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PHYSICAL REVIEW B 95, 054409 (2017)
Spin pseudogap in the S = 12 chain material Sr2CuO3 with
impuritiesG. Simutis,1,* S. Gvasaliya,1 N. S. Beesetty,2 T.
Yoshida,3 J. Robert,4 S. Petit,4 A. I. Kolesnikov,5 M. B. Stone,6
F. Bourdarot,7
H. C. Walker,8 D. T. Adroja,8 O. Sobolev,9 C. Hess,10 T.
Masuda,3 A. Revcolevschi,2 B. Büchner,10 and A. Zheludev11Neutron
Scattering and Magnetism, Laboratory for Solid State Physics, ETH
Zürich, Zürich, Switzerland
2Synthese, Proprietes et Modelisation des Materiaux, Universite
Paris-Sud, 91405 Orsay Cedex, France3Institute for Solid State
Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba
277-8581, Japan
4Laboratoire Léon Brillouin, CEACNRS, CEA-Saclay, F-91191
Gif-sur-Yvette, France5Chemical and Engineering Materials Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
6Quantum Condensed Matter Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831-6393, USA7Modélisation et
dexploration de la matiére, Université Grenoble Alpes et CEA,
INAC, 17 rue des Martyrs, 38054 Grenoble, France
8ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot,
Oxon OX11 OQX, United Kingdom9Forschungsneutronenquelle Heinz
Maier-Leibnitz (FRM-II), TU Munchen, D-85747 Garching, Germany
10Leibniz Institute for Solid State and Materials Research IFW
Dresden, P.O. Box 270116, D-01171 Dresden, Germany(Received 11
November 2016; revised manuscript received 23 January 2017;
published 7 February 2017)
The low-energy magnetic excitation spectrum of the Heisenberg
antiferromagnetic S = 1/2 chain systemSr2CuO3 with Ni and Ca
impurities is studied by neutron spectroscopy. In all cases, a
defect-induced spectralpseudogap is observed and shown to scale
proportionately to the number of scattering centers in the spin
chains.
DOI: 10.1103/PhysRevB.95.054409
I. INTRODUCTION
Defects play a special role in one dimension due to itsintrinsic
topology [1,2]. This is especially pronounced inquantum magnets
where both the ground states and theexcitations are modified upon
the introduction of imperfec-tions in magnetic lattices [3–10].
Recent experiments havedemonstrated that when impurities are added
to the proto-typical Heisenberg antiferromagnetic S = 1/2 chain
materialsSrCuO2 and Sr2CuO3, their magnetism is suppressed
[11–16].Long-range magnetic ordering arising from residual
three-dimensional interactions becomes inhomogeneous and occursat
lower temperatures [11,16]. Additionally, the
spin-latticerelaxation rate 1/T1 measured by NMR experiments
dropsat low temperatures, suggesting the depletion of
low-energyexcitations [12,15,17]. Defects also affect the heat
transportand compromise its ballistic nature [18,19].
The influence of defects on the magnetic excitation spec-trum
can be directly probed using neutron scattering. Wehave previously
employed this method to demonstrate theopening of a spin pseudogap
in the Heisenberg spin-chainsystem SrCuO2 with chain-breaking Ni2+
impurities [13].We proposed a simple chain-fragmentation model
basedon previous theoretical work [20] and derived
quantitativepredictions for the excitation spectrum in the presence
ofdefects. This model was able to account for the data overa wide
temperature range, on an absolute scale, and with noadjustable
parameters. Nevertheless, a few questions remainedto be answered:
(i) Does the size of the pseudogap indeed scalewith the impurity
concentration as predicted? (ii) What is therole of the
double-chain structure of SrCuO2? and (iii) Whateffect would other
types of impurities have?
The present paper aims to settle these outstanding issues.Here
we study a related spin-chain compound, namely,Sr2CuO3. It has the
benefit of having a simpler single-chain
*[email protected]
structure, as opposed to the paired chains in SrCuO2 [21–23].Two
types of impurities are investigated: S = 1 Ni2+ ions,which replace
the S = 1/2 Cu2+ ions in the chain [11,16,20];and Ca2+ ions, which
replace Sr2+ and therefore affect theCu2+ chains only indirectly
[17,19].
