Magnetic excitations in the trimeric compounds A 3 Cu 3 (PO 4 ) 4 (A = Ca, Sr, Pb)
MMAGNETIC EXCITATIONS IN THE TRIMERIC COMPOUNDSA Cu (PO ) (A = Ca, Sr, Pb)M. Georgiev and H. Chamati Abstract
We study the magnetic excitations of the trimeric magnetic compounds A Cu (PO ) (A= Ca, Sr, Pb). The spectra are analyzed in terms of the Heisenberg model and a generic spinHamiltonian that accounts for the changes in valence electrons distribution along the bondsamong magnetic ions. The analytical results obtained in the framework of both Hamiltoniansare compared to each other and to the available experimental measurements. The results basedon our model show better agreement with the experimental data than those obtained with the aidof the Heisenberg model. For all trimers, our analysis reveals the existence of one thin energyband referring to the flatness of observed excitation peaks. Molecular magnets possess unique properties and are ideal candidates for exploring the interplay ofthe quantum and the classical worlds. They may manifest a great variety of magnetic features de-termined from weakly interacting isolated fundamental structural units, such as dimers, trimers andtetramers [1] . The effect of quantum tunneling in single-molecule magnets [2,3] and the response ofspin-switching in the frustrated antiferromagnetic chromium trimmer [4] are some prominent exam-ples. With their short-range spin correlation the small spin clusters stand as elegant tools for studyingthe relevant coupling processes. Magnetic measurements on trimer copper chains A Cu (PO ) with(A = Ca, Sr), reported in Ref. [5] , show that the intertrimer exchange couplings are negligible and thusthe trimers might be considered as separate clusters. These results were confirmed via INS experi-ments [6,7] that shed light on the magnetic spectra with the aid of the antiferromagnetic Heisenbergmodel involving nearest and next-nearest intratrimer interactions, and later they were extended tothe compound Ca Cu (PO ) [1] . Moreover, it turns out that the interaction between edged spins inisolated trimers is also negligible. The difference in the magnetic properties among the compoundsCa Cu Ni(PO ) [8] and Ca Cu Mg(PO ) [9] is another demonstration for the richness of the physicalfeatures arising from a symmetrically trivial linear spin trimers, see e.g. Ref. [10] .In the present article we report a theoretical study of the magnetic spectra of magnetic clus-ters. We focus our attention on the trimeric compounds A Cu (PO ) with (A = Ca, Sr, Pb), forwhich the magnetic excitations are determined experimentally [6,7] . To describe the magnetism in thecompounds A Cu (PO ) we employ the approach devised in Refs. [11,12] . The approach is basedon a generic spin Hamiltonian that allows to compute effectively the changes of electron’s densitydistribution along the complex exchange bridges among magnetic centers. We compare the resultsof our study obtained in the framework of the named generic spin Hamiltonian and its Heisenbergcounterpart demonstrating their equivalence and differences.1 a r X i v : . [ phy s i c s . a t m - c l u s ] M a r he rest of this paper is structured as follows: In Section 2 we present the keystone relations forthe neutron scattering intensities and formulate explicitly the generic spin Hamiltonian. In Sections3 we explore the low-lying magnetic excitations of the compounds A Cu (PO ) (where A standsfor Ca, Sr, Pb) within the framework of the Heisenberg model and our Hamiltonian. A summary ofthe results obtained throughout this paper along with conclusions are presented in