**3.2 Room-temperature dielectric properties of multilayer C***f***/Si3N4 composites**

According to the classical transmission line theory, microwave complex permittivity (*<sup>ε</sup>* <sup>=</sup> *<sup>ε</sup>*′ <sup>−</sup> *<sup>j</sup> <sup>ε</sup>*″ ) is an important parameter to determine the absorbing performance. **Figure 5(a)** shows the real and imaginary permittivity of multilayer C*f*/Si3N4 composites at X-band, as well as as-prepared Si3N4 ceramics. Clearly, the dielectric constant of Si3N4 ceramics presents frequency-independent behavior. The mean real and imaginary parts of permittivity and dielectric loss (tan*<sup>δ</sup>* <sup>=</sup> *<sup>ε</sup>*″ /*ε*′ ) of pure Si3N4 ceramic were 7.7, 0.04, and 5.3 × 10<sup>−</sup><sup>3</sup> , respectively. The relatively low dielectric constant is considered to be helpful for microwave impedance matching with free space, which tends to reduce reflection of electromagnetic wave from the surface of material and enhance energy propagating in the material.

However, both the real permittivity and imaginary permittivity of C*f*/Si3N4 sandwich composites decrease markedly as frequency increases at X-band, varying from 12.3 and 5.1 to 7.9 and 1.2, respectively. This phenomenon is usually called frequency dispersion characteristic, which is acknowledged to be beneficial to broaden the microwave absorption bandwidth. The reflection loss (*R*) of Si3N4 and C*f*/Si3N4 sandwich composites was calculated according to the formula as follows:

$$R\,\mathrm{\dot{d}}\,\mathrm{dB}\,\mathrm{\dot{>}} = 20\,\log\left|\frac{Z\_{\mathrm{in}} - 1}{Z\_{\mathrm{in}} + 1}\right|\,\tag{1}$$

and

*Electromagnetic Materials and Devices*

**3. Microwave dielectric properties**

**Figure 2.**

**3.1 Structure of multilayer C***f***/Si3N4 composites**

The optical image of cross-section of multilayer C*f*/Si3N4 composites is shown in **Figure 3(a)**. As expected, three layers filled with short carbon fibers are uniformly embedded in the Si3N4 matrix. The microstructure of Si3N4 ceramic was formed by rod-like particles, which are evenly distributed and intercross with each other to form the main pores. Energy dispersive spectroscopy (EDS) analysis at spots A and B in **Figure 3(c)** demonstrates that the PyC/SiC interphase could effectively promote the chemical compatibility between carbon fibers and Si3N4 ceramic at high-temperature circumstance, which could be further proved by the XRD investigations (see **Figure 4**). As seen in **Figure 4**, in addition to the main *β*-Si3N4 peaks

*(a) Cross-section of multilayer Cf/Si3N4 composites, facture surface located at (b) Si3N4 matrix and (c) carbon* 

*Schematic diagram of complex permittivity test apparatus (reprinted with permission from Ref. [48]).*

**32**

**Figure 3.**

*fibers (reprinted with permission from Ref. [39]).*

$$Z\_{in} = \sqrt{\frac{\mu\_r}{\varepsilon\_r}} \tanh\left[j\left(\frac{2\pi}{c}\right)\sqrt{\mu\_r\varepsilon\_r}f d\right] \tag{2}$$

where *Zin* refers to input impedance, *j* is the imaginary unit (i.e., equals to √ \_\_\_ −1), *c* is the velocity of electromagnetic waves in free space, *f* is the microwave frequency,

**Figure 4.** *XRD patterns of as-prepared Si3N4 ceramics and Cf/Si3N4 composites (reprinted with permission from Ref. [39]).*

**Figure 5.**

*(a) The permittivity and (b) reflection loss curves of Si3N4 and Cf/Si3N4 sandwich composites over X-band (reprinted with permission from Ref. [35]).*

and *d* is the thickness of the samples. As depicted in **Figure 5(b)**, the microwave absorption ability of the C*f*/Si3N4 sandwich composites was significantly enhanced compared with pure Si3N4 ceramic. The reflection loss of the C*f*/Si3N4 sandwich composites gradually decreases from −3.5 dB to −14.4 dB with the increase of frequency, while that of the pure Si3N4 ceramic remains at −0.1 dB.

