Magnetic order and low-energy excitations in the quasi-one-dimensional antiferromagnet CuSe 2 O 5 with staggered fields
M. Herak, A. Zorko, M. Pregelj, O. Zaharko, G. Posnjak, Z. Jagličić, A. Potočnik, H. Luetkens, J. van Tol, A. Ozarowski, H. Berger, D. Arčon
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Magnetic order and low-energy excitations in the quasi-one-dimensionalantiferromagnet CuSe O with staggered fields M. Herak,
1, 2, ∗ A. Zorko,
1, 3
M. Pregelj, O. Zaharko, G. Posnjak, Z. Jagliˇci´c, A.Potoˇcnik, H. Luetkens, J. van Tol, A. Ozarowski, H. Berger, and D. Arˇcon
1, 9 Joˇzef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia Institute of physics, Bijeniˇcka c. 46, HR-10000, Zagreb, Croatia EN-FIST Centre of Excellence, Dunajska 156, SI-1000 Ljubljana, Slovenia Laboratory for Neutron Scattering, Paul Scherrer Institute, CH-5232 Villigen, Switzerland Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen, Switzerland National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA Institute of Physics of Complex Matter, EPFL, 1015 Lausanne, Switzerland Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia (Dated: October 6, 2018)Ground state and low-energy excitations of the quasi-one-dimensional antiferromagnet CuSe O were experimentally studied using bulk magnetization, neutron diffraction, muon spin relaxationand antiferromagnetic resonance measurements. Finite interchain interactions promote long-range antiferromagnetic order below T N = 17 K. The derived spin canted structure is charac-terized by the magnetic propagation vector k = (1 , ,
0) and the reduced magnetic moment m = [0 . , . , . µ B . The values of the magnetic anisotropies determined from thefield and angular dependencies of the antiferromagnetic resonance comply well with a previous elec-tron paramagnetic resonance study and correctly account for the observed magnetic ground stateand spin-flop transition. PACS numbers: 75.10.Pq, 75.50.Ee, 76.50.+g
I. INTRODUCTION
Experimentally obtained in-depth information aboutthe magnetic properties of one-dimensional (1D) quan-tum spin systems are important for testing predictionsof ground states and low-energy excitations in advancedquantum-mechanical theories. A much studied represen-tative of these systems is the spin S = 1 / H = J X i S i · S i +1 , (1)where i is the site index in the chain, and J is the intra-chain exchange coupling between the spins. The groundstate of the S = 1 / / The ex-citation spectrum of the S = 1 / which was observed experimentally in several 1Dspin systems. In real materials a finite interchain in-teraction, J IC , always exists, and can lead to long-rangeorder (LRO) below a finite temperature T N . When J IC is small compared to J , a system is called quasi-one-dimensional (quasi-1D). In the ordered state at T < T N the excitation spectrum of these systems at low energiesis usually described by a spin-wave theory, as for an or-dinary 3D antiferromagnet. At higher energies, however,where the chains start to decouple, a continuous excita- tion spectrum typical for a 1D system was observed. A crossover regime from 3D LRO to 1D TLL thus existsin these systems. When the crystal symmetry of quasi-1D spin sys-tems is sufficiently low, a staggered g tensor and/ora Dzyaloshinskii-Moriya interaction (DMI) can bepresent. The combined action of staggered g tensor andDMI leads to a staggered field in a finite applied field,opening a gap in the excitation spectrum of the quasi-1D chain. In the AFM ordered state, both may lead toa non-collinear staggering of the ordered moments. The problem of weakly coupled S = 1 / For instance, if a staggered g tensorand DMI are present, the staggered field can competewith the arrangement of spins favored by the inter-chain interaction. The magnetic arrangement, result-ing from such a competition, was recently studied by nu-clear magnetic resonance (NMR) in BaCu Si O wherethe presence of staggered fields was argued to cause un-usual spin reorientations. CuSe O is a novel Cu ( S = 1 /
2) quasi-1D HAF,which crystallizes in a monoclinic unit cell that belongsto the C /c space group (Fig. 1). The alternating CuO plaquettes form chains running along the c crystallo-graphic axis. The 1D nature of this system was ar-gued from the temperature dependence of the magneticsusceptibility which was satisfactorily described by theBonner-Fisher curve for S = 1 / J/k B = 157 K. Band structure calculations lead to similar
J/k B = 165 Kand reveal that the intrachain interaction is realizedthrough a double Cu-O-Se-O-Cu path (upper panel inFig. 1) with only one dominant interchain coupling of J IC /k B = 20 K, shown in the lower panel of Fig. 1by dashed red (dark) lines. Raman scattering measure-ments indicated that the spin-spin correlations set in be-low T ≈
100 K, which coincides with the maximumin the magnetic susceptibility. Contrary to what is ex-pected for a quasi-1D HAF, the temperature dependenceof the magnetic specific heat extracted from the Ramanscattering intensity exhibits no maximum, implying thepresence of classical spin dynamics originating from themoderate interchain interactions. On the other hand,the recent analysis of the electron spin resonance (ESR)linewidth in the paramagnetic state suggests that at tem-peratures
T > ∼ J IC /k B CuSe O essentially behaves as a1D antiferromagnet. The low crystal symmetry and the alternating arrange-ment of CuO plaquettes (Fig. 1) allow for staggered g -tensor and DMI with a DM vector confined in the a ∗ c plane. The early ESR study indeed confirmedthe presence of staggered fields and suggested an addi-tional symmetric anisotropic interaction. Bulk magnetic-susceptibility-anisotropy measurements showed signifi-cant deviation from the 1D HAF model, which weresuccessfully explained by the extension of S = 1 /
21D HAF to include staggered fields. At T N = 17 KCuSe O undergoes a phase transition to a LRO mag-netic state with so far yet unknown magnetic struc-ture. An important question that arises from these obser-vations is how the coexistence of the staggered field andthe symmetric anisotropic exchange affects the magneticLRO in the presence of J IC . We thus decided to per-form a detailed experimental study of the magneticallyordered state of CuSe O by employing bulk magneticmeasurements, neutron diffraction, muon spin relaxationand antiferromagnetic resonance measurements. II. EXPERIMENTAL DETAILS
The single crystalline CuSe O samples were synthe-sized by a standard chemical vapor transport method,as described previously, and characterized by X-raydiffraction. The samples had a platelet shape, elon-gated along the crystallographic c axis and with the a ∗ axis perpendicular to the platelet.The dc magnetic measurements were performed witha Quantum Design SQUID magnetometer in static mag-netic fields ranging from 1 kOe to 50 kOe in the tempera-ture interval between 2 K and 300 K. The measurementswere performed with the magnetic field applied along thecrystallographic a ∗ , b and c axes. The mass of the sam-ple was (2 . ± .
