Origin of Irradiation Synergistic Effects in Silicon Bipolar Transistors: a Review

To explore the behavior of the ISEs in PNP and NPN transistors, Song et al. (2019, 2020a) LM324N chips with 4 PNP-type input-stage transistors in each chip and 3DK9D NPN-type transistors were selected as the research object, respectively. These devices were chosen because they are sensitive to both ID and DD. The neutron irradiation hardly influences the dynamics of OTs in the oxide layer, Raymond and Petersen (1987) hence, we adopt a sequential neutron/$\gamma$-ray irradiation configuration. The processes of the sequential irradiation experiments for PNP transistors is shown in Fig. 2 (a). The transistors are first irradiated by neutron to the fluence as indicated in the figure along the vertical direction; then they are irradiated by $\gamma$-ray to the dose as indicated in the figure along the horizontal direction. By these irradiations the practical damages can be obtained. The summed damages as a reference are obtained by another two pure $\gamma$-ray irradiations. For each neutron-$\gamma$-ray condition, 8 PNP transistors (within 2 chips) are used to avoid possible misjudgment due to sample variability. Note, existing experiments Barnaby et al. (2001); Gorelick et al. (2004); Li et al. (2012a); Wang et al. (2016) usually adopt only a fixed displacement fluence and ionization dose, hence is very different from the present experimental setup. The radiation sources, irradiation bias, and testing systems are described in detail in Ref. Song et al., 2019. The ISE of NPN transistors is investigated by a very similar sequential neurton/$\gamma$-ray irradiation experiment, which is shown in Fig. 2 (b) and has been described in detail in Ref. Song et al., 2020a.

Artificial summed damages. The individual DDs of the PNP transistors are shown in Fig. 3, from which a sub-linear dependence of the input bias current ($I_{iB}$) on the neutron fluence is clearly seen. It is also seen that, the DDs of the 8 transistors under a same fluence are different due to sample-to-sample variability. To obtain the artificial summed damages, Nos. 1-2 and 9-10 chips were irradiated by a low and high dose rate $\gamma$ rays, respectively. The results for the former are displayed in Fig. 4 (a). An almost linear increase of the ID is seen and an average slope of $k_{0}^{L}=15.6$ nA/krad(Si) and $k_{0}^{H}=8.0$ nA/krad(Si) is extracted for the low and high dose rate case, respectively. The summed damage reads:

where $D_{0}$ is the initial DD before $\gamma$-ray irradiations. A typical result is plotted in Fig. 4 (c) as the dashed line.

Practical damages and behavior of the negative ISE. The $\gamma$-ray response of the $3^{rd}$-$8^{th}$ chips at low dose rate with different $D_{0}$’s are displayed in Figs. 4 (b)-(d). It is seen that, the practical damages of all 24 transistors are smaller than their corresponding summed damages. This means that we obtain a clear negative ISE in PNP transistors. Remarkably, the practical damages display ‘tick’-like curves: the input-bias currents first exponentially decrease for relatively small $\gamma$-ray dose and then linearly increase for relatively large $\gamma$-ray dose. Moreover, the slope of the linear increase is obviously smaller than that of the pure ID, $k_{0}^{L}$. The $11^{th}$-$16^{th}$ chips display similar negative ISEs at high dose rate $\gamma$-ray irradiations, see Ref. Song et al., 2019. To further obtain the $D_{0}$ and $\gamma$-ray dose rate dependences of the ISE, we plot six typical curves in Fig. 5, from which we can see that, the exponential decay of the practical damage at small $\gamma$-ray dose becomes stronger when the dose rate decreases; while the decrease of the slope at large $\gamma$-ray dose becomes bigger when the initial DD increases. These regular behaviors, especially the dose rate dependence of the ISE, cannot be explained by the traditional Coulomb interaction mechanism.

