**5.1.3 The third stage (second pulse injection)**

At the inception of the second injection, if the recorded currents are in the continuity of those of the first injection, all the charge Qst have remained trapped in the irradiated volume during the pause time Δt. If it is not the case, a fraction of the charge Qst has been removed from the irradiated zone as a result of discharging (i.e., detrapping and transport) during the pause time.

For the purpose of illustration, the curves of Fig. 7 display the recorded currents during the second injection, performed over the same area as the first injection and with identical experimental conditions, after a pause time Δt = 900 s. Here, the current curves are not in the continuity of those obtained at the end of the first injection (Fig. 5). As the second injection proceeds, the reference steady state is reached again. The ensuing quantity of net charge introduced during this stage is interpreted as the amount Qd that has been removed, given by:

$$\mathbf{Q}\_{\rm d} = \left[ -\int\_{0}^{t\_{\rm inj}} \mathbf{I}\_{\rm ind}(\mathbf{t}) \, \mathbf{d} \mathbf{t} \right]\_{\rm sec \, \rm cond \, injection} \tag{12}$$

In other words, this interpretation means that any loss of charges, during the pause, can be compensated by those introduced during the second irradiation to restore Qst.

Effects of the Microstructure Induced by Sintering on the Dielectric Properties of Alumina 567

We first try to find a kinetic description of the charges that remain in the irradiated volume

As an example, in Fig. 9, we report the best fit of the evolution of Qd with tp, for data obtained, at 473 K in polycrystalline alumina sample (d = 4.5 µm), by performing the measurements over different zones that are sufficiently distant from one another to avoid

Q (t ,T) Q (T) 1 exp (T) <sup>∞</sup>

temperature of 473 K is too low to provoke detrapping from the deeper ones.

In this equation, Q∞ is the asymptotic value reached by Qd. In fact, it stands for the maximum amount of charges going out of the irradiated volume in the discharging process at temperature T. In the example of Fig. 9 where T = 473 K, Q∞ is found equal 0.46 pC corresponding to about 80 % of Qst (hence, 20 % of the net charge is still inside the irradiated volume). This is an indication of the existence of different trapping sites and that the

Fig. 9. Evolution with the pause time at T = 473 K of the amount of charges that is removed from the irradiated volume Qd (in polycrystalline sample of 4.5 µm grain size). The solid line is the exponential fit of the data (Eq. 14). The asymptotic value Q∞(T) is attained after a

 **pause time tp (s)** 

= −− <sup>τ</sup>

Q (t ) Q Q (t ) l p st d p = − (13)

p

(14)

r t

**5.2 Definition of a recovery parameter for the evaluation of discharge** 

The fit is well described by an exponential with a time constant τr:

d p

**Q<sup>∞</sup>**

**removed charge Qd (pC)** 

after a pause time tp. This amount is given by:

where, Qd(tp) is given by Eq. 12.

any overlaps of irradiated volumes.

pause time of only 300 s.

Fig. 7. Current curves obtained during the second pulse injection performed after a pause time Δt = 900 s. Irradiation is carried out over the same area as in the first injection under identical experimental conditions (given in Fig. 5).

The three stages are summarized in Fig. 8, which illustrates the evolution of the amounts of charges remaining in the irradiated volume in the polycrystalline sample of grain diameter d = 4.5 µm at T = 473 K. The data concern the example of Figs. 5 and 7, for which some discharging occurs during the pause Δt. During the pause time tp (0 ≤ tp ≤ Δt), the net charge that still remains in the irradiated volume Ql(tp) evolves from its initial value, Ql(tp = 0) = Qst, to the final one, Ql(tp = Δt) = Qf. The form of the curve representing Ql(tp) will be justified below.

Fig. 8. Illustration of the time evolution of the amounts of charges remaining in the irradiated volume during the three stages. The first and the third curves correspond to Figs. 5 and 7 respectively. The net charge Ql(tp) during the pause evolves from Qst to Qf (which remains in the irradiated volume after a pause of Δt =900 s) as justified below by Eq. (15).

#### **5.2 Definition of a recovery parameter for the evaluation of discharge**

We first try to find a kinetic description of the charges that remain in the irradiated volume after a pause time tp. This amount is given by:

$$\mathbf{Q}\_{\rm l}(\mathbf{t}\_{\rm p}) = \mathbf{Q}\_{\rm st} - \mathbf{Q}\_{\rm d}(\mathbf{t}\_{\rm p}) \tag{13}$$

where, Qd(tp) is given by Eq. 12.