II. EXPERIMENT
Single crystals of Sr2CuO3 with Ni and Ca impurities weregrown
using floating zone furnaces as described in earlier re-ports
[15,19]. They crystalize in an orthorhombic Immm spacegroup, with
lattice parameters of a = 3.9089, b = 3.4940, andc = 12.6910 Å for
the pure compound [24]. The spin chainsformed by the Cu2+ ions run
along the crystallographic bdirection. The exchange interaction
between the spins hasbeen estimated as J = 241(11) meV [23]. While
the chainsare very well isolated, three-dimensional ordering still
takesplace due to residual three-dimensional interactions. In
thepure system the spins order at TN = 5.4 K [22] and the
orderingtemperature is rapidly decreased with the introduction
ofimpurities [11]. All of the measurements presented here
wereperformed above the three-dimensional ordering temperatures.The
measurement temperatures were much lower than thetemperature
corresponding to the intrachain exchange energy.
In order to study the effects of disorder, diligent attentionhad
to be given to controlling the level of impurities. Inaddition to
carefully monitoring the ingredients in the growthprocedure and
performing energy-dispersive x-ray analysismeasurements, we have
also measured the magnetic suscep-tibility in order to estimate the
extent of chain fragmentation.The small samples used for
susceptibility measurements werecut from the same crystals used for
neutron spectroscopy. Thecrystals were aligned with the b crystal
axis parallel to theapplied magnetic field. Data were taken with
the vibratingsample magnetometer option of the Quantum Design
PPMSinstrument.
The samples studied with neutron spectroscopy were madeof a few
coaligned crystals with the crystal a axis perpendicular
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Society
https://doi.org/10.1103/PhysRevB.95.054409
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G. SIMUTIS et al. PHYSICAL REVIEW B 95, 054409 (2017)
to the scattering plane. The sample with 1% Ni
impuritiesconsisted of three coaligned single crystals with a total
massof 8.6 g and an FWHM mosaic of 0.4◦ as measured at the 020and
002 peaks. The sample with 2% Ni impurities consistedof two single
crystals with a mass of 4.6 g and mosaic of 0.7◦.The sample with 5%
Ca impurities was made of two crystalswith a total mass of 3.5 g
and a mosaic of 0.3◦.
Neutron spectroscopy has been performed at a number ofuser
facilities. The existence of the spin pseudogap in Ni-substituted
samples was established using the 4F2 three-axisspectrometer at
LLB, using a combination of energy andmomentum scans with the final
neutron wave vector fixedat kf = 1.55 Å−1. Preliminary
measurements on the Ca-substituted compound were performed on the
PUMA three-
axis spectrometer at FRM-2, using kf = 2.66 Å−1. Most ofthe
measurements were then performed at the SEQUOIA [25]time-of-flight
spectrometer at ORNL using incident energiesof 12, 20, and 50 meV
with the high-resolution Fermi chopperfrequencies set to 240, 300,
and 420 Hz, respectively. Thebackground was measured by repeating
the experiment inthe identical configuration with the sample
removed andonly the sample can in the beam path. This background
wassubtracted from the data before analysis. Data for the
1%Ni-substituted crystal at various temperatures were
collectedusing the IN22 3-axis spectrometer [Collaborative
ResearchGroup (CEA) at the ILL, Grenoble, France] with the
final
neutron wave vector set to kf = 2.66 Å−1. Additional datafor
samples with Ni impurities were collected using theMERLIN [26]
spectrometer at ISIS with the high-resolutionFermi chopper with a
Gd-slit package rotating at 250 Hzand repetition rate
multiplication allowing access to incidentenergies of 10, 20, and
53 meV. In order to achieve quantitativecomparison with the theory,
the neutron time-of-flight spectrawere normalized using a standard
vanadium sample. Thethree-axis measurements at 8 K were then
compared withthe time-of-flight data at 6 K and a correction factor
thusobtained was used to normalize the three-axis data obtained
atother temperatures. All the data presented here were obtainedat a
temperature of T = 6 K. The only exception is themeasurement of
temperature dependence shown in Fig. 8,where the temperatures are
reported for each data set.
