PhD Thesis Arne Lüker final version V4 - Cranfield University

Sol-Gel derived Ferroelectric Thin Films for Voltage Tunable Applications
The three- and four-quantum mechanisms are the only two mechanisms which control
the intrinsic loss in centrosymmetric crystals [3]. In the case of practical interest of the
low-microwave-loss materials, the theory of phonon transport is applicable with quite
good accuracy. This means that, possessing the complete information on the phonon
spectrum and the constants of inter-phonon and ac field-phonon couplings, one can
calculate the intrinsic dielectric loss with an accuracy equal to the relative damping of the
typical phonons participating in the absorption of the ac field [3]. Thus, the result that the
intrinsic loss in centrosymmetric crystals is given by the sum of the three- and four-
quantum contribution can be considered as perfectly justified.
c) Quasi-Debye Loss Mechanism.
The origin of this mechanism is the relaxation of the phonon distribution function of
the crystal [3, 7, 8]. In noncentrosymmetric crystals, the phonon frequencies are linear
functions of a small electric field applied to the crystal. Thus, the oscillations of the ac
field result in time modulation of the phonon frequencies; the latter in turn induces a
deviation of the phonon distribution function from its equilibrium value. A relaxation of
the phonon distribution functions gives rise to dielectric loss in a similar way as a
relaxation of the distribution function of the dipoles gives rise to the loss in the Debye
theory [9]. This analogy is expressed by the name “quasi-Debye”. The frequency
dependence of the quasi-Debye contribution to the loss factor is of the Debye type, the
average relaxation time of the phonon distribution function playing the role of the Debye
relaxation time. In microwave materials for tunable application, which are typically
centrosymmetric, the quasi-Debye mechanism does not contribute to the loss in the
absence of the tuning bias. However, under a dc bias field, 0 Er , it becomes active due to
the breaking of the central symmetry so that one is dealing with the dc field-induced
quasi-Debye mechanism. An important feature of the quasi-Debye loss is that as a
function of frequency, similar to the Debye loss, its contribution to the loss tangent passes
through a maximum at ω ≅ Γ . For the materials used in this study, where the phonon
damping Г is of the order of 100 GHz , this means a certain slowing down of the linear
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