2) mg. Magnetic torque measurementswere performed on a home-built torque magnetometer at4 . λ = 2 .
316 ˚A. The mass of the crystal abcabc
FIG. 1. (Color online) Crystal structure of CuSe O . Lowerpanel shows dominant interaction paths. Solid line representsthe intrachain interaction J , red (dark) dashed line the dom-inant interchain interaction J IC and green (light) dashed linethe weak interchain interaction J IC,weak < ∼ . J IC accordingto Ref. 20. was around 10 mg. The single crystal was mounted in aCCR cooling device at a four-circle cradle. The data setswere collected at 6 K and 20 K. Neutron diffraction inmagnetic field was measured on the same crystal orientedwith the b axis vertical in the Oxford vertical cryomag-net.The muon spin relaxation ( µ SR) experiments were con-ducted on the General Purpose Surface-Muon (GPS) in-strument at the Swiss Muon Source (S µ S), Paul ScherrerInstitute (PSI), Switzerland, in the temperature rangebetween 1.8 and 25 K in zero applied magnetic field.Measurements on powders (500 mg) were performed in alongitudinal muon polarization mode – muon spins werepolarized almost parallel ( α ∼ ◦ ) to the beam ( z ) direc-tion. Measurements on a single crystal (10 × × . )were conducted in a transverse muon polarization mode– the muon polarization was rotated from the z direc-tion by α ∼ π/ y direction. The asymmetry ofdetected positrons, emitted after muon decays, was mea-sured with two sets of detectors; in the backward-forward( z ) direction and in the up-down ( y ) direction. The ini-tial asymmetry and the tilt angle α of the initial muonpolarization were calibrated for each experiment at 25 K;i.e., well above the ordering temperature T N = 17 K, inthe weak transverse magnetic field of 30 Oe applied in x direction. Measurements were performed in veto mode,leading to negligible background signal in the case of thepowder sample due to its large mass. The backgroundsignal could, however, not be avoided in the case of thesingle crystal due to the small thickness of the sample.The antiferromagnetic resonance (AFMR) measure-ments were performed on single-crystalline samples at T = 5 K. Measurements in X- (9.7 GHz) and in Q-(35 GHz) band were performed on a commercial Brukerspectrometers at the Joˇzef Stefan Institute in Ljubljana.AFMR measurements at frequencies from 50 GHz to450 GHz were performed using a custom-made transmis-sion type spectrometers at the National High MagneticField Laboratory (NHMFL) in Tallahassee, Florida. III. RESULTSA. Magnetization measurements
Temperature dependence of dc magnetic susceptibility( χ = M/H , where M is the sample magnetization) mea-sured in H = 10 kOe applied along a ∗ , b and c axes isshown in Fig. 2(a). In the paramagnetic state the mea-sured data can be described by the S = 1 / with J/k B = 156 K, if the g -factor values mea-sured by ESR ( g a ∗ = 2 . g b = 2 . g c = 2 . c direction a disagreement with themodel starts already below T ≈ T max , i.e., far abovethe transition to the magnetic LRO state. The pres-ence of the staggered DMI and the staggered g tensorleads to finite staggered field h proportional to the ap-plied field, h i = c s,i H ( i = a ∗ , b, c ), which results inthe anisotropic staggered susceptibility for T < J/k B , χ s,i ∝ c s,i . The temperature dependence of the stag-gered susceptibility reflects in the Curie-like term χ ∝ /T , which strongly varies with the direction of the mag-netic field, as has been observed for the 1D S = 1 / ) (H O) ] n (PM = pyrimidine). In-deed the measured magnetic susceptibility χ c can be sat-isfactorily described if the staggered susceptibility withcoefficient c s,c = 0 .