III. EXPERIMENTAL RESULTS
A. Low-temperature susceptibility
To verify the actual concentration of chain-breaking defectsin
our samples, we performed measurements of their
magneticsusceptibility. The results are plotted versus
temperaturein Fig. 1. For the Ni2+-substituted samples, impurities
areexpected to take the place of Cu2+ and directly fragment thespin
chains. Magnetic susceptibility curves for this scenariohave been
derived theoretically [20,27,28] The most completedescription is
given by Sirker et al. [28] and allows quantitativeestimation of
the actual number of chain breaks. As acharacterization tool, this
approach has also been validatedexperimentally [14,16]. By using
the same fitting equationas in Ref. [16], we get very good fits
(solid lines in Fig. 1)to our experimental data. The deviations
seen at very lowtemperatures (note the log scale on the abscissa)
can be
10 1001E-9
1E-8
1E-7
1E-6
2 % Ni1 % Ni5 % Ca
χΤ ( m
3 K/mol
)
T (K)
nominal x (%)
Sr2CuO3
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
fittedx(%)
FIG. 1. Magnetic susceptibility multiplied by temperature
plottedvs temperature for the studied compounds (symbols). Solid
lines arefits as described in the text. Inset: Extracted impurity
concentrationof Ni-substituted samples, for fit using all the data
(black circles)and measurements only above 10 K (blue squares). The
solid blackline corresponds to the number of impurities equal to
the nominalconcentration. In the case of Ca substitution, the
magnetic responseis much smaller as expected from substitution
outside the chain.
attributed to weak interchain coupling [14,16]. The
fittedchain-break concentration is an excellent match with
thenominal number of impurities, as shown in the inset in Fig.
1.
In the case of Ca defects, the Ca2+ ions are expectedto replace
Sr2+ and not enter the Cu2+ chains directly. Themagnetic
susceptibility measurements in Fig. 1 are consistentwith that
picture. Even though a large amount of Ca impuritiesis introduced
into the sample, their effect on the magneticresponse is minimal.
From the observed magnitude of theeffect in the 5% Ca-substituted
sample, we can say withcertainty that the number of broken links in
the Cu chainsis less than 0.6%.
B. Spin excitations
Time-of-flight neutron spectroscopy allows the measure-ment of
the magnetic dynamic structure factor in largeportions of the
energy-momentum space. When studyingone-dimensional systems, the
data quality can be improvedby integrating over the directions
perpendicular to the relevantdimension. In the case of Sr2CuO3, the
ratio of inter- andintrachain exchange constants has been estimated
[29] to beJ ′/J < 10−3, and therefore such integration can be
made useof. A projected slice of such a data set is shown in Fig. 2
forSr2CuO3 with 1% and 2% Ni impurities. The figure displaysthe
dynamic structure factor as a function of the energy transferh̄ω
and momentum transfer parallel to the direction of thespin chain
q‖. The vertical rods of intensity correspond tothe bottom of the
two-spinon continuum. Since the exchangeconstant in this material
is very large, no dispersion can beobserved and the width of the
observed scattering is due to theresolution of the experimental
setup. It is immediately visiblethat in both cases there is a
suppression of low-energy states.
054409-2
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SPIN PSEUDOGAP IN THE S = 12 CHAIN . . . PHYSICAL REVIEW B 95,
054409 (2017)
FIG. 2. Dynamical structure factor of the two
Ni-substitutedsamples as measured by the SEQUOIA time-of-flight
neutronspectrometer. The vertical rods at the antiferromagnetic
wave vectorare the bottom of the highly dispersive two-spinon
continuum. Thesuppression of low-energy states gives rise to a
pseudogap whichscales in energy with an increasing number of broken
links asdescribed in the text.
The scale of this pseudogap is larger in the compound withmore
impurities.
In order to quantify this effect, the dynamical structurefactor
was integrated with respect to the momentum transferalong the chain
axis. Since the peak width is given by theresolution of the
spectrometer, Gaussian peaks were used to fitthe momentum cuts, as
shown for a selection of cuts in Fig. 3and Fig. 4 for 1% and 2% Ni
impurity levels respectively.
