The long range electric fields associated with long-wave longitudinal phonons are responsible for the phenomenon of LO-TO splitting, that is, the removal of degeneracy (简并) between the LO and TO phonons at the Brillouin zone centre (gamma point).
In order to deal with this, one must first consider the behaviour of the dynamical matrix. It should be noted that the LO-TO splitting depends upon the direction in which one approaches the gamma-point, and this anisotropy is accessible to experiment. In a later chapter this direction dependence of the LO-TO splitting will be calculated for quartz. In the limit, the dynamical matrix may be split [81,98] into an analytic and non-analytic contribution:
(3-106)
(3-107)
which can be seen to correspond to 1/r3 behaviour in real space, corresponding to dipole-dipole interactions [81]. These dipoles are created when the ions within an ionic semiconducting or insulating crystal are displaced in a long-wave longitudinal phonon. It should perhaps be noted that it is possible for the Born effective charge to vanish by symmetry, and non-analytic contributions to the dynamical matrix still exist. This is because, although symmetry may forbid a dipole being formed by an atomic displacement, it does not forbid higher order moments, such as quadrupoles and octupoles being set up. However, these contributions shall not be considered in this work.
In order to simplify the analysis, it will be assumed that the LO eigendisplacements of are the same as those of , even if the corresponding frequencies are not identical. This allows the following relationship to be obtained:
(3.108)which directly connects the LO-TO splitting to the Born effective charge, as promised.
It is worth noting that the above analysis could be extended in order to obtain the Lyddane-Sachs-Teller relationship [81]. However, in this work, consideration will be restricted to the case of the Coulomb gauge, that is, the retardation of the Coulomb interaction will be neglected, and the above analysis of coupling between lattice degrees of freedom and electrical degrees of freedom will suffice.
In order to simplify the analysis, it will be assumed that the LO eigendisplacements of are the same as those of , even if the corresponding frequencies are not identical. This allows the following relationship to be obtained:
(3.108)which directly connects the LO-TO splitting to the Born effective charge, as promised.
It is worth noting that the above analysis could be extended in order to obtain the Lyddane-Sachs-Teller relationship [81]. However, in this work, consideration will be restricted to the case of the Coulomb gauge, that is, the retardation of the Coulomb interaction will be neglected, and the above analysis of coupling between lattice degrees of freedom and electrical degrees of freedom will suffice.
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