The enhanced microwave-absorbing performance could be mainly attributed to polarization relaxation. As we know, there exists migration of free electrons inside the electro-conductive carbon fibers, as well as charge accumulation at interfaces between short carbon fibers and insulated matrix when subjected to external electric field. As a result, the chopped carbon fibers are more inclined to be equivalent to micro-dipoles. With increase of frequency, the orientation of these dipoles could not keep up with change of electric field gradually, resulting in the real part of permittivity (*ε*′ ) of C*f*/Si3N4 sandwich composites decrease gradually. Furthermore, the scattering effect from defects and the crystal lattice on the back-and-forth movement of electrons under alternating electromagnetic waves predominately contributes to the dissipation of EM energy, which results in thermal energy.

For a deep-seated investigation of frequency-dependent dielectric responses of multilayer C*f*/Si3N4 composites (**Figure 6(a)**), here we proposed an equivalent RC circuit model, where each layer of carbon fiber plays a role of one electrode in a plane-parallel capacitor, while each layer of Si3N4 ceramic plays the role of the dielectric (**Figure 6(b)**). Considering the existence of leakage current, leakage resistances were applied in equivalent circuit (**Figure 6(c)**).

According to the circuit theory knowledge, the relationship between permittivity and frequency *ω* follows:

$$\begin{aligned} \text{O0100w.s.}\\ \text{e.e.} &= \frac{Q\left(C\_1 + C\_2\right)^2}{R\_1^2 \, C\_1 C\_2 \left(C\_1 + C\_2\right)} \cdot \frac{\mathbf{1}}{\left.\right.} + \frac{Q R\_1^2 \,^2 C\_1 \,^2 C\_2}{R\_1^2 \,^2 C\_1 C\_2 \left(C\_1 + C\_2\right)} \\ \text{e.o.} &\cdot \text{e.e.}^\circ = P \frac{\left(C\_1 + C\_2\right)^2 + R\_1^2 \,^2 C\_1 \,^2 C\_2 \cdot \left.\omega^2\right.}{\left(R\_1 + R\_2\right) \left(C\_1 + C\_2\right)^2 + R\_1^2 \,^2 R\_2 C\_1 \,^2 C\_2^2 \cdot \left.\omega^2\right.} \end{aligned} \tag{3}$$

**35**

(ε′

**Figure 7.**

**Figure 6.**

remains lower than 0.06.

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials*

permittivity declines faster than the real part with frequency increase, the reflection loss presents enhanced trend with frequency increase. This phenomenon is mainly attributed to the fact that microwave-absorbing efficiency is the combination of reflection from the material surface and attenuation inside the material. The lower the permittivity, the better its impedance matching between air and absorber. As a result, in order to achieve optimal reflection loss, one must lower reflection as much

*Experimental data and curves of (a)* ε ′ *and (b)* ω *⋅* ε ″ *versus* ω *2 (reprinted with permission from Ref. [35]).*

*(a) Cross-section morphology, (b) structural schematic diagram, and (c) equivalent circuit diagram of* 

*Cf/Si3N4 sandwich composites (reprinted with permission from Ref. [35]).*

Due to the fact that there will inevitably be some variation of electromagnetic

) shows no obvious change even though temperature rises up to 800°C. Likewise, thanks to the excellent electrical insulation of Si3N4 ceramics, the loss tangent (as shown in **Figure 8(b)**) is almost independent of frequency and temperature and

Herein, temperature coefficient κ is used to explicate the impact of temperature

temperature condition, the dielectric property would be supposed to dynamically change with temperature. How and to what extent does the permittivity dynamically change with temperature (increase or decrease)? All these are quite critical in parameter modification strategy for improving the accuracy of radar detection and guidance. Consequently, it is of utmost importance to explore the evolution of dielectric properties of Si3N4 ceramics used in high-temperature circumstances. Three-dimensional (3D) plots of the effect of temperature on permittivity of Si3N4

performance or even the mechanical property of materials served in high-

ceramics over X-band are shown in **Figure 8**. Clearly, the real permittivity

as possible and keep a modest loss tangent simultaneously.

on dielectric response of as-prepared Si3N4 ceramics:

**3.3 High-temperature dielectric behaviors of Si3N4 ceramics**

*DOI: http://dx.doi.org/10.5772/intechopen.82389*

where *P*, *Q,* and *C* are constants and are determined by the surface density of carbon fiber layers and thickness of Si3N4 layers. We have analyzed our experimental permittivity based on Eq. (3) using Trust-Region algorithm, which is illustrated in **Figure 7**. The points in **Figure 7** indicate the experimental data, while the results predicted by equivalent circuit model are given as solid line. Clearly, for multilayer C*f*/Si3N4 composites, both *<sup>ε</sup>*′ and (*<sup>ω</sup>* <sup>⋅</sup> *<sup>ε</sup>*″ ) are inversely proportional to the frequency square *ω*<sup>2</sup> , and the predicted results agree quite well with the measured data. Additionally, the experimental data show oscillation phenomena at high frequency, which may result from charge and discharge processes between C1 and C2 (**Figure 6**) with the increase of frequency. Note that, even though the imaginary part of

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials DOI: http://dx.doi.org/10.5772/intechopen.82389*

#### **Figure 6.**

*Electromagnetic Materials and Devices*

*(reprinted with permission from Ref. [35]).*

and *d* is the thickness of the samples. As depicted in **Figure 5(b)**, the microwave absorption ability of the C*f*/Si3N4 sandwich composites was significantly enhanced compared with pure Si3N4 ceramic. The reflection loss of the C*f*/Si3N4 sandwich composites gradually decreases from −3.5 dB to −14.4 dB with the increase of

*(a) The permittivity and (b) reflection loss curves of Si3N4 and Cf/Si3N4 sandwich composites over X-band* 

the scattering effect from defects and the crystal lattice on the back-and-forth movement of electrons under alternating electromagnetic waves predominately contributes to the dissipation of EM energy, which results in thermal energy. For a deep-seated investigation of frequency-dependent dielectric responses of multilayer C*f*/Si3N4 composites (**Figure 6(a)**), here we proposed an equivalent RC circuit model, where each layer of carbon fiber plays a role of one electrode in a plane-parallel capacitor, while each layer of Si3N4 ceramic plays the role of the dielectric (**Figure 6(b)**). Considering the existence of leakage current, leakage

According to the circuit theory knowledge, the relationship between permittiv-

⋅ \_\_1

where *P*, *Q,* and *C* are constants and are determined by the surface density of carbon fiber layers and thickness of Si3N4 layers. We have analyzed our experimental permittivity based on Eq. (3) using Trust-Region algorithm, which is illustrated in **Figure 7**. The points in **Figure 7** indicate the experimental data, while the results predicted by equivalent circuit model are given as solid line. Clearly, for multilayer

, and the predicted results agree quite well with the measured data. Additionally, the experimental data show oscillation phenomena at high frequency, which may result from charge and discharge processes between C1 and C2 (**Figure 6**) with the increase of frequency. Note that, even though the imaginary part of

*<sup>ω</sup>*<sup>2</sup> <sup>+</sup> *QR*<sup>1</sup>

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (*<sup>R</sup>*<sup>1</sup> <sup>+</sup> *<sup>R</sup>*2) (*C*<sup>1</sup> <sup>+</sup> *<sup>C</sup>*2)<sup>2</sup> <sup>+</sup> *<sup>R</sup>*<sup>1</sup>

<sup>2</sup>*C*<sup>1</sup> <sup>2</sup>*C*<sup>2</sup> <sup>2</sup> ⋅ *ω*<sup>2</sup>

<sup>2</sup>*C*<sup>1</sup> <sup>2</sup>*C*<sup>2</sup> 2 \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>R</sup>*<sup>1</sup> <sup>2</sup>*C*1*C*2(*C*<sup>1</sup> + *C*2)

> <sup>2</sup>*R*2*C*<sup>1</sup> <sup>2</sup>*C*<sup>2</sup> <sup>2</sup> ⋅ *ω<sup>2</sup>*

) are inversely proportional to the frequency

(3)

The enhanced microwave-absorbing performance could be mainly attributed to polarization relaxation. As we know, there exists migration of free electrons inside the electro-conductive carbon fibers, as well as charge accumulation at interfaces between short carbon fibers and insulated matrix when subjected to external electric field. As a result, the chopped carbon fibers are more inclined to be equivalent to micro-dipoles. With increase of frequency, the orientation of these dipoles could not keep up with change of electric field gradually, resulting in the real part of

) of C*f*/Si3N4 sandwich composites decrease gradually. Furthermore,

frequency, while that of the pure Si3N4 ceramic remains at −0.1 dB.

resistances were applied in equivalent circuit (**Figure 6(c)**).