17 is added to the 1D HAF model[dashed line in Fig. 2(a)], in rather good agreement withthe previous anisotropy results, c s = 0 . For the a ∗ and b direction the data are well described by the 1DHAF model combined with staggered susceptibility us-ing the previously obtained c a ∗ and c b [see Fig. 2(a)].Below ≈
22 K there is a drastic disagreement betweenthe data and the model even if the staggered susceptibil-ity is included [inset in Fig. 2(a)]. This was also observedin previous anisotropy measurement, however, no satis-factory explanation for this behavior can be given at themoment.On cooling below T N , χ measured along the b axisdecreases and saturates at the value of 9 · − emu/molbelow 4 K, while χ a ∗ and χ c slightly increase. This sug-gests an almost collinear spin arrangement with the b axisas the easy axis, which is in agreement with the mag-netization measurements at 4 . ( - e m u / m o l ) T (K) H = 10 kOe a* b c(a) a* b c M ( e m u O e / m o l ) H (kOe)T = 4.2K(b)
H = 1T
FIG. 2. (Color online) (a) Temperature dependence of mag-netic susceptibility measured in the field H = 10 kOe appliedalong a ∗ , b and c axis. Solid lines represent fits to the S = 1 /
21D HAF model with J = 156 K, while dashed line also in-cludes staggered susceptibility (see text). Error bar resultingfrom the uncertainty in the mass of the sample is shown onthe side. Vertical line represent T N = 17 K. Inset: Expandedregion around T N showing the disagreement between the ex-perimental data (red circles) and the model (dashed line) for c || H . (b) Field dependence of magnetization at T = 4 . where a spin-flop (SF) transition is observed for the field H SF ≈
13 kOe applied along the b axis. In contrast,the magnetization changes linearly with field for a ∗ and c directions up to the highest applied field of 50 kOe. B. Neutron diffraction measurements
Neutron diffraction is a powerful tool for determin-ing the magnetic order. Therefore, we have employedit to determine the magnetic structure of CuSe O moreprecisely. A refinement of the crystal structure at 6 Kconfirms the room-temperature structural model pub-lished previously but with slightly different values ofthe cell parameters; a = 12 . b = 4 . c = 7 . β = 112 . ◦ . Below T N = 17 K TABLE I. Irreducible representations Γ and Γ of the littlegroup for k = (1 , ,
0) in the space group C /c .Γ Γ x , y , z ( u, v, w ) u, v, w − x , y , − z + 1 / − u, v, w ) ( u, − v, w ) new reflections of magnetic origin appear. These corre-spond to the magnetic propagation vector k = (1 , , basireps program. The magnetic moment in CuSe O originatesfrom the Cu ions at the 4 a Wyckoff site. The twopossible irreducible representations, which connect theCu-sites related by a twofold screw axis, are given in Ta-ble I. The best agreement with the experimental datais obtained for the irreducible representation Γ with χ = 5 .
03 and R F = 21 .
6. The representation Γ can bediscarded due to much poorer agreement with the exper-imental data, χ = 21 . R F = 43 .
3. The agreementbetween the observed and calculated intensities for thosetwo models is shown in Fig. 3.The components of the magnetic moment were ob-tained from the data refinement using the fullprof program. The magnetic moment of the Cu ion atthe crystallographic position (0 , ,
0) in the crystallo-graphic ( abc ) coordinate system is m = ( m a , m b , m c ) =[0 . , . , . µ B . The value of the magneticmoment | m | = 0 . µ B is thus significantly smallerthan the full magnetic moment of 1 µ B for S = 1 /
2. Therefined magnetic structure is shown in Fig. 4. The mag-netic moments on a chain at the positions (0 , ,
0) and (cid:0) , , (cid:1) possess different b components in the represen-tation Γ , but equal a -components, which results in finitemagnetization on each chain. On the other hand, themagnetic moments at the positions (cid:0) , , (cid:1) and (cid:0) , , (cid:1) on the neighboring chain have the same symmetry as thefirst two, but are antiferromagnetically coupled to themso the total magnetic moment in the unit cell is zero.Finally, we have measured two magnetic reflections inthe magnetic field H = 40 kOe applied along the b axis totest the magnetic structure above the SF field. The com-parison of the integrated intensities for these reflections inzero field and H = 40 kOe is given in Table II. Althoughonly two reflections were measured, information aboutthe orientation of spins can be extracted. The best agree-ment between calculated and observed intensities is foundfor Γ and m = ( m a ∗ , m b , m c ) ≈ (0 . , , . µ B . Analmost orthogonal orientation of the spins in this fieldwith respect to zero field is expected, since the spin-floptransition was observed at H SF ≈
13 kOe. I c a l c I obs (a) R F = 43.3 I c a l c I obs (b) R F = 21.6 FIG. 3. Agreement of calculated and observed intensities ofmagnetic reflections for models (a) Γ and (b) Γ (see text). b ac FIG. 4. (Color online) Proposed zero-field magnetic struc-ture in the ordered state of CuSe O obtained from neutrondiffraction. C. Muon spin relaxation
For an independent proof of the refined magnetic struc-ture, a complementary local-probe technique was used. µ SR is a highly powerful method for detecting magnetismon a microscopic level. The almost 100% spin-polarizedmuons that stop in a sample probe the local magneticfield B µ , which leads to coherent oscillations of the muon TABLE II. Integrated intensities of magnetic reflections in H = 0 Oe and H = 40 kOe for (10¯1) and (¯10¯1) reflections. h k l (10¯1) (¯10¯1) H = 0 Oe 229 ±
10 117 ± H = 40 kOe 169 ±
12 37 ± polarization for a static field and a monotonic decay ofpolarization for fast fluctuations of the field. In the caseof a single quasi-static magnetic field the muon polariza-tion along the initial polarization will change in powdersamples as P z pwd ( t, B µ , λ L , λ T ) = 13 e − λ L t + 23 e − λ T t cos( γ µ B µ t ) (2)where longitudinal muon relaxation λ L and transverserelaxation λ T are taken into account ( γ µ = 85 .
