**7. Proposed tunable ZOR based on combined ferroelectric and HTS**

Ferroelectric thin films can easily be incorporated into the CRLH MTM microstrip structure shown in **Figure 9**. For frequency-agile microwave communications systems, ferroelectric tunable microstrip structures are potentially attractive. These tunable components allow an innovative class of components with large frequency-agile tunabilities. Additionally, attenuation losses can be minimized when combined with HTS conductor for low-temperature applications. SrTiO3 (STO) thin films with large dielectric tunability and low loss tangents at microwave *Tunable Zeroth-Order Resonator Based on Ferroelectric Materials DOI: http://dx.doi.org/10.5772/intechopen.98475*

#### **Figure 8.**

*Simulated phase responses and the phase shift of the one-unit CRLH-TL with two cases [28].*

frequencies have been the most promising ferroelectrics for integration with HTS circuits [32]. The popular ferroelectric tunable structures are based on conductor/ ferroelectric/dielectric two-layered microstrip.

The modified microstrip structure of **Figure 9** consists of a dielectric substrate (e.g., LAO or MgO, typically 254 to 500 um thick, LAO with permittivity 23.6 is selected to represent the base substrate [32], a ferroelectric thin-film layer

**Figure 9.** *Layout of ZOR with ferroelectric material (a) top view (b) side view.*

**7. Proposed tunable ZOR based on combined ferroelectric and HTS**

*Simulated responses of a one unit CRLH-TL with different surface resistances of the bias line [28].*

**Figure 6.**

**Figure 7.**

**114**

*Picture of the one-unit CRLH-TL [28].*

*Multifunctional Ferroelectric Materials*

Ferroelectric thin films can easily be incorporated into the CRLH MTM microstrip structure shown in **Figure 9**. For frequency-agile microwave communications systems, ferroelectric tunable microstrip structures are potentially attractive. These tunable components allow an innovative class of components with large frequency-agile tunabilities. Additionally, attenuation losses can be minimized when combined with HTS conductor for low-temperature applications. SrTiO3 (STO) thin films with large dielectric tunability and low loss tangents at microwave (thickness."*tferro*" varying between 300 and 2000 *nm* for various applications), a gold or YBCO thin film (2*μm*thick or 300–600 *nm* thick, respectively) for the top conductor, and a 2*μm* thick gold ground plane. The STO film is a lossy dielectric that has a complex permittivity with a dielectric constant *ε<sup>r</sup>* and a loss tangent *tan δ*. Both of these parameters are functions of the DC applied electric field (E) and the temperature (T), and are introduced in the analysis by a phenomenological model developed by Vendik et al. [32] for a single crystal, and are given by [33]:

$$
\varepsilon\_r(T, E) = \frac{\varepsilon\_{00}}{\Phi(T, E)} \tag{3}
$$

$$
\tan \delta = \tan \delta\_1 + \tan \delta\_2 + \tan \delta\_3 \tag{4}
$$

where

$$
\tan \delta\_1 = A\_1 (T/T\_0)^2 / \Phi(T, E)^{\frac{3}{2}} \tag{5}
$$

$$
\tan \delta\_2 = A\_2 \Psi(T, E)^2 / \Phi(T, E) \tag{6}
$$

$$\tan \delta\_3 = \frac{A\_3 n\_d}{\Phi(T, E)} \tag{7}$$

*and*

$$\Phi(T, E) = \left[ \left( \xi^2 + \eta^3 \right)^{\frac{1}{2}} + \xi \right]^{\frac{2}{5}} + \left[ \left( \xi^2 + \eta^3 \right)^{\frac{1}{2}} - \xi \right]^{\frac{2}{5}} - \eta \tag{8}$$

$$\Psi(T,E) = \left[\left(\xi^2 + \eta^3\right)^{\frac{1}{2}} + \xi\right]^{\frac{1}{\dagger}} - \left[\left(\xi^2 + \eta^3\right)^{\frac{1}{2}} - \xi\right]^{\frac{1}{\dagger}}\tag{9}$$

$$
\xi(E) = \sqrt{\xi\_s^2 + \left(E/E\_N\right)^2} \tag{10}
$$

**Figure 10.**

**Table 1.**

**117**

*tferro. (a) S21. (b) S11.*

*Parameters values of Figure 10.*

*Variation of the magnitude of the transmission coefficient with the frequency at a different ferroelectric thickness*

