PhD Thesis Arne Lüker final version V4 - Cranfield University

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40 Theoretical Considerations and Literature Review frequency dependence of the Quasi-Debye loss in the higher part of the microwave- frequency range. 2.2.2 Extrinsic Losses The role of the intrinsic mechanisms in the total balance of the dielectric loss of a material is strongly dependent on the dielectric permittivity of the material and the measuring frequency: typically, the higher the frequency and permittivity, the more important the intrinsic loss. In the case of tunable ferroelectric materials at microwave frequencies, the intrinsic and extrinsic contributions are comparable so that the dominating contribution to the loss may be extrinsic or intrinsic depending on the quality of the material. A kind of extrinsic/intrinsic crossover in loss may also take place under the action of a dc bias field, i.e. without the field, the extrinsic contribution dominates the loss, whereas under the field the intrinsic one does. Among the known extrinsic loss mechanisms those listed below are considered as significantly contributing to the loss in tunable microwave materials: (1) loss owing to charged defects, (2) universal relaxation law mechanism, (3) quasi-Debye contribution induced by random-field defects. a) Loss Owing to Charged Defects. Motion of charged defects caused by an ac electric field results in a generation of acoustic waves at the frequency of the applied field. This brings about an additional loss mechanism that was proposed by Schlöman [10], formulated for high-dielectric- constant materials by Vendik and Platonova [11], and developed by Garin [12]. The contribution of this mechanism to the loss tangent can be approximated as follows: ( ) ⎥ ⎥ 2 n ⎡ ⎤ d Z ω 1 tanδ ch = Fε ⎢1 − [Eq. 2.21] 3 2 2 2 ρυ t 4π ⎢⎣ 1+ ω / ωc ⎦ where Z and nd are the effective charge of the defects and their atomic concentration; ρ and vt are the density and average transversal sound velocity of the material; F is a material-dependent numerical constant of the order of unity; ωc = vt/rc where rc is the correlation length of the charge distribution in the material. The physical meaning of rc is
Sol-Gel derived Ferroelectric Thin Films for Voltage Tunable Applications the minimal distance at which the electroneutrality is maintained. For Schottky defects rc is of the order of the typical distance between the positively and negatively charged defects. This mechanism may play an essential role in thin film based tunable capacitors, where an elevated defect concentration compared to the bulk material is expected. In this case, the effect of semiconductor depletion of the carriers from the deep traps (caused by ferroelectric/electrode contact) [13, 14] will further increase the contribution of this mechanism via a strong reduction of ωc = vt/rc. This occurs in the depleted areas due to a strong increase of rc up to the depletion length. An essential feature of this mechanism is that its contribution to the loss tangent is proportional to the permittivity of the material. This implies that this contribution is inversely dependent on the applied dc field. b) Universal-Relaxation-Law Mechanism. For all of the loss mechanisms discussed above, a linear frequency dependence of the loss tangent is typical at least for microwave frequencies and below. In reality, this dependence is usually observed at microwave and higher frequencies. For lower frequencies, a much weaker frequency dependence is usually observed, which is consistent with the so-called universal relaxation law which in turn corresponds to the following expression for the complex dielectric permittivity ε * [15]: * n−1 n−1 ε = G( iω) = G(cos( nπ / 2) − i sin( nπ / 2)) ω [Eq. 2.22] where G is a frequency-independent constant and 0 < n < 1. In perovskite thin films in both frequency and time domains, dielectric relaxation corresponding to this equation (for n close to but smaller than unity) has been reported up to the microwave frequency range, e.g. in (Ba,Sr)TiO3 [16]. The physical origin of this behavior is attributed to a variation in charge transport barriers, e.g. at the grain boundaries [17], or to creep of the boundary of the near-by-electrode depletion layer [18]. No information is available on the dependence of G on the dielectric constant of the material. 41
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Intensity [arb. Units] (g) (f) (e)
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Relative diel. diel. constant Const
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Tunability [%] 100 90 80 70 60 50 4
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PhD Thesis Arne Lüker final version V4 - Cranfield University

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