16 kHz/Gis the muon gyromagnetic ratio). The former arises fromfinite dynamics of the internal field whereas the letteradditionally includes a distribution of local fields. Thenon-oscillating ” ”-tail signal corresponds to muons be-ing initially polarized along the internal field and is thusa fingerprint of the magnetic order, alongside the oscil-lating signal.The magnetic ordering in CuSe O is witnessed bya clear change of the polarization curve at 17 K. Belowthis temperature oscillations of the polarization appear[Fig. 5(a)]. The real part of a Fourier transform of thedata below 17 K, which directly gives the field distribu-tion at the muon sites, reveals four distinct components[inset in Fig. 5(b)]. We therefore fit the experimentalpolarization of the powder sample below T N with thefour-component function P z = X i =1 f i P z pwd ( t, B µi , λ L , λ T i ) , (3)where f i denotes the fraction of the i -th component.The fitting of the 1.8 K dataset [Fig. 5(b)] yields inter-nal magnetic fields B µ = 185(3) G, B µ = 297(5) G, B µ = 337(3) G and B µ = 533(3) G, transverse re-laxation rates λ T = 0 . µ s − , λ T = 1 . µ s − , λ T = 0 . µ s − and λ T = 0 . µ s − , the longitu-dinal relaxation rate λ L = 0 . µ s − and fractions f =0 . f = 0 . f = 0 . f = 0 . value. This demonstrates that thesample is 100% ordered and that the spin dynamics inthe ground state is marginal on the muon time scale.The observation of four distinct internal magneticfields reveals that muons stop at four different crystal-lographic sites, because the magnetic order does not re-duce the symmetry of the crystallographic unit cell. Thetemperature dependence of the four internal fields andthe transverse relaxation rates are shown in Fig. 6. Sincethe fields are proportional to the ordered magnetic mo-ment, their temperature dependence directly yield the
200 400 600024
18 K 17 K 16 K P z ( t ) (a) B B B B (b) P z ( t ) t ( s) FT ( a r b . un i t s ) B (G)
FIG. 5. (Color online) (a) Time dependence of muon polar-ization along the beam direction for longitudinal muon po-larization in the CuSe O powder sample at several selectedtemperatures close to the ordering temperature T N = 17 K.(b) Fit of the measured low-temperature polarization (circles)to the four-component model (solid line) given by Eq. (3). In-set: Real part of the Fourier transform of the 1.8 K dataset. temperature evolution of the magnetic order parameterin CuSe O . The relaxation rates, on the other hand, ev-idence an increasing spin relaxation rate when approach-ing the ordering temperature T N , which can be eitherexplained by an increased magnon density or by an in-creased width of local-field distribution with increasingtemperature.Measurements on the single crystal further allow usto set the actual direction of the internal fields at allfour stopping sites. We performed these measurementsat 1.8 K for two different orientations of the crystal; (a)with the a ∗ and b crystallographic axes oriented in the z and y directions, respectively, and (b) with the crystalrotated by π/ z direction so that the c crys-tallographic axis was pointing along the y direction. Dueto the symmetry of the magnetic space group, for eachmuon stopping site i four different directions of the mag-netic field are allowed; ( θ i , ϕ i ), ( θ i , ϕ i ), ( θ i , ϕ i ), and( θ i , ϕ i ). These are symmetry-related and are embeddedinto polarization functions P y,z sc ( t, B µi , θ i , ϕ i , λ Li , λ T i , α )(see Appendix B). For the second orientation the ϕ ji pa-rameters are increased by π/ θ ji parameters remain the same.We fitted simultaneously the powder data to Eq. (3) [see Fig. 5(b)] and the three single-crystal datasets recordedfor the two crystallographic orientations [see Fig. 7(a,b)]to equations P za ∗ = (1 − P bgd ) X i =1 f i P z sc ( t, B µi , θ i , ϕ i , λ L , λ T i , α ) + P bgd e − (Λ t )22 , (4a) P yb = (1 − P bgd ) X i =1 f i P y sc ( t, B µi , θ i , ϕ i , λ L , λ T i , α ) + P bgd e − (Λ t )22 , (4b) P yc = (1 − P bgd ) X i =1 f i P y sc ( t, B µi , θ i , ϕ i + π/ , λ L , λ T i , α ) + P bgd e − (Λ t )22 , (4c)where the two single-crystal polarization functions P y,z sc are given by Eq. (B8) with an additional background sig-nal P bgd e − (Λ t )22 . The above-reported powder muon relax-ation rates and fractions f i do not change when addingthe single-crystal datasets, while the background signalin these datasets amount to 26% (Λ = 0 . µ s − ). Sucha high background is not surprising for thin single crys-tals. The local magnetic fields at the four muon stoppingsites are summarized in Tab. III.Having determined the magnetic fields at the fourmuon stopping sites we next critically verify the mag-netic order determined in the neutron diffraction experi-ment (Fig. 4). To be able to perform this assessment,the knowledge of the muon stopping sites is needed.Muons possessing positive charge are likely to stop atelectrostatic-potential minima of the crystal structure,which has been shown before at several instances. In order to find the electrostatic-potential minima ofthe CuSe O crystal structure, we performed density-functional-theory (DFT) calculation, using the pwscf program of the Quantum Espresso software package. A self-consistent electron density distribution was calcu-lated, which yielded a spatial profile of the electrostaticpotential. A global electrostatic-potential minimum wasfound at R = (0 . , . , .
