Mechanisms of Dielectric Loss in Microwave Materials
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(see e.g.,I),
(iii) excitation of acoustic phonons at the frequency of the ac electric field (see e.g., -3), (iv) interaction with the ferroelectric domain walls (see e.g., 4) (v) loss related to the "universal relaxation law" (see e.g., 5), (vi) contribution of the Ohmic conduction to the loss. In principle, any of these mechanisms can substantially contribute to the dielectric loss, however, the better the quality of a low-loss material and the higher the frequencies of the ac field used, the more important the role of the fundamental phonon mechanisms, (ii). The present paper is devoted to a discussion of these mechanisms and their role in the absorption of ac electric field at microwave and higher frequencies.
FUNDAMENTAL PHONON LOSS MECHANISMS The origin of the fundamental loss is the interaction of the ac field with the phonons of the material. The theory of the loss stemming from this interaction has been developed for crystalline materials with a well defined phonon spectrum (see for a review'), i.e., for materials where the damping of phonons [average frequency of the inter-phonon collisions], F, is much smaller than their frequencies. According to this theory, in terms of quantum mechanics, the fundamental loss mainly corresponds to the absorption of the energy quantum of the electromagnetic field hwo [o) is the ac field frequency] in collision processes with the thermal 221 Mat. Res. Soc. Symp. Proc. Vol. 603 © 2000 Materials Research Society
phonons, which have much higher energies. This large difference in the energies makes it difficult to satisfy the conservation laws in these processes. In such a complicated situation, there exist three efficient schemes of absorption of the hco-quanta, which correspond to the three main fundamental loss mechanisms: 1) three-quantum, 2) four-quantum, and 3) quasi-Debye. Three-quantum mechanism The three-quantum mechanism corresponds to a process involving a h)o-quantum and two phonons. The theory of this contribution has been developed by different authors 1 ,6-9. Owing to a very big difference between the energy quantum of the electromagnetic field hco and those of the thermal phonons, three-quantum processes can take place only in the regions of the wave-vector space (k -space) where the difference between the frequencies of two different phonon branches is small, specifically, of the order of co and/or of the phonon damping
F.
These regions are usually located in the vicinity of the degeneracy lines of the spectrum, i.e. the lines in the k -space where the frequencies of different branches are equal. In crystals exhibiting a phonon spectrum with low-lying (soft) optical modes, the degeneracy lines formed with a participation of these modes are of primary importance for the loss. Since the degeneracy of the spectrum is mainly controlled by the symmetry of the crystal, the explicit temperature dependence [which does not take into account the temperature dependence of the dielectric permittivity] and frequency dependence of the three-quantum loss are very sens
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