One- and two-magnon spectra in quasi-one-dimensional triangular antiferromagnets

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    Journal of Magnetism and Magnetic Materials 140-144 (1995) 1955-1956 ~ Journal of

    magnetism and

    , i~ magnetic materials

    One- and two-magnon spectra in quasi-one-dimensional triangular antiferromagnets

    T. Ohyama *, H. Shiba Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan

    Abstract The nonlinear coupling among spin waves is taken into account to explain the observed spin excitation spectrum of

    CsNiC13. We found that the continuum due to the two-magnon excitation processes gives large peaks and also significantly affects the one-magnon spectrum. This is due to both the frustration and the quasi-one-dimensionality, which are important in CsNiCla-type antiferromagnets.

    Quantum fluctuations play an important role in low dimensional antiferromagnetic systems. Recently, neutron scattering experiments have revealed that the spin excita- tion spectrum of CsNiC13 cannot be described well by the conventional linear-spin-wave theory (LSW), even below the Nrel temperature [1-4]. In this compound, the S = 1 spins on Ni sites form a hexagonal lattice. The interaction is quasi-one-dimensional along the c-axis and the frustra- tion comes from the triangular-lattice structure in the c-plane.

    The disagreement with the experiments and LSW is present not only in the spin-wave dispersion. The excita- tion around the wave vector (0, 0, 1) in the xz mode is especially anomalous: the disappearance of the spin-wave excitation and the appearance of an additional branch with nearly degenerate energy with the y mode around 0.5 THz. Here the y and xz modes represent the fluctuations in and out of the plane of the 120 structure, respectively. Those discrepancies have been interpreted as due to either the effect of the Haldane gap or the non-collinearity. Phenomenological theories were proposed along these lines [5-7].

    In this contribution we show that the nonlinear spin- wave theory which includes the multi-maguon excitation processes can explain these experiments. The zero-field results were presented in Ref. [8] and the details of the finite-field case will be published elsewhere [9]. The Hamiltonian we consider is

    chains planes

    ,g"=2J o ~, S i 'S /+2 J 1 ~, Si'Sj-glXBH~_,S y. ( i , j ) ( i , j ) i


    * Corresponding author. Fax: +813-5734-2739; email: to-

    The Holstein-Primakoff transformation is carried out by assuming the 120 structure; then the first 1/S corrections are calculated, in which the two-magnon excitation pro- cesses are taken into account. The parameters are chosen as Jo = 310 GHz and J1 = 8 GHz to fit the dispersion relation and g = 2.2 is used.

    Fig. 1 shows the calculated dispersion relation at zero field together with the experimental data. Generally, three one-magnon branches are expected: one y mode and two xz (xz+ and xz_) modes. They agree with each other well, in contrast to the case of LSW. We also calculated the intensity of the one-magnon excitation and found a large increase and decrease in the y and xz modes, respectively around (0, 0, 1). The calculated dynamical correlation functions at (0, 0, 1) are presented in Fig. 2. Within the LSW, they consist of only delta-function-type spikes due to the one-magnon excitations, while the two-

    (0,0,1+r/) (r/,~,l) 1.6 ' 1 ' ' ' ' ' ' ' ' ' ,

    14~ ~z+ ----- XZ_ - . . . . . .

    l unpolarized x

    t 0.6



    Reduced Wavevector Component

    Fig. 1. Comparison between the theoretical dispersion relation (lines) and the experimental data (points) (taken from Ref. [3]).

    0304-8853/95/$09.50 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00721-7

  • 1956 T. Ohyama, H. Shiba /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1955-1956

    magnon excitation processes bring a continuum spectrum. Remarkably the spectrum in the xz mode at H = 0 shows two peaks: the lower one is the one-magnon excitation, which is fairly weak compared with LSW, and the upper one comes from the two-magnon excitation (Fig. 2(a)). The two-magnon peak remains well also at H :# 0 (Fig. 2(b)). The large peak in the y mode at H = 0 splits into two peaks at H = 10 T: the lower one-magnon and upper two-magnon peaks. (Note here we do not take into ac- count, for simplicity, the correction for the one- and two-magnon excitation energy, i.e., the peak positions are determined within the LSW.) We speculate that the upper peak in the xz mode could be assigned as the observed xz peak around 0.5 THz.

    The magnetic field dependence of the one-magnon excitation energy at (0, 0, 1) is shown in Fig. 3 together with the experimental data. The structure in the xz+ mode around 3 T is due to a hybridization with the two-magnon state. The observed highest branch may be the two-mag-

    12- -











    H = 10T

    I / II / / II


    q= (0,0,1) V XZ . . . .

    q = (0,0, i) V


    0.8 1.2 1.6

    Frequency (THz)

    Fig. 2. Frequency dependence of the dynamical correlation func- tions at (0, 0, 1) (a) for H = 0 and (b) for H = 10 T. The solid and dashed lines represent the y (a = y) and xz (a = x or z) modes, respectively.

    1! i i i i

    q = (0, 0, 1) l~ 0.8 o o

    o o

    0.( o o o u

    f 0.4

    xz_ -0 . . . . . . . . . t5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O

    0.2 . . . . . . . xg+

    0 4 ' 6 ' 8 ' 10

    Magnetic Field (T)

    Fig. 3. Comparison between the theoretical and experimental magnetic field dependence of the one-magnon excitation energies at (0, 0, 1) (taken from Ref. [4]).

    non peak. Experimentally, this was observed in the xz mode below 6 T, although at higher fields the polarization has not been determined. As shown in Fig. 2, two-magnon peaks exist not only in the xz mode but also in the y mode around 10 T. The experimental check on this point is highly desired.

    In conclusion, the two-magnon excitation is a conse- quence of the non-collinear spin structure and the quasi-one dimensionality makes the effect significant. We found that the two-magnon excitation processes are remarkably im- portant to describe the excitation spectrum.

    Acknowledgements: We express our thanks to W.J.L. Buyers and T. Inami for useful discussions.


    [1] W.J.L. Buyers, R.M. Morra, R.L. Armstrong, M.J. Hogan, P. Gerlach and K. Hirakawa, Phys. Rev. Lett. 56 (1986) 371.

    [2] Z. Tun, W.J.L. Buyers, R.L. Armstrong, K. Hirakawa and B. Briat, Phys. Rev. B 42 (1990) 4677.

    [3] K. Kakurai, M. Steiner, R. Pynn and J.K. Kjems, J. Phys. Condens. Matter 3 (1991) 715.

    [4] M. Enderle, K. Kakurai, K.N. Clausen, T. Inami, H. Tanaka and M. Steiner, Europhys. Lett. 25 (1994) 717.

    [5] I. Affleck, Phys. Rev. Lett. 62 (1989) 474, 1927(E); 65 (1990) 2477(E); 65 (1990) 2835(E).

    [6] I. Affleck and G.F. Wellman, Phys. Rev. B 46 (1992) 8934. [7] M.L. Plumer and A. Caill6, Phys. Rev. Lett. 68 (1992) 1042. [8] T. Ohyama and H. Shiba, J. Phys. Soc. Jpn. 62 (1993) 3277. [9] T. Ohyama and H. Shiba, J. Phys. Soc. Jpn. 63 (1994) 3454.


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