**4.1 Experimental set up**

The experiments are performed using a SEM (LEO 440), which is specially equipped (Fig. 3) to inject a controlled amount of charges Qinj with appropriate conditions (energy of incident electrons, charge and current beam densities, temperature). The electron beam is monitored by a computer system allowing the control of the beam parameters:


Fig. 3. Setting up for the measurement of the secondary electron current Iσ and induced current Iind. The secondary electron detector (diameter 15 cm) is positioned at a distance of about 2 cm from the sample surface.

Furthermore, the injection time, tinj (ranging from 10-3 to 1 s), is adjusted by the Electron Beam Blanking Unit "EBBU" using a function generator, allowing the turns off on the spot over the specimen surface with no electron track outside the investigated area when the beam is blanked.

The metallic sample holder is attached to a cooling-heating stage (temperature range 93 − 673 K). Hence, in situ thermal sample cleaning under vacuum (at T = 663 K during 180 minutes) and sample characterization at different temperatures are possible. Prior to electron irradiations, after the cooling that follows the cleaning step, the sample is held during 180 minutes at the testing temperature, so that the thermal equilibrium between the sample and the metallic holder is approached.

#### **4.2 The improved Induced Current Measurement method**

560 Sintering of Ceramics – New Emerging Techniques

In insulator material, the charges can be injected through interfaces, via an applied voltage, or created by irradiation via energetic particles. For instance, incident electrons, as they slowdown, can generate pairs of electrons and holes. These charge carriers can recombine, be trapped or move as a result of diffusion and/or field conduction. Concurrently, some of the electrons can be emitted from the sample surface and a distribution of trapped charges may develop within the irradiated volume. In our case, experiments are carried out using the electron beam of a SEM. The experimental set up, described below, provides means for measuring the evolution of the net amount of trapped charges Qt, which characterize the charging state of the insulator. For this purpose, we use the Induced Current Measurements "ICM" method (called also the Displacement Current Measurements), which we have

The experiments are performed using a SEM (LEO 440), which is specially equipped (Fig. 3) to inject a controlled amount of charges Qinj with appropriate conditions (energy of incident electrons, charge and current beam densities, temperature). The electron beam is monitored



**metallic holder** 

Fig. 3. Setting up for the measurement of the secondary electron current Iσ and induced current Iind. The secondary electron detector (diameter 15 cm) is positioned at a distance of

**trapped charges Qt** 

**objective** 

**EBBU**

**eσ e<sup>p</sup> -** 

**lens I<sup>σ</sup>**

**oscilloscope**

**amplifier**

**– +**

**Iind** 

**SEE detector +100V sample induced charges Qind** 

**function generator**

**amplifier** 

by a computer system allowing the control of the beam parameters:


focused and few hundred µm when it is defocused).

**4. Method for the characterization of the charging state of dielectrics** 

recently improved (Zarbout et al., 2008, 2010).

possibility of reaching a few µA),

**Iind(t)** 

**Iσ(t)** 

about 2 cm from the sample surface.

**4.1 Experimental set up** 

The ICM method is based on the measurement of the current Iind, produced by the variation of the induced charges Qind (in the sample holder) due to the trapped charges in the sample Qt (Liebault et al., 2001, 2003; Song et al., 1996). Since the influence coefficient in our experimental set up is close to one (Zarbout et al., 2008), the amount of the net trapped charges is given by:

$$\mathbf{Q}\_t(\mathbf{t}) = -\mathbf{Q}\_{\text{ind}}(\mathbf{t}) = -\int\_0^\mathbf{t} \mathbf{I}\_{\text{ind}}(\mathbf{t}) \, \mathbf{d}\mathbf{t} \tag{7}$$

The improvement brought to the ICM method consists in the concurrent measurement of Iind and the total secondary electron current Iσ due to the sole electrons emitted by the sample. For this purpose, as shown in Fig. 3, the SEM is specially equipped with a secondary electron low-noise collector located under the objective lens just above the sample. A biased voltage of 100 V is applied to it in order to collect all the electrons escaping from the sample surface.