Section 4. To determine the energy level structure and the transitions corresponding to the experimentally ob-served magnetic spectra one needs a number of parameters to account for all couplings in the system.It is cumbersome to apply a general approach with a unique set of parameters that can describe allpossible magnetic effects and in addition to distinguish between inter-molecular and intra-molecularfeatures. Usually, one starts with bilinear spin microscopic models, such as the Heisenberg Hamilto-nian [13] and depending on the exhibited magnetic features different interaction terms are included [14] .To obtain meaningful results one calculates the neutron scattering intensities I n (cid:48) n ( q ) integratedover the angles of scattering vector q of the neutron. For identical magnetic ions, represented by theoperators ˆ s αi and ˆ s αj , they read [15–18] I n (cid:48) n ( q ) ∝ F ( q ) (cid:88) α,β Θ αβ S αβ ( q , ω n (cid:48) n ) , (1)where F ( q ) is the spin magnetic form factor [19] , Θ αβ is the polarization factor and α, β, γ ∈{ x, y, z } . In (1) the magnetic scattering functions are explicitly written as S αβ ( q , ω n (cid:48) n ) = (cid:88) n,n (cid:48) ,i,j e i q · r ij p n (cid:104) n | ˆ s αi | n (cid:48) (cid:105)(cid:104) n (cid:48) | ˆ s βj | n (cid:105) δ ( (cid:126) ω n (cid:48) n − E n (cid:48) n ) , (2)where ω n (cid:48) n is the frequency of a magnetic excitation related to a transition between the states | n (cid:105) and | n (cid:48) (cid:105) with the corresponding energy E n and E n (cid:48) , respectively. Further, e i q · r ij is the structurefactor associated with the cluster geometry, p n = Z − e − E n /k B T is the population factor (with Z thepartition function). The distribution of coupled magnetic centers (ions) plays a crucial role in uniquely determining thescattering intensities. Even when these effective bonds are indistinguishable with respect to theirlengths and the total spin, according to (2), one can obtain different in magnitude neutron scatteringintensities. However, to identify each intensity one has to use an appropriate spin model leading toan energy sequence such that the δ function in the r.h.s of (2) defines the relevant spin bonds withrespect to the structure factors.To describe the magnetic spectra in the considered trimeric compounds we employ the proposedin Ref. [11,12] generic spin Hamiltonian ˆ H = (cid:88) i (cid:54) = j J ij ˆ σ i · ˆ s j , (3)2here the couplings J ij = J ji are effective exchange constants and the operator ˆ σ i ≡ (ˆ σ xi , ˆ σ yi , ˆ σ zi ) accounts for the differences in valence electron’s distribution with respect to the i th magnetic center.Let us note that model (3) was applied successfully to explore the magnetic spectra of the molecularmagnet Ni Mo [12] . Cu (PO ) (A =Ca, Sr and Pb)3.1 The Hamiltonian The magnetic compounds A Cu (PO ) (A = Ca, Sr, Pb) are convenient spin trimer systems fortesting the Hamiltonian (3). On Fig. 1 (a) we show a small fragment of the copper ions structurewith the relevant exchange pathways with respect to the arrangement of oxygen atoms. Whence theCu2 ion is surrounded by four oxygen atoms on a plane, while Cu1 and Cu3 ions are surrounded byfive oxygen atoms constructing a distorted square pyramid. For the sake of clarity the other elementsare not shown and only two oxygen atoms along the intratrimer Cu1–O1–Cu2 and intertrimer Cu2–O2–Cu4 pathways are labeled. In general, the exchange processes appear to be more complex anddepend on the global structure of the compounds [5] . Besides the superexchange interactions aresensitive [6] to the angle between Cu bonds and their lengths suggesting that the intertrimer Cu2–Cu4 interaction is much smaller than the intratrimer ones, i.e. Cu1–Cu2 and Cu3–Cu2. Thus, theintertrimer exchange can be neglected and the Cu sub-lattice is considered as a one-dimensionalarray of isolated spin trimers Fig. 1 (b).Cu1 Cu2 Cu3Cu4Cu5 O1O2 OCu(a) Cu2Cu3 Cu1 Cu4Cu5