566 Sintering of Ceramics – New Emerging Techniques

**Iind(t)** 

**Iσ(t)** 

Fig. 7. Current curves obtained during the second pulse injection performed after a pause time Δt = 900 s. Irradiation is carried out over the same area as in the first injection under

The three stages are summarized in Fig. 8, which illustrates the evolution of the amounts of charges remaining in the irradiated volume in the polycrystalline sample of grain diameter d = 4.5 µm at T = 473 K. The data concern the example of Figs. 5 and 7, for which some discharging occurs during the pause Δt. During the pause time tp (0 ≤ tp ≤ Δt), the net charge that still remains in the irradiated volume Ql(tp) evolves from its initial value, Ql(tp = 0) = Qst, to the final one, Ql(tp = Δt) = Qf. The form of the curve representing Ql(tp) will be

**0 200 400 600 800**

irradiated volume during the three stages. The first and the third curves correspond to Figs. 5 and 7 respectively. The net charge Ql(tp) during the pause evolves from Qst to Qf (which remains in the irradiated volume after a pause of Δt =900 s) as justified below by Eq. (15).

Fig. 8. Illustration of the time evolution of the amounts of charges remaining in the

**pause time (s)**

**Qf**

**0.1 0.2 0.3 0.4 0.5**

**Qst**

**Qt(pC)**

**0 10 20 30 40 50**

**injection time (ms)**

identical experimental conditions (given in Fig. 5).

**Ql(pC)** 

**0.1 0.2 0.3 0.4 0.5 0.6**

**0 10 20 30 40**

**injection time (ms)**

justified below.

**Qt(pC)** 

**0.0 0.1 0.2 0.3 0.4 0.5**

**Qst**

As an example, in Fig. 9, we report the best fit of the evolution of Qd with tp, for data obtained, at 473 K in polycrystalline alumina sample (d = 4.5 µm), by performing the measurements over different zones that are sufficiently distant from one another to avoid any overlaps of irradiated volumes.

The fit is well described by an exponential with a time constant τr:

$$\mathbf{Q}\_{\rm d}(\mathbf{t}\_{\rm p}, \mathbf{T}) = \mathbf{Q}\_{\rm \ast}(\mathbf{T}) \left[ 1 - \exp\left( -\frac{\mathbf{t}\_{\rm p}}{\mathbf{\tau}\_{\rm r}(\mathbf{T})} \right) \right] \tag{14}$$

In this equation, Q∞ is the asymptotic value reached by Qd. In fact, it stands for the maximum amount of charges going out of the irradiated volume in the discharging process at temperature T. In the example of Fig. 9 where T = 473 K, Q∞ is found equal 0.46 pC corresponding to about 80 % of Qst (hence, 20 % of the net charge is still inside the irradiated volume). This is an indication of the existence of different trapping sites and that the temperature of 473 K is too low to provoke detrapping from the deeper ones.

Fig. 9. Evolution with the pause time at T = 473 K of the amount of charges that is removed from the irradiated volume Qd (in polycrystalline sample of 4.5 µm grain size). The solid line is the exponential fit of the data (Eq. 14). The asymptotic value Q∞(T) is attained after a pause time of only 300 s.

Effects of the Microstructure Induced by Sintering on the Dielectric Properties of Alumina 569

The results of the two types of single crystals are reported in Fig. 10. The values of R in both Pi-Kem and RSA are zero up to 473 K, indicating a perfect stable trapping behavior. Above

**6.1 Charge spreading in reference materials (single crystals)** 

473 K, R increases but the enhancement is significant only for Pi-Kem.

Fig. 10. Fraction R of charges removed from the irradiated volume as a function of temperature for the two types of single crystals (Pi-Kem and RSA samples).

i. Cathodoluminescence spectra obtained in similar Pi-Kem samples (Jardin et al., 1995) have identified mainly the F and F+ centers. Stable trapping in these centers is expected due to their deep level in the band gap (3 eV, and 3.8 eV for the F and F+, respectively). The increase of R above 473 K might be due to the intervention of excited F centers whose energy levels are believed very near the edge of the conduction band (Bonnelle

ii. The RSA material displays a more stable trapping ability than the Pi-Kem one above 473 K. This stable trapping raises queries about the role of the defects induced by the impurities (Table 1) and in particular those by the dominant silicon. With such amount of silicon (290 ppm), the concentration of the defects associated with Si exceeds the others. Consequently, one can deduce that stable trapping may occur on these defects.