0.3 3.71 �0.95 3.43 �0.29 0.5 3.66 �0.97 3.5 �0.34

*f res*ð Þ *GHz* **S21 dB** *f res*ð Þ *GHz* **S21 dB**

*tferro*ð Þ μ*m* **Normal 300 K Super 77 K**

*Tunable Zeroth-Order Resonator Based on Ferroelectric Materials*

*DOI: http://dx.doi.org/10.5772/intechopen.98475*

$$\eta(T) = \left(\frac{\Theta}{T\_0}\right)\sqrt{\left(\mathbf{1}/\mathbf{16}\right) + \left(T/\Theta\right)^2} - \mathbf{1} \tag{11}$$

In the previous equations, *ε*<sup>00</sup> is a constant analogous to the Curie constant, *EN* is the normalizing applied electric field, *ξ<sup>s</sup>* is the rate of crystal strain, a measure of the density of defects, *Θ* is the effective Debye temperature, and *T*<sup>0</sup> is the effective Curie temperature. Numerical values for these model parameters for a single crystal of STO are given by [34]: *ε*00= 2080, *EN* = 19.3 KV/cm, *ξs*= 0.018, *Θ*= 17 K, and *T*0= 42 K. The change of dielectric constant with frequency is generally small in the microwave frequency range. In Eqs. (5), (6) and (7) *Α*1, *Α*2, *and Α*<sup>3</sup> are material parameters, and *nd* is the density of charged defects. For high-quality crystals, tan*δ*<sup>3</sup> is small with respect to tan *δ*<sup>1</sup> and tan *δ*<sup>2</sup> and can be neglected. Numerical values for *Α*<sup>1</sup> *and Α*<sup>2</sup> parameters for STO at a frequency of 10 GHz are given by [35]: *Α*<sup>1</sup> ¼ <sup>2</sup>*:*<sup>4</sup> <sup>∗</sup> <sup>10</sup>�<sup>4</sup> and *<sup>Α</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup> <sup>∗</sup> <sup>10</sup>�3.

**Figure 10** depicts the magnitude of the transmission scattering parameter (S21) for 0*Ω*YBCO/STO/LAO microstrip line CRLH ZOR shown in **Figure 9**, with a 0.3 *μm* STO thin film. It provides the STO permittivity dependence on its thickness. **Table 1** summarizes the plotting curves given in **Figure 10(a**, **b)**. As we see, for a given frequency, the attenuation increases with film thickness. At higher frequencies, because of the skin depth effect, more RF field is concentrated in the ferroelectric film and less is concentrated in the dielectric substrate, resulting in larger insertion loss. As the value of the *ε<sup>r</sup>* increases, the attenuation also increases. This is a consequence of mismatches resulting from the decrease in *Zo*.

**Figure 10.**

(thickness."*tferro*" varying between 300 and 2000 *nm* for various applications), a gold or YBCO thin film (2*μm*thick or 300–600 *nm* thick, respectively) for the top conductor, and a 2*μm* thick gold ground plane. The STO film is a lossy dielectric that has a complex permittivity with a dielectric constant *ε<sup>r</sup>* and a loss tangent *tan δ*. Both of these parameters are functions of the DC applied electric field (E) and the temperature (T), and are introduced in the analysis by a phenomenological model

*<sup>ε</sup>r*ð Þ¼ *<sup>T</sup>*, *<sup>E</sup> <sup>ε</sup>*<sup>00</sup>

2

*<sup>Φ</sup>*ð Þ *<sup>T</sup>*, *<sup>E</sup>* (3)

<sup>2</sup> (5)

� *η* (8)

(9)

(10)

*=Φ*ð Þ *T*, *E* (6)

*<sup>Φ</sup>*ð Þ *<sup>T</sup>*, *<sup>E</sup>* (7)

3

3

� 1 (11)