During charge injection, the currents, Iσ and Iind, are simultaneously amplified (Keithley 428) and observed on an oscilloscope (HP 54600B) where the material response is displayed after a short lag time. The primary current beam Ip (which is adjusted in a Faraday cage) and the current Iσ are always positive whereas Iind can be positive or negative depending on the Secondary Electron Emission "SEE" yield σ, which is equal to Iσ/Ip. Then if Iσ is higher than Ip (σ > 1), the sample charges positively and Iind is negative. In the other case (σ < 1), the sample charges negatively and Iind is positive. The general variation of σ with Ep is shown in Fig. 4.

Fig. 4. Schematic evolution of the SEE yield σ with primary beam energy Ep for uncharged insulator materials.

Effects of the Microstructure Induced by Sintering on the Dielectric Properties of Alumina 563

**10 ms** 

**100pC** 

Fig. 5. Current curves recorded during the first injection in polycrystalline alumina sample (d = 4.5 µm) at 473 K. The currents, –I0 = – 225 pA and I0 + Ip = 325 pA, are measured after a short lag time. The irradiation conditions are: Ep = 10 keV, Ip = 100 pA, tinj = 5 ms, Qinj = 5 pC

**injection time tinj**

**Iσ(t)** 

**I0+Ip**

**Iind(t)** 

**—I0**

At the beginning of injection, the current curves are affected by the rise time of the amplifiers. Therefore, a short lag time is required to display the actual current values –I0 and I0+Ip. As irradiation proceeds, the induced current, Iind, increases from –I0 to zero. Concurrently and since the current complementarity conditions are verified (as revealed by Eq. 8), the total secondary electron current Iσ decreases from (I0 + Ip) to Ip. This decrease has been associated with a progressive accumulation of positive charges that are distributed over a depth of about the escape length of secondary electrons λ. As suggested, this could be assigned to recombination of electrons with holes (in this zone), which should be otherwise emitted (Cazaux, 1986) or to field effect (Blaise et al., 2009). The other important feature is indeed that during injection, a negative charge distribution develops in the vicinity of the penetration depth of primary electrons Rp. As result, an internal electric field is established between the two charge distributions (holes near the surface and electrons around Rp) whose direction is oriented towards the bulk. Since the diameter Φ of the irradiated area is much larger than the penetration depth, a planar geometry can be used to evaluate this field

> <sup>2</sup> 0 0 0r r <sup>r</sup> 0 r 1 1 E(z,t) <sup>1</sup> (z,t)dz (z,t)dz (1 ) 4(1 )e

In this expression, where the contribution of image charges has been taken into account, e is the sample thickness, ε0 is the vacuum permittivity, εr the relative permittivity (taken equal to 10 for α-alumina) and ρ(z, t) is the density of charges at the depth z. The first term

When Iind reaches a zero value (Iσ(t) = Ip(t)), a steady state, which corresponds in fact to a self regulated flow regime, is achieved. There, on the average, for each incident electron one secondary electron is emitted. This state is characterized by some constant value of the

= + ρ+ρ

− Φ

−

<sup>1</sup> <sup>2</sup> e z

ε ε +ε + ε ε ε (10)

and J = 4×104 pA/cm2.

(Aoufi & Damamme, 2008) from the Gauss theorem:

corresponds to the surface electric field E (0, t).

The sign of the net sample charges (i.e., net trapped charges) depends mainly, for a given material, on the primary beam energy Ep and the primary current beam density J. In the case of α-alumina, a sign inversion of the net trapped charges is observed for J greater than 7×106 pA/cm2 (Thome et al., 2004).

