and the spin gap directly from the
and 3130 cm−1 (edge position). two-magnonpeak position. In order to obtain these val-
Using a simple model of thermal expansion we are ues from our spectra we have to make several assump-
able to understand the temperature dependence of these tions based on calculations: (i) Ab initio calculations10
peaks. Thestartingpointisasystemcontainingtwocop- yieldedJk/J⊥ =1.1forthecouplingconstantratio. This
perd-orbitalsandoneintermediateoxygenp-orbitalwith ratiois alsoclose to the isotropiclimit, whichcanbe de-
a 180◦ bond angle as found in flat ladder compounds, ducedfromgeometricalconsiderations. (ii) We include a
hence the model is not only applicable to SrCu2O3 but ring exchange of Jcycl/J⊥ =0.09−0.25. Cyclic ring ex-
also, e.g., to Sr14Cu24O41. The two copper atoms are changevaluesonthesameorder(Jcycl/J⊥ =0.18−0.30)
coupled by superexchange with the coupling constant have also been introduced for the comparable two-leg
J. We assume that the two-magnon energy scales lin- spin ladder compound (La,Ca)14Cu24O41 (Refs. 16,17).
early with J. The average distance d between two cop- (iii) Schmidt et al. have shown recently in a theoreti-
per atoms is a function of temperature. Due to an an- cal Raman study for a Heisenberg two-leg ladder with
harmonic interatomic potential the average distance in- Jk/J⊥ = 1.2 and Jcycl/J⊥ = 0.2 that the two-magnon
creases in first order as d = d + Aω/(exp(ω/T)−1), peakmaximumshouldbe locatedatE ≈2.6·J .18 Be-
0 0 ⊥
where d is the distance at T=0 K, A is a constant, rep- cause the ratio J /J = 1.2 is slightly larger than the
0 k ⊥
4
onestatedaboveforSrCu O wetakeE /J =2.6±0.1 NMR, the upper one to ∆ from neutron scattering.
2 3 0 ⊥
for a further analysis. (iv) Calculating the spin gap we
In conclusion we found the two allowed A phonons
use the expression ∆≈0.48·J −1.08·J which has g
⊥ cycl at 310 cm−1 and 550 cm−1 in SrCu O and studied
been derivedby Nunner et al.17 for a two-legHeisenberg 2 3
the magnetic excitations in this compound. A peak
ladder with ring exchange having adjusted for the dif- with an energy of 3160±10 cm−1 at low temperatures
ferent definitions of J . The expression for ∆ is valid
cycl starts to shift to lower energies with increasing temper-
approximately in the range J /J = 1.0−1.3. As a re-
k ⊥ ature. We identified this peak as the two-magnon peak
sult we obtain from our experiment J = 1170−1260
⊥ and confirmed this by its temperature dependence. We
cm−1 (1680−1810 K) for the coupling constant. Tak-
derived a simple expression which describes excellently
ing J = 1215 cm−1 results in ∆ = 260 − 470 cm−1
⊥ the peak energy as a function of temperature. For low
(380−680K)forthespingap. Thelattervalueisingood
temperatures we are able to explain the two-magnon
agreement with spin gap values measured with suscepti-
peak energy E =3160 cm−1 in accordance with exist-
bility(420K),6 NMR(680K)6,33,34 andneutronscatter- 0
ing theoretical and experimental results by including a
ing(380K)onSr Cu O .11 Incontrast,excludingthe
14 24 41 cyclic ring exchange using the values: J /J = 1.1,
spin exchange coupling would yield a spin gap value of k ⊥
J /J = 0.09−0.25, J ≈ 1215 cm−1 (1750 K) and
approximately840K,whichis outof scaleby a factorof cycl ⊥ ⊥
1.2−2.2whencomparingtothe experimentsmentioned. ∆=260-470cm−1 (380−680 K).
Ourexperimentthussupportsthepresenceofafourspin We thank K. P. Schmidt and G. S. Uhrig for helpful
ringexchangeontheorderofJ /J =0.09−0.25. The and valuable discussions. This work was supported by
cycl ⊥
lowerlimitofJ /J soobtainedcorrespondsto∆from the Deutsche Forschungsgemeinschaft,SPP 1073.
cycl ⊥
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