<sup>2</sup> � *ξ* h i<sup>2</sup>

> <sup>2</sup> � *ξ* h i<sup>1</sup>

*tan δ* ¼ *tan δ*<sup>1</sup> þ *tan δ*<sup>2</sup> þ *tan δ*<sup>3</sup> (4)

*<sup>=</sup>Φ*ð Þ *<sup>T</sup>*, *<sup>E</sup>* <sup>3</sup>

<sup>þ</sup> *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>3</sup> � �<sup>1</sup>

� *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>3</sup> � �<sup>1</sup>

2

developed by Vendik et al. [32] for a single crystal, and are given by [33]:

*tan δ*<sup>1</sup> ¼ *Α*1ð Þ *T=T*<sup>0</sup>

*tan <sup>δ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>Α</sup>*2*Ψ*ð Þ *<sup>T</sup>*, *<sup>E</sup>* <sup>2</sup>

*<sup>Φ</sup>*ð Þ¼ *<sup>T</sup>*, *<sup>E</sup> <sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>3</sup> � �<sup>1</sup>

*<sup>Ψ</sup>*ð Þ¼ *<sup>T</sup>*, *<sup>E</sup> <sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>3</sup> � �<sup>1</sup>

*<sup>η</sup>*ð Þ¼ *<sup>T</sup> <sup>Θ</sup>*

*ξ*ð Þ¼ *E*

*T*<sup>0</sup>

*tan <sup>δ</sup>*<sup>3</sup> <sup>¼</sup> *<sup>Α</sup>*3*nd*

<sup>2</sup> þ *ξ* h i<sup>2</sup>

> <sup>2</sup> þ *ξ* h i<sup>1</sup>

> > *ξ*2

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þþ <sup>1</sup>*=*<sup>16</sup> ð Þ *<sup>T</sup>=<sup>Θ</sup>* <sup>2</sup>

In the previous equations, *ε*<sup>00</sup> is a constant analogous to the Curie constant, *EN* is the normalizing applied electric field, *ξ<sup>s</sup>* is the rate of crystal strain, a measure of the density of defects, *Θ* is the effective Debye temperature, and *T*<sup>0</sup> is the effective Curie temperature. Numerical values for these model parameters for a single crystal of STO are given by [34]: *ε*00= 2080, *EN* = 19.3 KV/cm, *ξs*= 0.018, *Θ*= 17 K, and *T*0= 42 K. The change of dielectric constant with frequency is generally small in the microwave frequency range. In Eqs. (5), (6) and (7) *Α*1, *Α*2, *and Α*<sup>3</sup> are material parameters, and *nd* is the density of charged defects. For high-quality crystals, tan*δ*<sup>3</sup> is small with respect to tan *δ*<sup>1</sup> and tan *δ*<sup>2</sup> and can be neglected. Numerical values for *Α*<sup>1</sup> *and Α*<sup>2</sup> parameters for STO at a frequency of 10 GHz are given by [35]: *Α*<sup>1</sup> ¼

**Figure 10** depicts the magnitude of the transmission scattering parameter (S21)

for 0*Ω*YBCO/STO/LAO microstrip line CRLH ZOR shown in **Figure 9**, with a 0.3 *μm* STO thin film. It provides the STO permittivity dependence on its thickness. **Table 1** summarizes the plotting curves given in **Figure 10(a**, **b)**. As we see, for a given frequency, the attenuation increases with film thickness. At higher frequencies, because of the skin depth effect, more RF field is concentrated in the ferroelectric film and less is concentrated in the dielectric substrate, resulting in larger insertion loss. As the value of the *ε<sup>r</sup>* increases, the attenuation also increases. This is

a consequence of mismatches resulting from the decrease in *Zo*.

q

q

3

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>s</sup>* þ ð Þ *E=EN*

where

*Multifunctional Ferroelectric Materials*

*and*

<sup>2</sup>*:*<sup>4</sup> <sup>∗</sup> <sup>10</sup>�<sup>4</sup> and *<sup>Α</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup> <sup>∗</sup> <sup>10</sup>�3.

**116**

*Variation of the magnitude of the transmission coefficient with the frequency at a different ferroelectric thickness tferro. (a) S21. (b) S11.*


**Table 1.** *Parameters values of Figure 10.*