Experiments are performed with a 10 keV primary electron beam energy which is located, for the α-alumina materials, between the two crossover energies of primary electrons EpI and EpII (about 20 keV) for which σ is equal to 1. To probe a zone representative of the material we use a defocused beam over an area of 560 µm diameter Φ, which has been accurately measured using an electron resist (Zarbout et al., 2005). With the utilized value Ip = 100 pA, the primary current beam density is J = 4×104 pA/cm2. These experimental conditions give rise to net positive trapped charges (σ > 1) and then a negative induced current, as shown in the curves recorded by the oscilloscope in Fig. 3.

The positive surface potential developed by the trapped charges Qt does not exceed a few volts, which is very low to produce any disturbance of the incident electron beam and of the measurement process (Zarbout et al., 2008). Then, the good stability and reliability of the SEM ensure (for a biased voltage applied to the secondary electron collector greater than about 50 V) the current complementarity according to the Kirchhoff law:

$$\mathbf{I}\_{\mathbf{P}}(\mathbf{t}) = \mathbf{I}\_{\text{ind}}(\mathbf{t}) + \mathbf{I}\_{\mathbf{o}}(\mathbf{t}) \tag{8}$$

Hence, the presence of the collector with a sufficient biased voltage allows accurate measurements of the currents and therefore the precise determination of the quantity of trapped charges. Incidentally, this makes also possible a precise calculation of the SEE yield σ(t):

$$\sigma(\mathbf{t}) = \frac{\mathbf{I}\_{\sigma}(\mathbf{t})}{\mathbf{I}\_{\mathbf{p}}(\mathbf{t})} = \frac{\mathbf{I}\_{\mathbf{p}}(\mathbf{t}) - \mathbf{I}\_{\mathrm{ind}}(\mathbf{t})}{\mathbf{I}\_{\mathrm{ind}}(\mathbf{t}) + \mathbf{I}\_{\sigma}(\mathbf{t})} = \mathbf{1} - \frac{\mathbf{I}\_{\mathrm{ind}}(\mathbf{t})}{\mathbf{I}\_{\mathrm{ind}}(\mathbf{t}) + \mathbf{I}\_{\sigma}(\mathbf{t})} \tag{9}$$

The SEE yield could be deduced directly from the secondary electron current Iσ(t) and the primary one Ip(t). However, the use of Iσ(t) and Iind(t) in Eq. 9 is more appropriate as it provides the opportunity to circumvent any uncontrolled fluctuation of the primary electron current during the different phases of the experimental process.

It is worth noting that after the fabrication process of polycrystalline samples (with firing temperatures above 1863 K) or the thermal treatment of single crystals (at 1773 K for 4 hours), the final thermal cleaning stage under vacuum of all the samples prior to electron irradiation will completely remove initial charges and surface contamination that could interfere with the generated charges.

#### **4.3 The charging kinetic**

The measurement of the foregoing currents gives means to follow the net quantity of trapped charges during irradiation. The current curves Iσ(t) and Iind(t) of Fig 5 are a typical example of the recorded currents. As pointed out in the experimental conditions, irradiation is performed with a 10 keV electron beam energy, which ensure a net amount of positive charges in the sample.

The sign of the net sample charges (i.e., net trapped charges) depends mainly, for a given material, on the primary beam energy Ep and the primary current beam density J. In the case of α-alumina, a sign inversion of the net trapped charges is observed for J greater than

Experiments are performed with a 10 keV primary electron beam energy which is located, for the α-alumina materials, between the two crossover energies of primary electrons EpI and EpII (about 20 keV) for which σ is equal to 1. To probe a zone representative of the material we use a defocused beam over an area of 560 µm diameter Φ, which has been accurately measured using an electron resist (Zarbout et al., 2005). With the utilized value Ip = 100 pA, the primary current beam density is J = 4×104 pA/cm2. These experimental conditions give rise to net positive trapped charges (σ > 1) and then a negative induced

The positive surface potential developed by the trapped charges Qt does not exceed a few volts, which is very low to produce any disturbance of the incident electron beam and of the measurement process (Zarbout et al., 2008). Then, the good stability and reliability of the SEM ensure (for a biased voltage applied to the secondary electron collector greater than