In low-energy transfers, the intensity of the peaks is low
andextracting the correct intensity becomes difficult. Thereforethe
position and width of the peak were obtained from higherenergy
cuts, where the intensity is substantial, and were thenheld fixed
for the low-energy cuts. The momentum-integrated(local) dynamical
structure factor thus obtained, correctedfor the anisotropic form
factor [30] of Cu2+, is shown inFig. 5. Most of the emphasis was
drawn to the low-energy
FIG. 3. Constant-energy cuts of the measured dynamical
struc-ture factor in Sr2CuO3 with 1% Ni impurities. Data sets are
offsetby 25 mb/meV/sr/f.u. for clarity. Solid lines are Gaussian
fits, asdescribed in the text.
FIG. 4. Constant-energy cuts of the measured dynamical
struc-ture factor in Sr2CuO3 with 2% Ni impurities (symbols). Data
setsare offset by 25 mb/meV/sr/f.u. for clarity. Solid lines are
Gaussianfits, as described in the text.
part of the spectrum where the pseudogap is observed,
andmeasurements with 50-meV incident energy were performedwith
shorter counting times. This is especially evident for
thehigh-energy part of the spectrum for the 2% sample where,due to
the smaller mass of the sample and shorter countingtime, the error
bars are substantially larger. Nevertheless, thehigh-energy
measurements complement the low-energy dataset and demonstrate that
the dynamical structure factor staysconstant in high-energy
transfers.
As shown in Fig. 5, the system with a higher amount ofimpurities
has a more pronounced suppression of low-energystates. However, the
decrease in intensity is gradual andtherefore the extraction of the
gap size is not straightforward.
FIG. 5. Symbols: momentum-integrated dynamical structure fac-tor
of 1% (squares) and 2% (circles) Ni-substituted samples. Thedashed
line shows the prediction for an intact chain, whereas solidlines
show the scattering for spin chains with 1% (black) and 2%(blue)
broken links.
054409-3
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G. SIMUTIS et al. PHYSICAL REVIEW B 95, 054409 (2017)
FIG. 6. Constant-energy cuts of the dynamical structure factorin
Sr2CuO3 with 5% Ca impurities for selected energy
transfers(symbols). Data sets are offset by 25 mb/meV/sr/f.u. for
clarity.Solid lines are Gaussian fits as described in the text.
The simplest way to assign the value is by quoting the energy
atwhich the intensity is halved compared to the expected value.Such
an estimate would give values of ≈3 meV for 1% Niimpurities and ≈6
meV for 2% Ni impurities. A much betterway is to consider the whole
line shape by introducing a modelbased on the distribution of gap
sizes as detailed in Sec. IV.
Deviations from the dynamical structure factor of an intactchain
were observed at all temperatures. It is best visible, how-ever, at
low temperatures, where the momentum-integratedspectrum of an ideal
chain is constant in the measured energyrange. Therefore most of
the discussion is dedicated to themeasurements performed at 6 K,
where the observed reductionof low-energy states is directly due to
the introduction ofimpurities.
A series of similar measurements was additionally per-formed at
different temperatures. The results are plotted inFig. 8(a),
together with previous results for 1% Ni-substitutedSrCuO2 [13].
Here we used scaled coordinates, with h̄ω/2kBTon the abscissa. The
y axis is the momentum-integrated imagi-nary part of the
susceptibility multiplied by the correspondingmaterial’s exchange
constant (J = 241 meV [23] for Sr2CuO3and J = 221 meV [30] for
SrCuO2), for direct comparison.
The same type of measurements was carried out for thesample with
Ca impurities. The data, while of poorer qualitydue to the smaller
sample, show a clear suppression oflow-energy states (Fig. 6). Its
value can be estimated fromthe momentum-integrated dynamical
structure factor shownin Fig. 7. Somewhat surprisingly, it is
similar in magnitude tothe pseudogaps observed in the samples with
1% and 2% Niimpurities.
IV. DISCUSSION
A. Pseudogap and impurity concentration
The data shown in Fig. 2 are a vivid illustration of
thepseudogap dependence on the impurity concentration. Asargued
earlier [13], the pseudogap arises from the confinedmotion of
spinons in chain fragments bound by defect sites.
FIG. 7. Measured low-energy excitations in Ca-substitutedSr2CuO3
in a momentum-integrated form (symbols). The solid linerepresents a
prediction for 2.1% broken links.