The practical neutron/$\gamma$-ray damage (the base currents) of the 3DK9D NPN transistors are shown in Fig. 6 (a) as a function of the $\gamma$-ray dose at a dose rate of 10 rad/s (see the dots). Besides, the summed damages for three typical samples are obtained from a direct sum of their DDs and the average ID of five extra samples (see the dashed curves). From the results it is clear that a positive ISE arises for the NPN transistors. From Fig. 6 (a) it is also seen that, the practical damages increase sublinearly as the ionization dose increases; meanwhile, the positive ISE becomes larger when the initial DD increases. The practical and summed damages are shown in Fig. 6 (b) for the low dose rate (10mrad/s) case. A second stage arises when the dose exceeds 2 krad. However, the two different stages are found to show similar dependences on the $\gamma$-ray dose and neutron fluence as the high dose rate case.

To sum up, we have designed a variable neutron-fluence and $\gamma$-ray-dose irradiation experiment to explore the behaviors of the negative and positive ISEs in PNP and NPN transistors. The results show that both ISEs display regular dependences on the $\gamma$-ray dose, dose rate, and neutron fluence. The obtained features are quite different for PNP and NPN transistors.

As identified by Deep Level Transient Spectroscopy (DLTS), Lang (1974a, b) the main defect type in neutron irradiated p-type silicon is donor-like divacancies ($\rm V_{2}^{0/+}$), Cheng et al. (1966); Lee and Corbett (1976); Lindström et al. (1999) see Fig. 7 (a). We notice that, the diffusion of $\rm V_{2}$ will be greatly enhanced when the damaged Si samples are further subjected to a $\gamma$-ray irradiation. Song et al. (2020a) The microscopic picture of such a carrier-enhanced defect diffusion is as followings. The $\gamma$-ray-induced charge carriers will shift the quasi-Fermi energy in the Si samples. Accordingly, the $\rm V_{2}$ defects can change from one charge state at its equilibrium site to another that is not at equilibrium. Bar-Yam and Joannopoulos (1984a, b); Baraff and Schlüter (1984); Car et al. (1984) The latter will easily diffuse to its equilibrium site without having to overcome a diffusion barrier. These behaviors largely enhance the diffusion of $\rm V_{2}$ defect. Besides, many oxygen interstitials ($\rm O_{i}$), which are electrically-inactive, have been introduced in the Si sample in its growth process. The moving $\rm V_{2}$ defects will be trapped by $\rm O_{i}$ where they react to form another defect $\rm V_{2}O$ that is electrically-active, Mikelsen et al. (2005); Markevich et al. (2014)

On the other hand, the energy liberated upon nonradiative recombination will be converted into vibrational energy that can be utilized to promote the defect reactions, which include the dissociation of clusters and re-emission of self-interstitials in damaged Si. Lang and Kimerling (1974); Weeks et al. (1975); Kimerling (1978) The charge-enhanced defect diffusion also applies to the emitted self-interstitials. The moving divacancies and self-interstitials can collide and annihilate each other Song et al. (2020a)

The rate equation of $\rm V_{2}$ can be derived from Eq. (2). The result is $\rm\partial V_{2}(t)/\partial t=-k_{1}O_{i}V_{2}(t)-k_{2}Si_{i}V_{2}(t)$, where $k_{1}$ and $k_{2}$ are the transformation and annihilation reaction velocities in Eqs. (2a) and Eq. (2b), respectively, see Fig. 7 (a). To simplify the calculations, we replace $V_{2}(t)$ and $\rm Si_{i}$ in the last term by their initial concentrations, which are directly related to the irradiation fluence of neutron, $F=V_{2}(0)$. This replacement can be done because the annihilation rate $k_{2}$ is relatively small Mikelsen et al. (2005); Markevich et al. (2014) and the concentration of $\rm Si_{i}$ is directly proportion to the total amount of vacancies. The above equation becomes