22) (Wyckoff position 8 f ) TABLE III. The local magnetic field B µi , the polar angle θ i and the azimuthal angle ϕ i (the former is given with respectto the a ∗ crystallographic axis and the latter with respectto the c axis) at four muon stopping sites i at 1.8 K. Thecorresponding component of the magnetic-field vectors B µi in the a ∗ bc orthogonal system are also given. i B µi (G) θ i ϕ i B µi (G)1 185 1.22 1.97 (62, -159, 67)2 297 2.24 2.63 (-185, 112, 203)3 337 2.25 2.25 (-211, 211, 156)4 533 1.71 5.08 (-77, -494, -187) B ( G ) (a) site 1 site 2 site 3 site 4 (b) T ( s - ) T (K) FIG. 6. (Color online) Temperature dependence of (a) in-ternal fields and (b) transverse muon relaxation rate for fourcrystallographically nonequivalent muon stopping sites in theCuSe O powder sample. and three local minima at R = (0 . , . , . f ), R = (0 , . ,
0) (4 b ) and R = (0 , . , .
25) (4 e ).We further calculated the dipolar magnetic field atthese sites by taking into account all spins within a spherelarge enough to assure convergence of these calculations.Since CuSe O is an insulator and all the potential min-ima are located outside the exchange paths, the dipolarcontribution to the magnetic field at these sites is ex- P y ( t ) (a) a* axis(b) P z ( t ) t ( s) FIG. 7. (Color online) Simultaneous fit (solid lines) of muonpolarization data (symbols) to Eqs. (4) along three crystallo-graphic axes of the single-crystal CuSe O sample, measuredin transverse muon polarization mode (a) along and (b) per-pendicular to the beam direction. pected to be by far dominant. The dipolar fields at allthese sites (except R ) are in the range between 189 and654 G, thus seemingly well suiting the experimental fields B µ − = 185 −
533 G. However, the direction of the cal-culated B calc and experimentally determined fields B µi are very different thus yielding large relative deviations σB µi > except for site R , where the calculated fieldis the closest to the measured field ( σB µ = 0 . TABLE IV. The muon stopping sites P i found at 8 f Wyck-off positions, the corresponding dipolar magnetic fields B i ,the distance r P i − R j to the closest electrostatic-potential min-imum R j and r P i − O to the closest oxygen site. All the fieldsare within σ/B µi = 5% of the experimentally determinedvalues B µi and are given in the a ∗ bc orthogonal system. i P i B i (G) r P i − R j r P i − O Therefore, the muons do not seem to stop atthe electrostatic-potential minima of the unperturbedCuSe O structure. In order to determine the possi-ble muon stopping sites we calculated the dipolar mag-netic field on a 100 × ×
100 mesh of the unit cell andsearched for positions where the calculated fields matchthe experimental fields B µ − . These positions are sum-marized in Tab. IV, where the distances to the closestelectrostatic-potential minima and to the closest oxygensite are also shown.The position P corresponding to the smallest ex-perimentally determined local field B µ is found only0.21 ˚A away from the global electrostatic-potential min-imum at R . A slight local modification of the electro-static potential by the positively charged muon and/orslightly different magnetic moment m are likely rea-son for the small mismatch between P and R . Theother three sites are found further away from the min-ima. However, they are all positioned 1.03(5) ˚A awayfrom oxygen. The muon is well-known for its affinityof ”bonding” to the oxygen ion with the correspondingbond length of about 1.0 ˚A, which is in nice agree-ment with our determination of the muon stopping sites.We stress that the site with the largest internal field P is also found 0.36 ˚A away from the global electrostatic-potential minimum. This can explain its dominant oc-cupation f = 61%. The accordance of the observeddistances between the muon stopping sites and oxygensites and the reported muon-oxygen ”bond” length sug-gest that the muon perturbs the electrostatic potentialof CuSe O . However, it preferentially remains relativelyclose to the global electrostatic-potential minimum of theunperturbed structure. The convincing agreement of themeasured and the calculated magnetic fields at the de-termined muon stopping sites sets a firm confirmation onthe magnetic structure determined by the neutron scat-tering experiment. D. Antiferromagnetic resonance
Next, we decided to perform AFMR measurements,which can provide additional information of the spinHamiltonian responsible for the onset of the above-determined magnetic order. In the AFMR theory themagnetic order and the low-energy excitations of a spinsystem are described within a molecular–field approxi-mation. The resonant frequencies of the sublattice mag-netizations induced by a microwave field in the finiteapplied magnetic field are associated to the exchangeand anisotropy molecular fields felt by the sublatticemagnetizations. Studying the AFMR is thus an al-ternative way of obtaining information about the long-range order, superexchange and magnetic anisotropy of asample, with a high precision characteristic of magnetic-resonance experiments. Below T N the paramagnetic ESRsignal quickly disappears in CuSe O and is replaced by ashifted temperature dependent resonances (Fig. 8), sug- S i gna l ( a r b . un i t s ) H (kOe)22K18K 5K7K10K13K15K16K
FIG. 8. (Color online) The temperature dependence of theantiferromagnetic resonance line shape measured at frequency ν = 9 . b axis. Paramagneticspectra at T > T N are also shown for comparison. a* ( G H z ) b c H (10 Oe)
FIG. 9. (Color online) The field dependence of the AFMRfrequency measured at T = 5 K. Solid and dashed lines showthe results of calculations for parameters J = 157 K, J IC =0 . J , D ∗ a = − . D c = 0 . δ a ∗ = 0 . δ b = 0 and δ c = − .