A finite length L of such fragments ensures a gappingand overall
discretization of the excitation spectrum withthe spacing of � =
3.65 × J/L, where J is the intrachainexchange constant [20]. The
observed spectrum is defined bythe statistical length distribution
of fragments in which thequasiparticles are trapped. The resulting
dynamic structurefactor can be factored into that of a defect-free
chain and adefect-related envelope function [13]. The envelope
functiondepends on the defect concentration x and can be
expressedas [13]
F (ω) =(
�0 × x2h̄ω
)2sinh−2
(�0 × x
2h̄ω
), (1)
where �0 = 3.65 × J .The envelope function effectively describes
the mod-
ification of the spin-chain spectrum in a bulk materialdue to
randomly distributed impurities. Different distri-butions would
lead to different shapes of the envelopefunction.
Since the magnetic dynamical structure factor of a defect-free
chain is known in absolute units, and the envelope functionis
uniquely expressed through the exchange constant and thedefect
concentration, the model has no adjustable parameters.Nevertheless,
as shown by solid lines in Fig. 5 it providesan excellent
description of our data on the absolute scale. Itis noteworthy that
there seems to be a deviation of the datapoints from the prediction
at energy transfers around 10 meV.While the origin of this
deviation is unclear, there are twopossibilities for such behavior.
First, it is conceivable thatour assumption of a random
distribution of the defects is toosimplistic. A reduced probability
of very short chain segmentswould lead to the observed spectrum
with an earlier saturation.Second, this deviation could be due to
an additional peak,which may warrant further attention from the
theoretical pointof view.
As pointed out in Ref. [13], the envelope function isindependent
of the temperature. For this reason, the model
054409-4
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SPIN PSEUDOGAP IN THE S = 12 CHAIN . . . PHYSICAL REVIEW B 95,
054409 (2017)
FIG. 8. The dynamical structure factor expressed in a
universalscaling form (a) before and (b) after correction by the
pseudogapfunction. Open (filled) symbols correspond to
time-of-flight (three-axis) measurements. The solid line is the
theoretical prediction forthe S = 1/2 chain. While the raw data in
(a) display the breakdownof scaling behavior, it can be restored by
taking into account theT-independent envelope function. The
corrected data set in (b) showsscaling at all studied
temperatures.
is expected to work even at elevated temperatures. Inparticular,
after a normalization by the envelope function,finite-temperature
data are expected to obey the scalingrelations for spin
correlations in defect-free chains. In thelatter, the imaginary
part of the local susceptibility, χ ′′(ω) =S(ω)/[n(ω) + 1], is a
universal function of ω/T [31–33].The scaling function is known
[31,32] and is plotted as thesolid line in Fig. 8. The raw data for
our Ni-substitutedsamples clearly violate scaling. However, when
normalized bythe respective envelope function, they produce a
convincingdata collapse across all materials, temperatures, and
defectconcentrations.
Additional confidence in this interpretation of the pseudo-gap
is provided by the recent NMR experiments on 1% and
2%Ni-substituted Sr2CuO3, where the characteristic temperatureof
the drop in the spin-lattice relaxation rate was found toscale with
the impurity concentration [15]. Since the relaxationmechanism in
such an experiment is due to the scatteringof thermally excited
spinons, the observation in Ref. [15]also implies scaling of the
pseudogap with the impurityconcentration. We believe that this
scaling of the pseudogapwith the impurity concentration should
persist up to higherconcentrations until clustering of impurities
becomes relevant.Experimentally, it would become more challenging
to observe
it, since at higher energies, the phonons start to obscure
theneutron scattering spectrum and NMR measurements wouldrequire a
high-temperature apparatus.
B. Defects vs scattering centers
The spectrum measured in the 5% Ca-substituted com-pound at
first glance seems to be at odds with this picture.As mentioned
above, off-chain Ca2+ defects have virtually noeffect on the
magnetic susceptibility. At the same time, neutronscattering
clearly shows a pseudogap (Figs. 6 and 7). Usingthe previously
proposed model, and treating the number ofchain breaks as a fitting
parameter for the neutron data, wededuce as much as 2.1(3)% chain
defects, well above the
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