The initial conditions for the two defects are $\rm V_{2}(0)$ $=F$ and $\rm V_{2}O(0)$$=0$. Solving the resulting coupled equations with these conditions, we obtain the $\gamma$-ray dose dependence of the concentration of $\rm V_{2}$ and $\rm V_{2}O$

Here we have changed $\partial/\partial t$ to $R\partial/\partial D$, where $R$ is the dose rate. From the pair of solutions it is clear that, the concentration of $\rm V_{2}$ is an exponential decay function of the $\gamma$-ray dose, while the concentration of $\rm V_{2}O$ is both an asymptotic growth function and a linear decay behavior of the $\gamma$-ray dose. Song et al. (2020a) The neutron fluence dependence of the defect concentrations is indicated by the factor $F$ in Eq. (4). Song et al. (2020a)

Comparison with the annealing experiments. Evolution of displacement defects also happens in traditional annealing process induced by elevated temperature. However, the self-interstitial emission mechanism described in Eq. 2 (b) is absent for this case, Kimerling (1978); Pellegrino et al. (2001) as the energy is not enough to decompose the clusters. As a result, the defect evolution is only due to defect diffusion and transformation induced by elevated temperatures, as described by Eq. (2a). Then, the defect concentrations as a function of the annealing time ($t$) reduces to

These results are the same as those in Ref. Mikelsen et al., 2005. It is noticed that, here the increase of $\rm V_{2}O$ equals exactly to the decrease of $\rm V_{2}$, and we will not observe a linear decay term of the defect concentration as described in Eq. (4b).

To show the behavior of the ISE in NPN transistors, we relate the defect concentrations in p-type Si to the SRH base current in the NPN transistor, which are found to be in direct proportion to each other. Pierret and Neudeck (1987); Adell and Boch (2014) The donor levels of the defects in p-type Si have been identified by DLTS. Mikelsen et al. (2005); Markevich et al. (2014) As shown in Fig. 7 (a), there is one active donor level for the $\rm V_{2}$ defect, $\rm V_{2}^{0/+}$ at $E_{v}+0.19$ eV, and two active donor levels for the $\gamma$-ray induced $\rm V_{2}O$ defect, $\rm V_{2}O^{0/+}$ at $E_{v}+0.24$ eV and $\rm V_{2}O^{+/+2}$ at $E_{v}+0.09$ eV. The contribution factors of each defect charge state to the base current can be denoted as $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$, see Fig. 7 (a). Note, the same contribution factor of $\lambda_{1}=\lambda_{3}$ is often assumed Ganagona et al. (2014) as $\rm V_{2}^{0/+}$ and $\rm V_{2}O^{0/+}$ have almost the same donor levels and hole capture cross sections. Srour et al. (1973); Myers et al. (2008); Liu et al. (2015) The $\gamma$-ray-induced changes of the base current corresponding to the evolution of defects in Eq. (4) read Song et al. (2020a)

Here the measurable initial DD due to the neutron-irradiation-induced $V_{2}$ is denoted as $f=\lambda_{1}F$, the ratio between the $\gamma$-ray-irradiation-induced annihilation and transformation velocities is denoted as $\eta=\kappa_{1}^{-1}\kappa_{2}/\lambda_{1}$, $D_{c}=\kappa_{1}^{-1}R$ is a characteristic dose of the transformation from $\rm V_{2}$ to $\rm V_{2}O$, and $\lambda_{21}=\lambda_{2}/\lambda_{1}$ is a ratio between the contribution factors of the first and second donor levels of $\rm V_{2}O$. The total change of the base current is the sum of the above three components, which reads Song et al. (2020a)

This is the desired analytical model of the positive ISE in NPN transistors, which is based on the ionization-irradiation-induced dynamics of displacement defects in p-type silicon and shows the explicit dependence of the applied $\gamma$-ray dose and initial DD (neutron fluence). These four current profiles are plotted in Fig. 7(b) by using different colors.

