ROLE OF PHONONS IN ORIENTATIONAL INTRA-PLANE ORDERING IN GRAPHITE INTERCALATION COMPOUNDS

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ROLE OF PHONONS IN ORIENTATIONAL INTRA-PLANE ORDERING IN GRAPHITE

INTERCALATION COMPOUNDS

C. Horie, H. Miyazaki, S. Igarashi, S. Hatakeyama

To cite this version:

C. Horie, H. Miyazaki, S. Igarashi, S. Hatakeyama. ROLE OF PHONONS IN ORIENTA- TIONAL INTRA-PLANE ORDERING IN GRAPHITE INTERCALATION COMPOUNDS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-298-C6-300. �10.1051/jphyscol:1981686�. �jpa-00221622�

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JOURNAL DE PHYSIQUE

ColZoque C6, supple'ment au nO1 2, Tome 42, de'cembre 1981

ROLE OF PHONONS IN ORIENTATIONAL INTRA-PLANE ORDERING IN GRAPHITE INTERCALATION COMPOUNDS

C . Horie, H . Miyazaki, S . Igarashi and S . Hatakeyama

Department o f Applied Physics, Tohoku University, Sendai, Japan

Abstract.- Theoretical study of orientational intra-plane

ordering in the second stage graphite intercalation compounds is presented with particular emphasis on the role of phonons. It is shown that the rotation angle of intercalant layers relative to the graphite layers is predominantly determined by i,ntra=

plane transverse modes and also depends on the ratio of lattice constants of both intercalant and graphite layeas.

1. Introduction.- X-ray diffraction studies of CZ4Cs prepared from single crystal graphite demonstrate that a triangular arrangement of Cs atoms in the real space is non-registered with graphite layers in macroscopic domains, generating a hexagonal sextetof diffraction spots rotated by + 1 4 O about the c-axis with respect to the <loo> graphite direction at temperatures 50KzTz165K [l]. Similar orientational intra-plane ordering has also been studied on C24Rb [2,3,41 and other compounds. In the present paper, a microscopic theory of the orientational intra-plane ordering in the second stage compounds is developed with particular emphasis on the role of phonons. The theoretical procedure follows the one developed by Novaco [ 5 1 for mono-layer films adsorbed on solid surfaces, but is extended so as to be applicable to our system consisting of intercalant layers sandwitched between a pair of graphite layers. It is shown that characteristic phonon dispersions modulated by a coupling between intercalant atoms and carbon atoms in the adjacent layers play a crucial role in determining the rotation angle in the orientational ordering.

2. Model.- We assume that intra-plane structure of an intercalant layer is a triangular lattice with a lattice constant dI and is non=

registered and incommensurate with the adjacent hexagonal graphite layers with a lattice constant d G. Then, the atomic displacements from the virtual lattice sites are modulated each other by means of the interatomic potential between intercalant and carbon atoms. This modulation yields displacements of atoms to new stable positions of intercalant and/or carbon atoms, giving rise to a rotation of the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981686

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reciprocal lattice of intercalants relative to that of carbon lattice.

The free energy, then, turns out to be given by Eph+EMDW,where E ph denotes the renormalized phonon energy and EMDW corresponds to the energy gain due to the static displacement of atoms and is called the mass density wave(MDW) energy hereafter. Within the self-consistent harmonic phonon approximation, EfifDFJ is given in a form

-f 3

where E.(q) are phonon frequencies of j-th modes, and g.(q) involves

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factors , ⁽⁶⁾, , ⁾ ⁾ and the Kronecker's delta 6+ + + which

j G+q , ^T

select the relevant phonons to stabilize the mass density wave. 2 and are the reciprocal lattice vectors of graphite and intercalant layers, respectively, and 2.

( G )

is the polarization vector of (j ,6)

3

phonon. The g.

( G )

depends also on the Debye-tialler factor, and the Fourier transform of the interaction potential between intercalant 3 and carbon atoms, which consists of the Born-Mayer type repulsive part and the screened Coulomb interaction.

3. Results and Conclusions.- Figure 1 shows the renormalized phonon dispersion curves for modes with polarization parallel to the basal plane in C24Cs. Other modes polarized parallel to the c-axis are omitted because they have no effect on the orientational rotation under consideration. Frequencies at the r point are determined on a simplified one-dimensional model, in which atomic arrangements and masses are taken to meet with layer stacking A a A A a A A

...

^{and with}

the in-plane stoichiometry of C12M. Use is also made of the neutron data of CsCs and C K in order to estimate the effective force con-

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stants. Details of the derivation of the phonon dispersion curves and Emw will be published elsewhere.

Fig. 1: Phonon dispersion curves for C2qCs. Ll! L2 and L3 represent longl- tudinal modes. T1, T2 and T3 represent transverse modes polarized parallel to the basal plane.

r W A V E VECTOR 4

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C6-300 JOURNAL DE PHYSIQUE

The EmW for C 2 4 C ~ is calculated by using phonon dispersion curves shown in Fig. 1, and is plotted against the rotation angle 0 between + rloo and dlOO in Fig. 2. We have taken dI = 6.02; and dG = 2.47;

( z = dI/dG = 2.44) which correspond to the layer stoichiometry C12Cs.

ROTATION ANGLE 8 ( degrees)

----/-

Fig. ² The 0-dependence of MDW energy for C24Cs calculated by uslng the dispersion curves given in Fig. 1. Solid curves represents the total MDW energy. The dash-dotted and dashed curves indicate contributions to EMD~J from T and L modes, respectively.

The minimum of EMDW is found at about 14O in good agreement with ex- periment. It seems interesting to observe from Fig. 2 that the lowest longitudinal mode always plays a role to stabilize energy at 0=19.1°, but the lowest transverse mode is more effective in stabilizing MDW at angles below 19.1°. The similar feature is also seen in the case of C24Rb. In this case, however, the EMDW shows a flat minimum over a certain range of 8 around 12O for z =2.45. It is concluded from our results that the rotation angle in the orientational ordering is determined not only by the value of z, but also by the detailed phonon dispersion curves characteristic to the compounds.

References

113 Clarke R. and Caswell N., Solin S.A. and Horn P.W., Phys. Rev.

Letters 4 5 (1979) 2018; Physica (1980) 457.

[ 2 ] ~uematsuH., Suzuki M., Ikeda H., and Endoh Y., Synthetic Metals

2 (1980) 133.

131 Fambe N., Dresselhaus G. and Dresselhaus M.S., Phys. Rev. B A (1980) 3491.

[ 4 ] Yamad; Y., Naiki I., Watanabe T., Kiichi T., and Suematsu H.,

Physica (1981) 277.

[ 5 ] Novaco A.D., Phys. Rev. (1979) 6493.