Hence, the presence of the collector with a sufficient biased voltage allows accurate measurements of the currents and therefore the precise determination of the quantity of trapped charges. Incidentally, this makes also possible a precise calculation of the SEE yield

> p ind ind I (t) I (t) I (t) I (t) (t) <sup>1</sup>

The SEE yield could be deduced directly from the secondary electron current Iσ(t) and the primary one Ip(t). However, the use of Iσ(t) and Iind(t) in Eq. 9 is more appropriate as it provides the opportunity to circumvent any uncontrolled fluctuation of the primary electron

It is worth noting that after the fabrication process of polycrystalline samples (with firing temperatures above 1863 K) or the thermal treatment of single crystals (at 1773 K for 4 hours), the final thermal cleaning stage under vacuum of all the samples prior to electron irradiation will completely remove initial charges and surface contamination that could

The measurement of the foregoing currents gives means to follow the net quantity of trapped charges during irradiation. The current curves Iσ(t) and Iind(t) of Fig 5 are a typical example of the recorded currents. As pointed out in the experimental conditions, irradiation is performed with a 10 keV electron beam energy, which ensure a net amount of positive

<sup>−</sup> σ= = = − + +

p ind ind

σ σ

I (t) I (t) I (t) I (t) I (t)

p ind I (t) I (t) I (t) = + <sup>σ</sup> (8)

(9)

current, as shown in the curves recorded by the oscilloscope in Fig. 3.

about 50 V) the current complementarity according to the Kirchhoff law:

σ

current during the different phases of the experimental process.

interfere with the generated charges.

**4.3 The charging kinetic** 

charges in the sample.

7×106 pA/cm2 (Thome et al., 2004).

σ(t):

Fig. 5. Current curves recorded during the first injection in polycrystalline alumina sample (d = 4.5 µm) at 473 K. The currents, –I0 = – 225 pA and I0 + Ip = 325 pA, are measured after a short lag time. The irradiation conditions are: Ep = 10 keV, Ip = 100 pA, tinj = 5 ms, Qinj = 5 pC and J = 4×104 pA/cm2.

At the beginning of injection, the current curves are affected by the rise time of the amplifiers. Therefore, a short lag time is required to display the actual current values –I0 and I0+Ip. As irradiation proceeds, the induced current, Iind, increases from –I0 to zero. Concurrently and since the current complementarity conditions are verified (as revealed by Eq. 8), the total secondary electron current Iσ decreases from (I0 + Ip) to Ip. This decrease has been associated with a progressive accumulation of positive charges that are distributed over a depth of about the escape length of secondary electrons λ. As suggested, this could be assigned to recombination of electrons with holes (in this zone), which should be otherwise emitted (Cazaux, 1986) or to field effect (Blaise et al., 2009). The other important feature is indeed that during injection, a negative charge distribution develops in the vicinity of the penetration depth of primary electrons Rp. As result, an internal electric field is established between the two charge distributions (holes near the surface and electrons around Rp) whose direction is oriented towards the bulk. Since the diameter Φ of the irradiated area is much larger than the penetration depth, a planar geometry can be used to evaluate this field (Aoufi & Damamme, 2008) from the Gauss theorem:

$$\mathbf{E}(\mathbf{z},\mathbf{t}) = \frac{-1}{\varepsilon\_0 \varepsilon\_r (1+\mathfrak{e}\_\mathbf{r})} \left(1 + \frac{\mathfrak{d}^2}{4(1+\mathfrak{e}\_\mathbf{r})\mathfrak{e}^2} \right)^{-1} \int\_0^\mathbf{e} \mathfrak{p}(\mathbf{z},\mathbf{t}) d\mathbf{z} + \frac{1}{\mathfrak{e}\_0 \mathfrak{e}\_\mathbf{r}} \int\_0^\mathbf{z} \mathfrak{p}(\mathbf{z},\mathbf{t}) d\mathbf{z} \tag{10}$$

In this expression, where the contribution of image charges has been taken into account, e is the sample thickness, ε0 is the vacuum permittivity, εr the relative permittivity (taken equal to 10 for α-alumina) and ρ(z, t) is the density of charges at the depth z. The first term corresponds to the surface electric field E (0, t).