001 of the Hamiltonian (6). gesting that these resonances belong to AFMR modes.The field dependence of the resonant frequency for a ∗ , b and c directions measured at T = 5 K is shown in Fig. 9.Finally, we have also measured angular dependencies ofthe resonance field H res at T = 5 K and ν = 240 GHzin the a ∗ b and the a ∗ c plane, and at ν = 35 GHz and ν = 9 . a ∗ b plane. The results are summa-rized in Fig. 10. At 240 GHz the anisotropy is muchlarger in the a ∗ c plane than in the a ∗ b plane. For X- andQ-band measurements the AFMR modes are observedonly for the b direction due to the experimental limita-tions. H r e s ( k O e ) Angle (deg)a* bc = 240 GHzT = 5 K H
FIG. 10. (Color online) The angular dependence of theAFMR resonance modes in the a ∗ c and the a ∗ b planes mea-sured at different frequencies. The solid and dashed linesrepresent a fit to the model described in the text using pa-rameters J = 157 K, J IC = 0 . J , D ∗ a = − . D c = 0 . δ a ∗ = 0 . δ b = 0 and δ c = − .
001 in Hamiltonian (6).
E. Torque magnetometry
The torque magnetometry can be a useful tool fordetecting spin reorientations even in magnetic fieldswhich are substantially smaller than the critical field ofreorientation. This is because the measured magnetictorque Γ = V M × H ( V = is the sample volume) issensitive to the direction of the induced magnetizationin the sample which changes when the field approachesthe critical field if the direction of the field does not co-incide with the easy axis (if it does then a spin flop isobserved at the critical field H SF ). In the case of a uni-axial antiferromagnet in a field H ≪ H SF the angulardependence of the measured component of torque Γ z forfield H rotating in xy plane is given byΓ z = m M mol H ∆ χ xy sin(2 φ − φ ) (5)where m is the mass, M mol is the molar mass, ∆ χ xy = χ x − χ y , φ is the goniometer angle and φ is the angle -2-1012 -202-4-2024 -5050 50 100 150 200-10-50510 0 50 100 150 200 250-2-1012 T o r que ( - d y n c m ) H = 2 kOeb a*
H = 3 kOeb a* T o r que ( - d y n c m ) H = 4 kOeb a*
H = 5 kOeb a* T o r que ( - d y n c m ) Angle (deg)
H = 6 kOeb a*
Angle (deg)
H = 8 kOeb a*
FIG. 11. (Color online) Torque measured at T = 4 . a ∗ b plane. Dashed blue line is obtained from Eq. (5) for χ a ∗ and χ b taken from susceptibility measurements, Fig. 2(a). Solidred line is the result of calculations taking into account thesame parameters as in Figs. 9 and 10. The values obtainedby calculations are rescaled to match the observed torque am-plitude in the same way as magnetization in Fig. 2(b). the x axis makes with the goniometer zero angle. Thetorque measured at T = 4 . a ∗ b plane in differ-ent magnetic fields is shown in Fig. 11. We have also plot-ted the expected angular dependence (dashed blue lines),Eq. (5), where x = a ∗ , y = b , φ = 132 ◦ and the valuesof χ a ∗ and χ b are taken from the magnetic susceptibilityresults at T = 4 . H ≥ a ∗ c plane, where even thetorque in the highest applied field of H = 8 kOe obeysEq. (5). F. Modeling
In order to describe the above presented experimentalresults we start with the spin Hamiltonian H = X all chains H D + H IC , (6)where the single-chain Hamiltonian H D and theinterchain-interaction Hamiltonian H IC are given by H D = J X i S i · S i +1 + X i ( − i D · ( S i × S i +1 ) ++ X i S i · ˆ δ J · S i +1 − µ B X i S i · ˆ g i · H , (7a) H IC = J IC X S i · S j . (7b)The sum in (7a) runs over spins on one chain, whilethe sum in (7b) runs over spins i and j which resideon neighboring chains. D is the DM vector which,due to crystal symmetry of CuSe O , is restricted to D = ( D a ∗ , , D c ). The DM vector is staggered which istaken into account by ( − i in the DM term. The tensorˆ δ represents the symmetric anisotropic exchange and isassumed to be diagonal in the a ∗ bc coordinate systemˆ δ = δ a ∗ δ b
00 0 δ c . (8)The g tensor in CuSe O is staggered, ˆ g i = ˆ g u +( − i ˆ g s , where ˆ g u is the uniform and ˆ g s the staggeredcomponent. To model the experimentally observed AFMR modes,we first transform the Hamiltonian [Eq. (7)] into mag-netic free energy F per Cu site by applying the molecularfield approximation. Neutron diffraction suggests fourmagnetic sublattices so we write our F as (see Fig. 12), F = 2 X i =1 (cid:0) J ′ M i − · M i + J ′ IC M i · M i +2 ++ D ′ · M i − × M i + M i − · ˆ δ J ′ · M i (cid:1) −− X i =1 M i · (ˆ g u − ( − i )ˆ g s ) /g · H , (9)where the factor 2 emerges from the boundary condi-tions. Eq. (9) represents the simplest expression in whicheach spin has two intrachain neighbors and two interchainneighbors (see Fig. 12), which seems to be a good approx-imation for CuSe O , as mentioned above. In Eq. (9) thestaggered DM interaction and the staggered g tensor aretaken into account. The sublattice magnetizations aregiven by M i = − N gµ B h S i i , where N is the numberof Cu ions on the i -th sublattice, g = 2 . JJ IC
12 34
FIG. 12. (Color online) Basic cell describing the interactionspresent in CuSe O . Numbers indicate the four magneticsublattices. free-electron g factor and h ... i indicates the thermal av-eraging. The relation between molecular-field constantsand the interaction constants of the Hamiltonian (6) aredefined by J ′ = JN ( gµ B ) , (10a) J ′ IC = J IC N ( gµ B ) , (10b) D ′ = 1 N ( gµ B ) D . (10c)The ground state of the system described by the ex-pression (9) is obtained by numerical minimization ofthe free energy. Using the parameters suggested previ-ously from the ESR analysis in the paramagnetic phase, J = 157 K, J IC = 0 . J , | D ∗ a | = 0 . | D c | = 0 . δ a ∗ = δ b = 0, | δ c | = 0 . , and taking µ eff = 0 . µ B from the neutron scattering data we correctly predict themagnetic structure if δ c is negative. The effective fieldacting on the i -th sublattice magnetization is calculatedfrom B eff,i = − ∂ F /∂ M i . The AFMR modes are thenobtained from the equation of motion for M i precessingaround B eff,i . The four sublattice model predicts fourAFMR modes, where only the lowest in energy is ex-perimentally observed. We note that our previous EPRlinewidth analysis could not determine the sign of theanisotropy parameter. However, these parameters fail topredict the precise value of the SF field and some detailsof the AFMR modes.We have managed to describe both the field as wellas the angular dependence of the AFMR data by tun-ing only the symmetric anisotropic exchange parameters; δ a ∗ = 0 . δ b = 0, δ c = − . b directionat the field of H SF ≈ . abc ) coordinate system obtained for these param- eters are M , = ( − . , ± . , − . M , =(0 . , ∓ . , . b be-ing the easy axis, as well as a finite magnetization of asingle chain. Furthermore, our model also correctly pre-dicts the magnetic order above the spin-flop transition.At H = 40 kOe our calculations yield | m a ∗ | = 0 .