*<sup>ε</sup>*′ <sup>=</sup> *<sup>Q</sup>*(*C*<sup>1</sup> <sup>+</sup> *<sup>C</sup>*2)<sup>2</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>R</sup>*<sup>1</sup> <sup>2</sup>*C*1*C*2(*C*<sup>1</sup> + *C*2)

*<sup>ω</sup>* <sup>⋅</sup> ε″ <sup>=</sup> *<sup>P</sup>* (*C*<sup>1</sup> <sup>+</sup> *<sup>C</sup>*2)<sup>2</sup> <sup>+</sup> *<sup>R</sup>*<sup>1</sup>

and (*<sup>ω</sup>* <sup>⋅</sup> *<sup>ε</sup>*″

**34**

square *ω*<sup>2</sup>

permittivity (*ε*′

**Figure 5.**

ity and frequency *ω* follows:

C*f*/Si3N4 composites, both *<sup>ε</sup>*′

*(a) Cross-section morphology, (b) structural schematic diagram, and (c) equivalent circuit diagram of Cf/Si3N4 sandwich composites (reprinted with permission from Ref. [35]).*

**Figure 7.** *Experimental data and curves of (a)* ε ′ *and (b)* ω *⋅* ε ″ *versus* ω *2 (reprinted with permission from Ref. [35]).*

permittivity declines faster than the real part with frequency increase, the reflection loss presents enhanced trend with frequency increase. This phenomenon is mainly attributed to the fact that microwave-absorbing efficiency is the combination of reflection from the material surface and attenuation inside the material. The lower the permittivity, the better its impedance matching between air and absorber. As a result, in order to achieve optimal reflection loss, one must lower reflection as much as possible and keep a modest loss tangent simultaneously.

## **3.3 High-temperature dielectric behaviors of Si3N4 ceramics**

Due to the fact that there will inevitably be some variation of electromagnetic performance or even the mechanical property of materials served in hightemperature condition, the dielectric property would be supposed to dynamically change with temperature. How and to what extent does the permittivity dynamically change with temperature (increase or decrease)? All these are quite critical in parameter modification strategy for improving the accuracy of radar detection and guidance. Consequently, it is of utmost importance to explore the evolution of dielectric properties of Si3N4 ceramics used in high-temperature circumstances. Three-dimensional (3D) plots of the effect of temperature on permittivity of Si3N4 ceramics over X-band are shown in **Figure 8**. Clearly, the real permittivity (ε′ ) shows no obvious change even though temperature rises up to 800°C. Likewise, thanks to the excellent electrical insulation of Si3N4 ceramics, the loss tangent (as shown in **Figure 8(b)**) is almost independent of frequency and temperature and remains lower than 0.06.

Herein, temperature coefficient κ is used to explicate the impact of temperature on dielectric response of as-prepared Si3N4 ceramics:

**Figure 8.**

*Three-dimensional plots of complex permittivity of Si3N4 ceramics versus frequency and temperature (reprinted with permission from Ref. [39]).*

$$
\kappa = \frac{1}{q\rho} \cdot \frac{\Delta \varphi}{\Delta T},
\tag{4}
$$

where *T* is the temperature and *φ* refers to either the dielectric constant or loss tangent. As summarized in **Table 1**, the temperature coefficients of both dielectric constant and loss tangent remain around 10<sup>−</sup><sup>4</sup> °C<sup>−</sup><sup>1</sup> . From this perspective, the as-prepared Si3N4 ceramics exhibit excellent thermo-stability of dielectric response within the range of evaluated temperatures. This weak temperature dependence further corroborates as-prepared Si3N4 ceramics to be a competitive candidate as the matrix of high-temperature microwave-absorbing materials.

It should be noted that the real permittivity increases slightly with frequency increase, which is contrary to the ordinary frequency dispersion effect described by the Debye model [49–55]. In order to further expound this peculiar frequency dispersion characteristic, it is essential to explore the details of electronic polarizing processing of Si3N4 ceramics. Considering the covalent bonding, the electronic polarization in Si3N4 ceramics mainly results from the bound charge's displacement deviated from the equilibrium position. The motion equation of bound charge driven by an external electric field *E*0*e j*ω*t* can be expressed as:


$$m\frac{\partial^2 \mathbf{x}}{\partial t^2} = qE\_0 e^{j\alpha t} - f\mathbf{\hat{x}} - \mathbf{\mathcal{D}}\eta \frac{\partial \mathbf{x}}{\partial t} \tag{5}$$