When Iind reaches a zero value (Iσ(t) = Ip(t)), a steady state, which corresponds in fact to a self regulated flow regime, is achieved. There, on the average, for each incident electron one secondary electron is emitted. This state is characterized by some constant value of the

Effects of the Microstructure Induced by Sintering on the Dielectric Properties of Alumina 565

To characterize the degree of spreading (of discharge), we set up a protocol allowing its evaluation. The procedure consists in analysing the states of charging deduced from two pulse electron injections over the same area, separated by some lapse of time as explained in

The first stage is intended to achieve a charging reference state, which is attained once σ reaches the constant value of 1, i.e. the situation where for each electron entering one is emitted. This charging reference state is associated with the trapping of a maximum quantity of charges Qst (cf. Fig. 6). The embedded charges can be either indefinitely trapped or just localised for some time lapse (seconds, minutes, hours or more). If all the charges were trapped in a stable way, the foregoing reference state would stay unchanged when irradiation is turned off. In the case of partially localised charges, some fraction of Qst could manage to spread out from irradiated region, through conduction. To take into account this fact, a pause time long enough to appraise the degree of charge spreading is chosen for the

During the pause time Δt, the stability of the amount of charges Qst is determined by the efficiency of traps associated with impurities and lattice defects as well as the charge transport properties. Then, if one wants to evaluate the extent of discharging, one has to perform, with identical conditions, a second injection over the same area as the first one for

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

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:

<sup>d</sup> ind <sup>0</sup> second injection

In other words, this interpretation means that any loss of charges, during the pause, can be

Q I (t) dt = − (12)

inj t

compensated by those introduced during the second irradiation to restore Qst.

**5.1 Experimental protocol** 

the three following stages.

pause stage.

pause time.

**5.1.2 The pause stage** 

**5.1.1 The first charging stage (first pulse injection)** 

the purpose of restoring the reference steady state.

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

electric field as well as by a maximum amount of net trapped charge in the sample, Qst, equal to inj <sup>t</sup> ind <sup>0</sup> <sup>−</sup> I (t) dt . The steady state is interesting because it will be taken as a reference state and then the practical choice of the injection time tinj is determined by the achievement of this state. The evolution of the net trapped charges Qt during injection, which can be deduced from Eq. 7, is shown in Fig. 6.

Fig. 6. Evolution with time, at 473 K, of the net charge, Qt, during the first pulse injection in polycrystalline alumina sample (d = 4.5 µm). The quantity Qt is derived from the currents of Fig. 5 via Eq. 7. The solid line represents the exponential fit of the data, as given by Eq. 11.

The best fit of data represented in Fig. 6, leads to an exponential time evolution:

$$\mathbf{Q}\_t(\mathbf{t}) = \mathbf{Q}\_{\rm st} \left[ 1 - \exp\left( -\frac{\mathbf{t}}{\mathbf{r}\_c} \right) \right] \tag{11}$$

where τc is the charging time constant (found equal to 2.07x10-3 s), which characterizes the charging kinetic of the material in the used experimental conditions.

#### **5. Measurement of the ability to spread charges**

The method described above gives the opportunity to evaluate the quantity of trapped charges during irradiation. If the probed zone is initially uncharged, Qt(t) characterizes the quantity of charges that accumulates after an irradiation time t. Depending on the insulator conduction properties, the accumulated charges can either remain localized or partially (or even totally) spread out of the irradiated volume (discharge phenomenon). Hence, the measurement of the ability of the material to spread charges is of technological interest. In this paragraph, we give details of the experimental procedure developed for the measurement of this ability and we define a recovery parameter allowing the quantitative evaluation of the extent of discharge.