J J ′ (b)1Figure 1: (a) Exchange pathways in A Cu (PO ) (A = Ca, Sr, Pb). Copper colored circles representcopper ions, the red ones stand for oxygen atoms. The solid (black) and dashed (gray) lines representthe intratrimer and intertrimer exchange pathways, respectively. (b) Schematic representation of theintratrimer J and intertrimer J (cid:48) magnetic interactions in the array of isolated trimers.Taking into account that Cu1-Cu2 and Cu2-Cu3 are bonded by a single oxygen ion we set J ij → J = J and perform a study of the magnetic excitations. Owing to the trimer symmetry we applythe coupling scheme | s − s | ≤ s ≤ | s + s | , where s and s (with | s − s | ≤ s ≤ | s + s | ) arethe trimer and Cu1-Cu3 coupled pair spin quantum numbers, respectively. Thus, the Hamiltonian(3) reads ˆ H = J ( ˆ σ · ˆ s + ˆ σ · ˆ s + ˆ σ · ˆ s + ˆ σ · ˆ s ) . (4)With respect to the selected spin coupling scheme the total spin eigenstates are denoted by | s , s, m (cid:105) .3 .2 Energy levels The isolated trimer is described by four quartet and four doublet eigenstates. The eigenvalues of (4)are denoted by E ms ,s . Further, analyzing the energy spectrum we obtain the ground state energy for s = 1 , s = . The respective doublet states are (cid:12)(cid:12) , , ± (cid:11) with corresponding energies E ± / , / = − J. (5)The second pair of doublet states is associated with the first excited energy level, see Fig. 2. Theedged spins of the isolated trimer are coupled in a singlet, with corresponding state (cid:12)(cid:12) , , ± (cid:11) . Now,using (4) we end up with E ± / , / = − J a , , (6)where the parameter a , ∈ R account for the variations of electrons spatial distribution alongthe Cu1-Cu3 exchange bridge. To fully characterize the experimentally observed transitions forPb Cu (PO ) one requires at least three excited energy levels. Bearing in mind that the quartetlevel is four-fold degenerate, we deduce that the corresponding coefficient may take only two val-ues a , ∈ { c , c } . Further, the observed excitations spectra [6] are not broadened signaling that (cid:12)(cid:12) c − c (cid:12)(cid:12) ≈ . Therefore, taking into account (6) we get E ± / , / = − J c n n = 1 , . For all four quartet eigenstates (cid:12)(cid:12) , , m (cid:11) , with m = ± , ± , we have E ± / , / = E ± / , / = J. E n e r gy [ m e V ] E E E E − . . . Ca Cu (PO ) − . . . Sr Cu (PO ) − . . . . Pb Cu (PO ) Cu (PO ) (A = Ca, Sr, Pb). The blue arrowsshow the ground state transitions, while the red arrow stands for the excited transition. The energylevels corresponding to the ground state are designated by blue lines. The initial energy level of theexcited transition is depicted by a red line, while by analogy to Pb Cu (PO ) the dashed red linesstand for a presumed second sub level of the excited doublet level.4 .00.10.20.30.40.50.60.70.8 0 20 40 60 80 100 120 q = 1.72 Å −1 I n t en s i t y [ a r b . un i t s ] Temperature [K] I Pb I Pb I Sr I Sr I Ca I Ca I Pb q = 1.72 Å −1 I n t en s i t y [ a r b . un i t s ] Temperature [K]