46 and | m c | = 0 .
19, in good agreement with the neutron experi-ment.Finally, we also calculated angular dependence of thetorque in the a ∗ b plane. The results are shown withfull red lines in Fig. 11. The amplitude of the measuredtorque depends on the mass of the sample, so calculatedtorque was rescaled to match the observed torque ampli-tude in the same way as magnetization. The agreement isexcellent. Since the modeling with the experimental ∆ χ does not work, the complementary free-energy approachclearly demonstrates that the spins progressively rotateaway from the b axis in the a ∗ c plane with rotation ofthe applied field. IV. DISCUSSION
Magnetic structure, AFMR, magnetization and angu-lar dependence of the magnetic torque were correctly re-produced using the Hamiltonian proposed in the previ-ous ESR study of the PM state. Present experimentsallow for some further improvement. First, in the previ-ous ESR study it was impossible to determine the signof δ c . The free energy analysis presented here showedthat in order to reproduce the correct ground state inagreement with the neutron diffraction and µ SR mea-surements, it is necessary to have δ c <
0. Furthermore,we were able to precisely assess the values of the sym-metric anisotropic exchange, which was in the previousstudy least defined. The new values δ a ∗ = 0 . δ b = 0and δ c = − .
001 are significantly smaller than previouslyproposed δ a ∗ = δ b = 0, δ c = − .
04. The most likelyreason for the discrepancy is that in the previous ESRstudy this parameter was derived based on the assump-tion that the observed linear temperature increase of theESR linewidth results solely from the 1D spin-spin cor-relations below T ≤
100 K, as suggested by theory for1D HAF. In fact, part of the observed linear behaviormight as well arise from the spin-phonon line broadening,which was observed at higher temperatures.The magnitude of the ordered Cu magnetic mo-ment obtained from the present measurements is µ eff ≈ . µ B . The moment is thus significantly reduced with re-spect to 1 µ B . The measured value is in good agreementwith the predictions of the coupled quantum spin-chainapproach, which gives µ eff = 0 . µ B for J ic = 0 . J . The present results, similarly as previous ESR analysis,thus point to strong quantum spin fluctuations as antic-ipated in the quasi-1D systems. This conclusion opposesthe conclusions drawn from Raman scattering, which in-dicate more classical 3D spin dynamics. Our susceptibility measurements along the three crys-1tal directions allowed us to determine the anisotropicstaggered field coefficient in CuSe O , which is a re-sult of both the staggered g tensor and DMI. The de-rived staggered field coefficient c c =0.15(2) is compara-ble to those found in related Cu-based 1D compounds[PMCu(NO ) (H O) ] n ( c c ” ≈ . c b ≈ . and Cubenzoate ( c c = 0 . The staggered g tensor and theobserved DMI in CuSe O place this system to a subclassof quasi-1D spin systems where a staggered field can beinduced by a magnetic field. The frequency span accessi-ble in present experiments, however, seems not sufficientto directly probe the low-energy excitations characteristicof decoupled TLL–chains. In addition, strong staggeredfields may drive CuSe O further away from the TTLstate explaining why only spin-waves were observed aslow-energy excitations. The crystal symmetry and theordered state found in CuSe O are such that the com-petitive case should be realized when a finite magneticfield is applied below T N . The staggered field coeffi-cient found in CuSe O amounts to c c ≈ . J IC /k B ≈
20 K(1 K ∼ . H ∼ O to induce the competition between the stag-gered field h = cH and the staggered field originatingfrom LRO structure. The observed spin-flop transitionat H SF ≈
13 kOe applied along the easy axis directionthus presents a classical spin-flop transition which orig-inates from the competition of underlying anisotropiesand is not driven by the staggered field, such as the spinreorientation transitions observed in BaCu Si O . Wenote that the latter is characterized by a similar stag-gered field coefficients but with much smaller interchaininteraction, on the other hand. The observed decrease of susceptibility [inset ofFig. 2(a)] and susceptibility anisotropy below 22 Kwhich cannot be explained in 1D HAF model even whenincluding staggered fields deserves a comment, albeit ofa speculative nature. It is possible that this is connectedto the crossover from 1D to 3D behavior of a 1D HAF ina staggered field. Recent NMR results on BaCu Si O were satisfactorily explained using a Ginzburg-Landaufree energy expansion in the vicinity of T N where thiscrossover was revealed. Further investigations of themagnetic response of CuSe O in the vicinity of T N byusing other local probes, such as nuclear magnetic reso-nance, should prove very informative in this respect. V. CONCLUSION