**37**

**Figure 9.**

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials*

where *m*, *q,* and *x* are the mass, charge, and displacement of single bound charge, respectively; *f* is the coefficient of restoring force; and *η* is the damping coefficient. Taking Lorentz correction [56–58] into consideration, the real permittivity of Si3N4 ceramic containing *N* polarized bound charges could be obtained as:

> (*ω*<sup>0</sup> <sup>2</sup> − *ω*<sup>2</sup> ) 2 + 4*η*<sup>2</sup>*ω*<sup>2</sup>

*N* ∝ exp(−*Ea* ⁄*RT*) (7)

*η* ∝ exp(−*Eb*⁄*RT*) (8)

where *Ea* and *Eb* are the activation energy of electrons and lattice, respectively. The dependence of real permittivity of Si3N4 ceramics on temperature at three representative frequencies and the best fitting diagram according to Eqs. (6)–(8)

It can be clearly seen that the dielectric constant gradually increases as temperature increased, starting from room temperature to 800°C, and results show that the real permittivity is well distributed on the predicted curves with coefficient of deter-

the temperature dependence of permittivity at three representative frequencies by the Trust-Region algorithm are also listed in **Figure 8**. The activation energy of electrons *Ea* is distributed between 15.46 and 17.49 KJ/mol, while the activation energy of

*Dependences of real permittivity of Si3N4 ceramics on temperature (reprinted with permission from Ref. [48]).*

) ranging from 0.91 to 0.93. The characteristic parameters fitted from

where *ω*0 is the resonant frequency of Si3N4 ceramics, and *ε*0 and *εs* are the vacuum permittivity and static dielectric constant of Si3N4 ceramics, respectively. Theoretical results have shown that the order of magnitude of resonant frequency *ω*0 is around 10 eV (~1015 Hz) [59, 60] which is considerably larger than the tested frequency (~1010Hz). Combining with Eq. (4), the real permittivity increases slightly with frequency increase, which is closely coincident with the experimental results. Furthermore, taking the effect of temperature into consideration, *N* and *η*

<sup>2</sup> − *ω*<sup>2</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_

(6)

\_\_\_\_ *<sup>ε</sup>*0*<sup>m</sup>* <sup>⋅</sup> *<sup>ω</sup>*<sup>0</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.82389*

*<sup>ε</sup>* <sup>=</sup> *<sup>ε</sup><sup>s</sup>* <sup>+</sup> *Nq*<sup>2</sup>

are shown by solid lines in **Figure 9**.

should follow:

mination (*R2*

#### **Table 1.**

*Temperature coefficient of permittivity and loss tangent at selected frequency (reprinted with permission from Ref. [39]).*

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials DOI: http://dx.doi.org/10.5772/intechopen.82389*

where *m*, *q,* and *x* are the mass, charge, and displacement of single bound charge, respectively; *f* is the coefficient of restoring force; and *η* is the damping coefficient. Taking Lorentz correction [56–58] into consideration, the real permittivity of Si3N4 ceramic containing *N* polarized bound charges could be obtained as:

by \$\mathbf{S}\$: \$\mathbf{S}\$\_3\$N\$\_4\$ етапис сопалин \$\mathbf{N}\$ ройлгээс вочиа спагре сочи ве отаплеа за 
$$\boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}\_s + \frac{N\boldsymbol{q}^2}{\boldsymbol{\varepsilon}\_0 \boldsymbol{m}} \cdot \frac{\boldsymbol{\alpha}\boldsymbol{o}\_0^2 - \boldsymbol{o}^2}{\left(\boldsymbol{\alpha}\boldsymbol{o}\_0^2 - \boldsymbol{o}^2\right)^2 + 4\boldsymbol{\eta}^2 \boldsymbol{o}^2} \tag{6}$$

where *ω*0 is the resonant frequency of Si3N4 ceramics, and *ε*0 and *εs* are the vacuum permittivity and static dielectric constant of Si3N4 ceramics, respectively. Theoretical results have shown that the order of magnitude of resonant frequency *ω*0 is around 10 eV (~1015 Hz) [59, 60] which is considerably larger than the tested frequency (~1010Hz). Combining with Eq. (4), the real permittivity increases slightly with frequency increase, which is closely coincident with the experimental results. Furthermore, taking the effect of temperature into consideration, *N* and *η* should follow:

$$N \propto \exp\left(\cdot^{E\_4}/\text{RT}\right) \tag{7}$$

$$
\eta \propto \exp\left\{ \cdot \text{E}\_{\text{V}} \langle \text{r} \rangle \right\} \tag{8}
$$

where *Ea* and *Eb* are the activation energy of electrons and lattice, respectively. The dependence of real permittivity of Si3N4 ceramics on temperature at three representative frequencies and the best fitting diagram according to Eqs. (6)–(8) are shown by solid lines in **Figure 9**.