Figure 3: Scattering intensities I A20 , I A30 and I Pb31 with (A = Ca, Sr, Pb) as a function of the temperature,calculated with the Hamiltonian (3). The blue squares, the green circles and red triangles correspondto the values of the intensities given in Tab. 2.The energy sequence consists of four levels. Henceforth we denote these levels as follow E = − J, E = − J c , E = − J c , E = J. (7)Therefore, we have at hand the parameters J and c n . The coupling J accounts for the interactionalong Cu1-Cu2 and Cu2-Cu3 bridges and c n will indicate any changes in the interaction betweenedged ions. However, we take further actions and derive the following relation J c n = J (cid:0) c n + (cid:1) ,where J c n represents the exchange constant between the next-nearest neighbors. Based on the selection rules ∆ s = 0 , ± , ∆ s = 0 , ± and ∆ m = 0 , ± and the aid of the identities S αβ ( q , ω n (cid:48) n ) + S βα ( q , ω n (cid:48) n ) = 0 , S αα ( q , ω n (cid:48) n ) = S ββ ( q , ω n (cid:48) n ) , where α (cid:54) = β and n, n (cid:48) = 0 , , , ,we may compute the scattering functions. Moreover, taking into account the cluster structure, wehave (cid:80) α Θ αα = 2 . The analysis of intensities reported in [6] allows us to determine the observedfirst magnetic excitation. It corresponds to the transition between the ground state energy E and E with scattering functions S αα ( q , ω ) = [1 − cos(2 q · r )] p , where r is the vector of the average distance r between neighboring ions with r = 2 r . Thedegeneracy of the quartet energy level is four–fold and hence the second ground state excitationrefers to transition from the doublet (cid:12)(cid:12) , , ± (cid:11) to the quartet states (cid:12)(cid:12) , , m (cid:11) , where m = ± , ± .Hence, for E → E we get S αα ( q , ω ) = [3 + cos(2 q · r ) − q · r )] p . The excited peak is indicated by the transition E → E . The corresponding scattering functions are S αα ( q , ω ) = [1 − cos(2 q · r )] p . I ∝ γ (cid:104) − sin(2 qr )2 qr (cid:105) F ( q ) , I ∝ γ (cid:104) sin(2 qr )6 qr − sin( qr )3 qr (cid:105) F ( q ) ,I ∝ γ (cid:104) − sin(2 qr )2 qr (cid:105) F ( q ) , (8)where γ = p , γ = p and γ = p . Moreover, for dications Cu the form factor reads F ( q ) = 256 / (16 + q r ) , where q is the magnitude of the scattering vector, r o = 0 .
529 ˚A is theBohr radius.
Taking into account (7) and (8) for the transition energies we get E = J (cid:0) − c (cid:1) , E = 3 J, E = J (cid:0) c (cid:1) . (9)Neutron scattering experiments performed on Pb Cu (PO ) with T ≥ K [6] show the presenceof a third peak at about . meV, which may be related to the excited transition energy E . Thevalues of c , c and J , according to INS experiments [6] performed on polycrystalline samplesA Cu (PO ) (A = Ca, Sr, Pb) are shown in Tab. 1. In addition, for the compound Ca Cu (PO ) we have c = − . and J ≈ .
741 meV based on INS data at T = 1 . K [7,13] .Table 1: The values of the coupling constants and the quantities c , c for our model applied toA Cu (PO ) (A = Ca, Sr, Pb) obtained by taking into account the experimental data in Ref. [6] .A E E E c c J J c J c Ca 9.335 14.174 − − -0.317 4.725 0.058 –Sr 9.936 15.064 − − -0.319 5.021 0.054 –Pb 9.005 13.693 4.9 -0.284 -0.315 4.564 0.062 0.168The temperature dependence of the integrated scattering intensities for each compound is shownon Fig. 3. On Fig. 4a we present the scattering intensities for Pb Cu (PO ) computed with ourHamiltonian and the Heisenberg model along with the experimental data taken from Ref. [6] . Let uspoint out that our results are in better agreement with their experimental counterpart for I Pb20 and I Pb30 ,while for I Pb31 we have a qualitative agreement. The averaged magnitudes of the scattering vector q and the distance r between neighboring ions are taken from Ref. [6] , q = 1 .