The magnetic ground state and the low-energy excita-tions in quasi-1D HAF CuSe O were studied experimen-tally by the neutron diffraction, static magnetic measure-ments, µ SR and AFMR. All experimental results werecoherently explained with the same Hamiltonian as de-rived previous from the analysis of the EPR linewidthwithin the theory for 1D HAF. The antiferromagnet- ically ordered ground state below T N = 17 K is char-acterized by the reduced Cu ( S = 1 / µ B , which is in line with the expected strongquantum fluctuations emerging from the underlying one-dimensionality of the system. Staggered magnetic fieldsarising from the staggered g tensor and DMI govern theground state and low-energy magnetic properties of thesystem, however, within experimentally accessible mag-netic field they are too small to prevail over the inter-chain interaction and thus induce the TLL physics. Nev-ertheless, future studies of this system in the vicinity ofthe phase transition could provide intriguing new insightabout the influence of the staggered fields on a dimen-sional crossover, expected in quasi-1D systems with long-range order. ACKNOWLEDGMENTS
M. H. acknowledges financial support by the Postdocprogram of the Croatian Science Foundation (Grant No.O-191-2011), the Slovene Human Resources Developmentand Scholarship fund under grant No. 11013-57/2010-5 and the Croatian Ministry of Science, Education andSports under Grant No. 035-0352843-2846. A. Z., M. P.and D. A. acknowledge the financial support of the Slove-nian Research Agency (projects J1-2118 and BI-US/09-12-040). Neutron diffraction experiments were performedat SINQ, Paul Scherrer Institute, Villigen, Switzerland.
Appendix A: Comparison of calculated and observedintensities in neutron diffraction for Γ and Γ models of magnetic structure in CuSe O In Fig. 3 the agreement of the two possible models,Γ and Γ (see Tab. I), with the observed intensities,is shown. In Table V we list the magnetic intensitiesmeasured at 6 K and compare them to the calculatedintensities for the Γ and Γ model. Appendix B: Muon polarization in the
CuSe O single crystal We set the orthogonal coordinate system so that its x , y and z axes corresponds to the right, up and back-ward direction with respect to the muon-beam direction,respectively. In general, the initial muon polarization istilted by an angle α from the z axis in the yz plane, P = (0 , sin α, cos α ), and the local field is character-ized by the polar angle θ and the azimuthal angle ϕ , B µ = B µ (sin θ cos φ, sin θ sin φ, cos θ ). The parallel andthe perpendicular component of the muon polarizationwith respect to the field are then derived from the equa-2 TABLE V. Observed and calculated magnetic intensities I obs and I calc , respectively, of CuSe O single crystal at 6 K cor-responding to the models Γ and Γ discussed in the text. h k l I obs I calc (Γ ) I calc (Γ )1 0 0 2.00 3.17 1.66-1 0 1 172.42 162.88 156.871 0 1 140.45 115.9 146.330 -1 0 13.48 0 9.52-3 0 1 108.31 22.27 133.15-1 0 2 9.01 2.56 7.772 -1 0 11.88 1.05 5.442 -1 1 63.93 88.95 68.973 0 1 73.92 38.41 109.25-1 0 3 145.91 123.63 96.95-3 0 3 100.76 98.27 94.64-5 0 1 53.88 2.5 91.671 2 1 17.85 96.27 14.19-5 0 3 64.05 45.59 75.13-1 0 5 64.12 49.88 39.462 1 -1 52.95 116.69 54.530 -3 0 5.00 0 3.102 1 0 2.97 1.05 5.441 2 0 4.24 0.08 5.700 -1 2 10.92 1.36 7.23 tion of motion d P ( t ) / d t = γ µ P ( t ) × B µ , P k = ( P · B µ ) B µ B µ , (B1) P ⊥ ( t ) = P ⊥ cos( γ µ B µ t ) + P ⊥ sin( γ µ B µ t ) , (B2)respectively, where the two orthogonal perpendicular vec-tors are P ⊥ = P − P k , (B3) P ⊥ = P ⊥ × B µ B µ . (B4)If the longitudinal muon relaxation rate λ L and the trans-verse relaxation rate λ T are taken into account, the total muon polarization at a given time will be given by P ( t ) = P k e − λ L t + P ⊥ ( t )e − λ T t (B5)The muon polarization measured by the backward-forward and the up-down sets of detectors is then chang-ing with time as P z ( t, B µ , θ, ϕ, λ L , λ T , α ) = P ( t ) · (0 , , , (B6) P y ( t, B µ , θ, ϕ, λ L , λ T , α ) = P ( t ) · (0 , , , (B7)respectively.In the CuSe O single crystal each muon stoppingsite i gives four different orientations of the given mag-netic field, because of the symmetry of the magnetic littlegroup (only the inversion symmetry leaves the field un-changed). Therefore, the polarizations are given by P y,z sc ( t, B µi , θ i , ϕ i , λ Li , λ T i , α ) =14 X j =1 P y,z ( t, B µi , θ ji , ϕ ji , λ Li , λ T i , α ) . (B8)The four polar and azimuthal angles for each site aresymmetry related. 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