As mentioned in paragraph 2, the dissolution of silicon into Al2O3, is expected to be compensated by a negatively charged cationic vacancy, ''' VAl (Eq. 4). In this context, the positively charged substitutional silicon Al Si• may act as electron trapping site while the cationic vacancy ''' VAl as hole trap. Upon trapping one electron during irradiation, Al Si• induces a donor level associated to Al Si<sup>×</sup> , which is estimated at 1.59 eV below the edge of the conduction band (Kröger, 1984). With regard to ''' VAl , hole trapping will give an acceptor level (associated to '' VAl ) located at 1.5 eV above the valence band (Kröger, 1984).

In order to interpret these behaviors we can consider the following results:

& Jonnard, 2010; Jonnard et al. 2000; Kröger, 1984).

From Eq. 14, we can associate to Q∞(T) the amount of charges Qf(T) that still remain in the irradiated volume, Q∞(T) = Qst – Qf(T). Therefore, the remaining quantity of charges Ql(tp, T) can be obtained from Eqs. 13 and 14:

$$\mathbf{Q}\_{\rm l}(\mathbf{t}\_{\rm p}, \mathbf{T}) = \left(\mathbf{Q}\_{\rm st} - \mathbf{Q}\_{\rm l}(\mathbf{T})\right) \exp\left(-\frac{\mathbf{t}\_{\rm p}}{\sigma\_{\rm r}(\mathbf{T})}\right) + \mathbf{Q}\_{\rm l}(\mathbf{T}) \tag{15}$$

The first term in this expression, which corresponds to the curve of the pause stage in Fig. 8, expresses charge decay under the internal electric field. Generally, the time constant τr can be set equal to ε/γ, where ε is the dielectric permittivity (ε = εrε0) and γ is the electric conductivity of the material (Adamiak, 2003; Cazaux, 2004). The value of τr deduced from Fig. 9 is 82 s giving a conductivity of about 10-14 Ω-1cm-1 which can be expected for this material, in agreement with the experimental value of the resistivity obtained for this sample in our laboratory (1.2 1014 Ωcm).

The asymptotic value Q∞ at T = 473 K is reached after only 300 s (Fig. 9). Therefore, one can anticipate that at temperatures within the range of interest (300 − 663 K), the condition for reaching the asymptotic value Q∞(T) are met for the chosen pause time Δt of 900 s.

The measured value of Q∞(T) is the result of detrapping of charges and their subsequent transport under the internal electric field. During this transport, a fraction of the detrapped charges can undergo a retrapping in deeper traps in the irradiated volume and eventually a recombination. The overall effect is a variation of charge distribution in the volume of interest, which affects the electric field. Consequently, since in the considered temperature range the experimental results do not reveal any significant dependence of Qst on temperature, the ratio R(T) = Q∞(T)/Qst can be expressed in terms of the measured currents:

$$\mathbf{R(T)} = \frac{\mathbf{Q\_{os}(T)}}{\mathbf{Q\_{st}}} = \frac{-\left\| \int\_{0}^{t\_{\text{inj}}} \mathbf{I\_{ind}(t)} \mathbf{dt} \right\|\_{\text{second injection}}}{-\left[ \int\_{0}^{t\_{\text{inj}}} \mathbf{I\_{ind}(t)} \mathbf{dt} \right]\_{\text{first injection}}} \tag{16}$$

This experimental parameter, which measures the fraction of charge removed from the irradiated volume, also characterizes the extent of discharging.

The ratio R(T), which can vary between 0 and 1, corresponds to an evolution from either the dominance of stable charge trapping (low values of R) or of charge spreading (high values of R, with R = 1 for a complete recovery of the uncharged state). The rate at which charges are detrapped depends usually on an attempt escape frequency and an activation energy linked to the trap depth. As a result, one can expect that, the variation of R with temperature could shed some light on the discharge process.

#### **6. Effect of microstructure induced by sintering on the ability of a dielectric to trap or spread charges**

As it will be pointed out, trapping and spreading are intimately linked to the microstructure and defects. Sintering not only leads to the creation of new interfaces but also to important phenomena such as segregation at grain boundaries and defects association.