72 ˚A − and r = 3 . .The explicit expressions of the scattering intensities for each transition are I A20 ( T ) ∝ . Z − e − E A0 kBT , I A30 ( T ) ∝ . Z − e − E A0 kBT , I Pb31 ( T ) ∝ . Z − e − E Pb1 kBT , (10)where A = Ca, Sr, Pb. As T vanishes the scattering intensities I A20 and I A30 are equal by about a factorof 2, see Tab. 2. For
T > K a third peak sets in, but the evaluated intensity I Pb31 remains smallerthan the experimentally observed one [6] . In contrast to the functions I Pb30 and I Pb20 the intensitiesof the ground state transitions for A = Ca, Sr decrease slowly with temperature. The predicted6 q = 1.72 Å −1 I n t en s i t y [ a r b . un i t s ] Temperature [K] I Pb I Pb I Pb q = 1.72 Å −1 I n t en s i t y [ a r b . un i t s ] Temperature [K] (a) Scattering intensities for the compound Pb Cu (PO ) as a function of the temperature, along with ex-perimental results from Ref. [6] . The solid and dashed lines show the calculated intensities for the Heisenbergmodel and Hamiltonian (3), respectively. I n t en s i t y [ c oun t s / k m on ~ m i n .] Scattering vector [Å −1 ] I Pb I Pb I Pb I n t en s i t y [ c oun t s / k m on ~ m i n .] Scattering vector [Å −1 ] (b) Calculated intensities as a function of the scattering vector for Pb Cu (PO ) along with the experimentaldata of Ref. [6] . The dashed lines depict the intensities obtained from the Hamiltonian (3). The solid red andorange lines correspond to the Heisenberg model. I Pb20 and I Pb30 correspond to the ground state transitions at T = 8 K. The intensity I Pb31 stands for the excited transition at T = 60 K. Figure 4: Scattering intensities.peak for Pb Cu (PO ) is in concert with the experimental findings [6] . Unfortunately there are noexperimental data confirming the presence of this third peak for the compounds Ca Cu (PO ) andSr Cu (PO ) and hence the energy level E could not be included in determining the sequence ofenergy spectrum. On Fig. 2 the presumed energy levels E Ca1 and E Sr1 are illustrated with dashed redlines. For all compounds the scattering intensities as a function of the magnitude of the scatteringvector are represented in Fig. 4b. 7able 2: Calculated values of integrated scattering intensities I A n (cid:48) n [arb. units], using the Hamiltonian(3), for the trimers A Cu (PO ) (A = Ca, Sr, Pb) at temperatures 8, 60 and 125 K, depicted on Fig.3. T [K] 8 60 125 I Ca20 I Ca30 I Sr20 I Sr30 I Pb20 I Pb30 I Pb31
We propose an study for the magnetic excitations of the compounds A Cu (PO ) with (A = Ca, Sr,Pb). To this end, we use a generic bilinear spin Hamiltonian (3) that accounts for the variations in theelectron’s spatial distributions along the exchange bridges. Alongside with the named Hamiltonianwe compute the magnetic spectrum in the framework of the Heisenberg Hamiltonian and comparethe outcome from both models, see Figs. 4a and 4b. We found that the results obtained with ourmodel are in better agreement with the INS experimental data [6,7] than the Heisenberg model. On theother hand our results for the Heisenberg model coincide with those reported by other authors [6,7,13] .With respect to the energy levels sequence and relevant eigenstates the Heisenberg and ourHamiltonian (3) lead to similar values. For the investigated compounds, the ground state energyis related to the Cu1-Cu3 triplet bond, and the neutron energy loss, associated to both ground statemagnetic excitations, is due to the local triplet-singlet transition. However, the spin Hamiltonian (4),with the intrinsic parameter a , , identifies the experimentally observed third peak (about 4.9 meV)for the compound Pb Cu (PO ) [6] accurately, while the Heisenberg model is enable to reproduceit. We obtain one thin energy band composed