A theoretical study of magnetic excitations in disorderedspin-Peierls systems

M. Saito* , H. Fukuyama

Department of Physics, University of Tokyo, Tokyo 113-0033, Japan

Abstract

It is now established that disorder in a spin-Peierls system can result in long-range-ordered staggered moments, whoseamplitudes vary spatially. In such a system two kinds of magnetic excitations are expected due to antiferromagnetic orderingsand dimerization. We investigate them phenomenologically and obtain gapless antiferromagnetic spin wave mode and gappeddimerization mode with intrinsic broadenings. These results are compared with experiments.q 1999 Elsevier Science Ltd. Allrights reserved.

Keywords:A. Magnetic materials; A. Oxides; D. Defects; D. Magnetic properties; D. Phase transitions

1. Introduction

Recently there has been much interest in the effects ofimpurities on the singlet ground states such as spin-Peierls(SP), spin ladder, Haldane systems and high temperaturesuperconducting cuprates. In the case of the SP system,CuGeO3 [1], neutron scattering experiments on CuGe12y-

SiyO3 (y � 0.007) [2] demonstrated the coexistence of theBragg spot intensities at low temperature for both latticedimerization and antiferromagnetic (AF) orderings. Thetwo Bragg spots are resolution limited, which implies thatthe two orderings are long ranged. The result was verysurprising since the two orderings had been considered tobe exclusive.

We proposed a theoretical possibility of the coexistenceusing the phase Hamiltonian technique atT � 0 [3]. It isshown that in this case both orderings are truly long rangedbut have spatial variations reflecting distribution of impuri-ties. The essence of the proposal is that once the quantumcoherence, which is inevitable in the singlet ground state, isperturbed, the hidden AF long range order will certainlyappear. This phenomenon is seen to be common in all sing-let ground states such as two-leg ladder [4], Haldanesystems and high temperature superconducting cuprates[5]. We have also seen, using field theoretical techniques,

that the exotic ground state results in a very characteristicmagnetism in the system [6,7].

The coexistence state in disordered SP systems has beenobserved also for another concentration [8] and in sampleswith other impurities, e.g. in samples where Cu atoms arereplaced with a few percent of Zn atoms as well [9]. Thecoexistence and the spatial variation of moments predictedtheoretically have been also confirmed by ESR [10,11] andmSR measurements [12,13], respectively.

In the present paper magnetic excitations in the systemare investigated. We study two excitation modes due to twoorderings separately and find that each mode has each char-acteristic broadening even atT� 0. In Section 2 we reviewour theoretical results about the coexistence state briefly. InSection 3 we study the magnetic excitations. Comparisonwith experimental results and discussions are given inSection 4.

2. Coexistence of AF and SP long range orderings

In our original proposal of the possibility of the coexis-tence, we assumed that the lattice distortion is reduced atimpurity sites and showed that the reduction results in theappearance of the AF moment by using phase Hamiltoniantechniques [3]. Then true long range AF ordering is obtainedby considering the three-dimensionality of the system impli-citly. In this case the magnitudes of both dimerization and

Journal of Physics and Chemistry of Solids 60 (1999) 1113–1115

0022-3697/99/$ - see front matterq 1999 Elsevier Science Ltd. All rights reserved.PII: S0022-3697(99)00063-3

* Corresponding author.

AF moments are spatially varying as schematically shown inFig. 1.

Later we extended the theory in terms of the effect ofimpurity so that we analyzed the problem under the assump-tion that an impurity changes couplings around it, such assuperexchange interaction,J, spin–lattice coupling,l , andelastic constant,K [7]. These changes of couplings areshown to cause the reduction of lattice distortion and leadto the coexistence as in the original paper. The phasediagram given by such an elaborate analysis, however, isdifferent from the original result and has phase transitionfrom the coexistence phase to Ne´el state at a critical density,

which is suggested by experiments [8,14]. The transition isvery recently indicated to be first-order by Masuda et al. inMg-substituted CuGeO3 [14], about which we are nowstudying in detail by using the extended theoretical frame-work [15].

3. Two kinds of magnetic excitations

Magnetic excitations in disordered SP systems areexpected to have characteristic features reflecting the uniquenature of the ground state. The two long range ordersemphasized in the previous section will lead to two kindsof excitations, AF spin waves and gapped dimerizationmode. We have studied AF mode by spin wave theory anddimerization mode by the phase Hamiltonian. Randomnessof the system causes broadenings of both modes [6].

Spin wave excitation due to the AF long range order isfound to beq-linear along the chain as in the Ne´el state inclean systems. The excitation spectrum, however, stronglydepends on the spatial variation of AF moments. It becomeslarge, in general, as impurity increases and has intrinsicbroadening even atT � 0. The broadening is caused bythe random distribution of impurities and is shown to beproportional toq2, which is completely different from thatby finite temperature effects and is seen to be particular todisordered systems.

The gapped mode due to SP long range ordering is alsoobtained by using the phase Hamiltonian. Its frequency isreduced by the reduction of the average lattice distortion andis broadened. This broadening is due to the spatial variationof the order parameter and is large around a smallq region incontrast to the broadening of the spin wave mode.

These results about the two modes are schematicallyshown in Fig. 2 [6].

4. Discussion

Neutron scattering studies on magnetic excitations ofdisordered SP systems, CuGe12ySiyO3 with y � 0.007 [2],Cu12yZnyGeO3 with y� 0.032 [16] were carried out, whichactually disclosed the existence of the two modes. Above allMartin et al. [16] discussed theq-dependence of broaden-ings qualitatively. They indicated that the broadening of thespin wave mode and dimerization mode are large at largeqand smallq, respectively.Their results are consistent withour results.Very recently Hirota et al. investigated the spinwave mode in CuGe12ySiyO3 with y � 0.011 andy � 0.03,especially noting theq-dependence of the broadening ofspin waves [17]. They concluded that the broadening ofthe spin wave is scaled well withq2 for small q regionand their results strongly support our prediction, thoughthey measured only perpendicular to the chain. Quiterecently the neutron scattering studies on the spin wavepropagating along the chain have also been carried out,whose results are in good agreement with our theory [18].

M. Saito, H. Fukuyama / Journal of Physics and Chemistry of Solids 60 (1999) 1113–11151114

Fig. 1. The spatial variation of (a) the spin magnetic moment,kSz

l l=s0, and (b) the lattice distortion,ul/u0, in the region betweentwo impurities.

Fig. 2. Two magentic excitations in disordered SP systems.

In the present paper we considered two modes separately.We are, however, interested in their strong mutual relationsin terms of the spectral weight and are trying to construct atheoretical framework to study them together.

In summary, we have investigated magnetic excitations indisordered SP systems, which are very characteristic,reflecting the exotic ground state. We have shown thattwo magnetic excitations—spin wave mode and gappeddimerization mode—exist due to coexistence of two longrange orderings. We have disclosed that randomness of thesystem leads to broadenings of both modes even atT� 0 indifferent regions. Our results, includingq-dependence of thebroadening of the spin wave mode, qualitatively agree withexperimental results.

Acknowledgements

We are grateful especially to G. Shirane and K. Hirota forvaluable discussions and to S. H. Lee for providing theirexperimental results prior to publication. M.S. also thanksJSPS Research Fellowships for Young Scientists.

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