The derived model predicts that, the positive ISE of NPN transistors would possess one asymptotic growth term and one linear decay term as a function of the $\gamma$-ray dose, Song et al. (2020a) see the black curves in Fig. 7 (b). The asymptotic growth term of the ISE stems from the ionization-irradiation-induced asymptotic growth of the $\rm V_{2}O^{+/+2}$ defect, see the magenta solid curves in Fig. 7 (b). The linear decay term of the ISE results from the two ionization-irradiation-induced linear decaying components of $\rm V_{2}O^{0/+}$ and $\rm V_{2}O^{+/+2}$, see the blue and magenta straight solid lines in Fig. 7 (b). As shown by the red and blue dashed curves in Fig. 7 (b), the remaining asymptotic growth of $\rm V_{2}O^{0/+}$ and the exponential decay of $\rm V_{2}^{0/+}$ cancel each other. The reason is the one-to-one transformation between $\rm V_{2}$ and $\rm V_{2}O$ defects and the same contribution factors of their first donor levels. The proposed model also predicts an almost linear dependence of the magnitude of the asymptotic growth term on the initial DD ($f$), and a quadratic dependence of the slope of the linear decay term on the initial DD. Song et al. (2020a) The reasons are the one-to-one correspondence between $\rm V_{2}$ and $\rm V_{2}O$ in the transformation and the binary behavior of interstitials and vacancies in the annihilation. Song et al. (2020a) The proposed defect dynamics model also predicts a dose-rate dependence of the two independent synergistic terms.

To testify the above proposed model for NPN transistors, we try to fit the obtained data in Sec. II.C. by using a fitting model

where $j$ denotes the sample index. The first term in the fitting model contains both the linear decay term in the proposed physical model and the IDs of the samples. As the ID varies among different samples, the physical model, Eq. (7), cannot be directly used. In stead, the presence of the linear decay term can be identified by the statistical difference between the extracted $k_{j}$ and $k_{0}$ (the slopes of pure ID). The correspondence between the fitting and physical parameters is $k_{j}-k_{0}=-(1+\lambda_{21})\eta f^{2}/D_{c}$. The second term in the fitting model is exactly the asymptotic growth term in the proposed physical model. The correspondence between the parameters is $A_{j}=\lambda_{21}(f+\eta f^{2})$ and $D_{c}^{j}=D_{c}$. The fitting curves for the high dose rate case are plotted in Fig. 6 (a), which are found to work very well for all 9 samples. The separated linear component (blue dashed) and asymptotic growth term (red dashed) of a typical sample are shown in Fig. 8 (a). A significant decrease is found in the linear component relative to the average slope of pure ID (black dashed). For all 9 samples’ statistical results, see $k_{j}$ plotted in Fig. 9 (a). Similar perfect fitting can be done for the low dose rate case within the first stage, see the fitting curves plotted in Fig. 6 (b). These facts verify the predicted asymptotic growth and linear decay terms in the ISE of NPN transistors, reflecting the proper underlying defect evolution physics in the prediction of dose dependence of the ISE.

To further testify the initial DD dependence of the proposed model, we plot the fitting parameters of Eq. (8) ($k_{j}$, $A_{j}$, and $D_{c}^{j}$) in Fig. 9 as a function of the base current. A decreasing trend is found in Fig. 9 (a) for the slope of the linear term $k_{j}$ as a function of the initial DD, $D_{0}$, which starts from the pre-irradiation base current, $\rm I_{B}^{0}\approx 2mA$. Further investigation shows that a quadratic dependence is satisfied, i.e., $k_{j}=k_{0}-\alpha D_{0}^{2}$. This behavior is consistent with the model’s prediction. Calculations show that $\alpha=5\times 10^{-3}\rm mA^{-1}krad^{-1}$ and $17\times 10^{-3}\rm mA^{-1}krad^{-1}$ for the high and low dose rate $\gamma$-ray irradiation, respectively. A linearly increasing trend is found in Fig. 9 (b) for the amplitude of the asymptotic growth term $A_{j}$ as a function of the initial DD. This feature is also the same as the prediction of the model. The characteristic dose of the asymptotic growth term $D_{c}^{j}$ shown in Fig. 9 (c) is insensitive to the DD. These results further confirm the novel initial DD dependence predicted by the proposed model, reflecting the proper defect evolution physics underlying the model.