of two very close energy levels that corresponds to theCu1-Cu3 singlet state (see e.g. Fig. 2). The energy band width signals for the small change in theelectrons distribution along the Cu1-Cu2-Cu3 bridge due to the temperature. Thus, the intensitiesindicated by dashed lines on Fig. 4a decrease rapidly than in the case of the Heisenberg model. Inother words, the inequality (cid:12)(cid:12) c n (cid:12)(cid:12) < for n = 1 , , shows that in the doublet level, the spatial distribu-tion of the electrons common to the edge ions is such that the exchange becomes negligible. Further,it points out that the next-nearest neighbor coupling slightly varies with respect to the temperaturetaking two values J ∈ (cid:8) J c , J c (cid:9) , see Tab. 1. On the other hand the difference (cid:12)(cid:12) c − c (cid:12)(cid:12) = 0 . explains the sharpness of the experimentally observed peaks [6,7] . Acknowledgments
The authors are indebted to Prof. N. Ivanov and Prof. J. Schnack for very helpful discussions, andto Prof. M. Matsuda for providing us with the experimental data used in FIGs. 4a and 4b. This workwas supported by the Bulgarian National Science Fund under contract DN/08/18.8 eferences [1] A. Furrer and O. Waldmann, “Magnetic cluster excitations,” Rev. Mod. Phys. , 367 (2013).[2] W. Wernsdorfer, N. Aliaga-Alcalde, D. N. Hendrickson, and G. Christou, “Exchange-biasedquantum tunnelling in a supramolecular dimer of single-molecule magnets,” Nature , 406(2002).[3] R. Schenker, M. N. Leuenberger, G. Chaboussant, D. Loss, and H. U. Güdel, “Phonon bot-tleneck effect leads to observation of quantum tunneling of the magnetization and butterflyhysteresis loops in (Et N) Fe F ,” Phys. Rev. B , 184403 (2005).[4] T. Jamneala, V. Madhavan, and M. Crommie, “Kondo Response of a Single AntiferromagneticChromium Trimer,” Phys. Rev. Lett. , 256804 (2001).[5] M. Drillon, M. Belaiche, P. Legoll, J. Aride, A. Boukhari, and A. Moqine, “1D ferrimagnetismin copper(II) trimetric chains: Specific heat and magnetic behavior of A Cu (PO with A =Ca, Sr,” J. Magnet. Magnet. Mater. , 83 (1993).[6] M. Matsuda, K. Kakurai, A. A. Belik, M. Azuma, M. Takano, and M. Fujita, “Magneticexcitations from the linear Heisenberg antiferromagnetic spin trimer system A Cu (PO ) (A= Ca, Sr, and Pb),” Phys. Rev. B , 144411 (2005).[7] A. Podlesnyak, V. Pomjakushin, E. Pomjakushina, K. Conder, and A. Furrer, “Magnetic ex-citations in the spin-trimer compounds Ca Cu − x Ni x (PO ) ( x = 0 , , ),” Phys. Rev. B ,064420 (2007).[8] M. Ghosh and K. Ghoshray, “Spin trimers in Ca Cu Ni(PO ) ,” Low Temp. Phys. , 645(2012).[9] M. Ghosh, M. Majumder, K. Ghoshray, and S. Banerjee, “Magnetic properties of the spintrimer compound Ca Cu Mg(PO ) from susceptibility measurements,” Phys. Rev. B ,094401 (2010).[10] M. Ghosh, K. Ghoshray, M. Majumder, B. Bandyopadhyay, and A. Ghoshray, “NMR study ofa magnetic phase transition in Ca CuNi (PO ) : A spin trimer compound,” Phys. Rev. B ,064409 (2010).[11] M. Georgiev and H. Chamati, “Magnetic exchange in spin clusters,” , 020004 (2019).[12] M. Georgiev and H. Chamati, “Magnetic excitations in molecular magnets with complexbridges: The tetrahedral molecule Ni Mo ,” EPJ B (The European Physical Journal B) (2019),accepted.[13] A. Furrer, “Magnetic cluster excitations,” Int. J. Mod. Phys. B , 3653 (2010).[14] H. Chamati, “Theory of Phase Transitions: From Magnets to Biomembranes,” Adv. PlanarLipid Bilayers Liposomes , 237 (2013).[15] S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter: Polarization Effects andMagnetic Scattering , International Series of Monographs on Physics, Vol. 2 (Oxford UniversityPress, Oxford, New York, 1986).[16] M. F. Collins,
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