It can be clearly seen that the dielectric constant gradually increases as temperature increased, starting from room temperature to 800°C, and results show that the real permittivity is well distributed on the predicted curves with coefficient of determination (*R2* ) ranging from 0.91 to 0.93. The characteristic parameters fitted from the temperature dependence of permittivity at three representative frequencies by the Trust-Region algorithm are also listed in **Figure 8**. The activation energy of electrons *Ea* is distributed between 15.46 and 17.49 KJ/mol, while the activation energy of

**Figure 9.** *Dependences of real permittivity of Si3N4 ceramics on temperature (reprinted with permission from Ref. [48]).*

*Electromagnetic Materials and Devices*

*(reprinted with permission from Ref. [39]).*

**Figure 8.**

*κ* = \_\_1 *<sup>φ</sup>* <sup>⋅</sup> \_\_\_ 

*Three-dimensional plots of complex permittivity of Si3N4 ceramics versus frequency and temperature* 

constant and loss tangent remain around 10<sup>−</sup><sup>4</sup>

driven by an external electric field *E*0*e*

*<sup>m</sup>* <sup>∂</sup><sup>2</sup> \_\_\_*<sup>x</sup>*

**Temperature (°C)** *κε*′ **(×10<sup>−</sup><sup>4</sup>**

matrix of high-temperature microwave-absorbing materials.

∂ *t*

where *T* is the temperature and *φ* refers to either the dielectric constant or loss tangent. As summarized in **Table 1**, the temperature coefficients of both dielectric

as-prepared Si3N4 ceramics exhibit excellent thermo-stability of dielectric response within the range of evaluated temperatures. This weak temperature dependence further corroborates as-prepared Si3N4 ceramics to be a competitive candidate as the

It should be noted that the real permittivity increases slightly with frequency increase, which is contrary to the ordinary frequency dispersion effect described by the Debye model [49–55]. In order to further expound this peculiar frequency dispersion characteristic, it is essential to explore the details of electronic polarizing processing of Si3N4 ceramics. Considering the covalent bonding, the electronic polarization in Si3N4 ceramics mainly results from the bound charge's displacement deviated from the equilibrium position. The motion equation of bound charge

*j*ω*t*

<sup>2</sup> <sup>=</sup> *qE*<sup>0</sup> *<sup>e</sup><sup>j</sup><sup>t</sup>*

**°C<sup>−</sup><sup>1</sup>**

 0.46 0.45 2.08 2.84 0.22 0.93 1.79 3.11 0.45 0.031 2.83 4.83 1.04 1.02 3.02 3.91 0.16 1.31 2.87 4.89 0.40 1.11 4.98 6.92 0.97 1.70 4.68 5.97 1.16 1.94 3.49 4.34

*Temperature coefficient of permittivity and loss tangent at selected frequency (reprinted with permission from* 

°C<sup>−</sup><sup>1</sup>

can be expressed as:

− *fx* − 2*η* \_\_\_ <sup>∂</sup>*<sup>x</sup>*

**)** *κ***tan***δ* **(×10<sup>−</sup><sup>4</sup>**

**8.2 GHz 12.4 GHz 8.2 GHz 12.4 GHz**

*T*, (4)

. From this perspective, the

<sup>∂</sup>*<sup>t</sup>* (5)

**°C<sup>−</sup><sup>1</sup> )**

**36**

**Table 1.**

*Ref. [39]).*

lattice *Eb* is distributed between 33.29 and 40.40 KJ/mol. The activation energy of electronic *Ea* is less than that of lattice *Eb*, which is mainly attributed to the binding force between the electrons and nucleus being lower than the covalent bonding force of lattice. Another important feature to be noticed is that the real permittivity of Si3N4 ceramics shows symmetrical features between the heating-up and cooling-down periods. The excellent thermo-stability of dielectric properties of Si3N4 ceramics has established the foundation for high-temperature radar absorbing materials.