The significant difference between the red and blue dots in Fig. 9 also reflect the predicted dose-rate dependence of the ISE. For a fixed total dose, the carrier-enhanced defect diffusion and recombination-enhanced defect reactions become stronger for decreasing dose rate, which means dramatically increased action time. This fact readily explains the dose rate dependence of the ISE in NPN transistors, which, however, cannot be explained by the Coulomb interaction mechanism.

The physical parameters in Eq. (7) can be calculated from their correspondence to the fitting parameters. The results are shown in Table 1 and some valuable facts can be inferred. The values of $\lambda_{21}$ at different dose rates indicate that the second donor level of $\rm V_{2}O$ contributes 5 or 10 times less than the first donor level of $\rm V_{2}O$ and $\rm V_{2}$. The values of $\eta$ implies that $\eta f\sim 1\times 10^{-3}\ll 1$, which supports $\eta f^{2}\ll f$ in Eq. (7) hence the linear dependence of the growth term on the initial DD. The dose rate sensitivity of both the dynamic parameters ($D_{c}=\kappa_{1}^{-1}R$ and $\eta=\kappa_{1}^{-1}\kappa_{2}/\lambda_{1}$) and the contribution factor ($\lambda_{21}=\lambda_{2}/\lambda_{1}$) of the defects result in the dose rate dependence of the ISE.

Similar to the p-type silicon case, $\rm V_{2}$ in n-type silicon as acceptor levels will transform to $\rm V_{2}O$ due to the above mentioned carrier-enhanced diffusion of $\rm V_{2}$ under the $\gamma$-ray irradiation, Kimerling et al. (1975); Kimerling (1976); Bar-Yam and Joannopoulos (1984a, b); Baraff and Schlüter (1984); Car et al. (1984) see Fig. 7 (a). $\rm V_{2}$ will also be annihilated by self-interstitials which are emitted due to recombination-enhanced defect reactions. Lang and Kimerling (1974); Weeks et al. (1975); Kimerling (1978) The ionization dose and neutron fluence dependence of the concentrations of $\rm V_{2}$ and $\rm V_{2}O$ are given by Eq. (4).

Different from the p-type silicon case, both $\rm V_{2}$ and $\rm V_{2}O$ in n-type silicon have two acceptor levels: $\rm V_{2}^{-/-2}$ at $\rm E_{c}-0.23eV$, $\rm V_{2}^{0/-}$ at $\rm E_{c}-0.42eV$, $\rm V_{2}O^{-/-2}$ at $\rm E_{c}-0.20eV$, and $\rm V_{2}O^{0/-}$ at $\rm E_{c}-0.46eV$, Mikelsen et al. (2005) see the upper half in Fig. 7 (a). Denoting the contribution factors of these acceptor charge states to the base current of PNP transistors as $\lambda_{4}$, $\lambda_{5}$, $\lambda_{6}$, and $\lambda_{7}$, respectively. Note, the same contribution factor of $\lambda_{4}=\lambda_{6}$ ($\lambda_{5}=\lambda_{7}$) can be assumed, because $\rm V_{2}^{0/-}$ and $\rm V_{2}O^{0/-}$ ($\rm V_{2}^{-/-2}$ and $\rm V_{2}O^{-/-2}$) have almost the same acceptor levels and electron capture cross sections. Mikelsen et al. (2005) Multiplying the defect concentrations, Eq. (4), by these factors, we obtain the corresponding components of the base current in PNP transistors, which are plotted as the red and blue profiles in Fig. 7 (c). As the exponential decay curve of $\rm V_{2}^{0/-}$ ($\rm V_{2}^{-/-2}$) and the asymptotic growth curve of $\rm V_{2}O^{0/-}$ ($\rm V_{2}O^{-/-2}$) cancel each other, we can find that the net result due to the ionization-induced evolution of $\rm V_{2}$ displacement defects in n-type Si is a linear decay term of the ISE in PNP transistors

Here $f_{1}=(\lambda_{4}+\lambda_{5})F$ is the measurable initial DD due to $\rm V_{2}$ in PNP transistors and $\eta_{1}=\kappa_{1}^{-1}\kappa_{2}/(\lambda_{4}+\lambda_{5})$ is the ratio between the annihilation and transformation reactions for $\rm V_{2}$ in n-type silicon; these factors are different from the $f$ and $\eta$ factors in Eq. (7) for NPN transistors.

Different from p-type silicon, neutron-irradiation-induced VO is also an effective recombination center in n-type silicon, see the upper half in Fig. 7 (a). So, besides $\rm V_{2}$ the $\gamma$-ray-irradiation-induced evolution of VO should also be considered for PNP transistors. Due to the above mentioned carrier-enhanced diffusion of $\rm VO$ Kimerling et al. (1975); Kimerling (1976); Bar-Yam and Joannopoulos (1984a, b); Baraff and Schlüter (1984); Car et al. (1984) and recombination-enhanced emission of self-interstitial, Lang and Kimerling (1974); Weeks et al. (1975); Kimerling (1978) the transformation and annihilation reactions of VO will happen

According to Ref. Pellegrino et al., 2001, the transformation reaction happens because the moving VO will be trapped by electrically-inactive $\rm O_{i}$, which forms an electrically non-active defect $\rm VO_{2}$, see Fig. 7(a). The annihilation reaction happens because the moving vacancy-oxygen complex and self-interstitials can collide and generates electrically-inactive $\rm O_{i}$. Eqs. (2) and (10) make up the full defect dynamics accounting for the negative ISE in PNP transistors.

Using the same modeling approach as $\rm V_{2}$ and $\rm V_{2}O$ in Sec. III B, the concentrations of VO and $\rm VO_{2}$ can be obtained as functions of the $\gamma$-ray dose and initial DD

Here $\kappa_{3}=k_{3}\rm O_{i}$ and $\kappa_{4}=k_{4}\rm Si_{i}/VO$ are effective reaction velocities of the transformation and annihilation processes of the defects in n-type Si, respectively, see Fig. 7 (a). The used initial conditions are $\rm VO(0)$ $=G$ and $\rm VO_{2}(0)=0$. The results indicate that, similar to the $\rm V_{2}$ defect, the concentration of VO will exponentially decay under ionization irradiation, and similar to $\rm V_{2}O$ defect, the concentration of $\rm VO_{2}$ will asymptotically increase and linearly decay as a function of the $\gamma$-ray dose.

The faded VO defect contributes an acceptor level $\rm E_{c}-0.17eV$ Svensson et al. (1991) and the generated $\rm VO_{2}$ defect is non-active in n-type silicon. Denoting the contributing factor of VO to the recombination current as $\lambda_{8}$, see Fig. 7 (a). The base current in PNP transistors due to VO and $\rm VO_{2}$ can be solved as a function of the $\gamma$-ray dose and initial DD. The net effect is found as an exponential decay of the initial base current ($f_{2}$) contributed by VO

see the magenta curve in Fig. 7 (c). Here $f_{2}=\lambda_{8}G$ is the measurable initial DD due to VO defect, $\eta_{2}=\kappa_{3}^{-1}\kappa_{4}/\lambda_{8}$ is the rate ratio between the annihilation and transformation reactions of VO in Si, and $D_{c}^{\prime}=\kappa_{3}^{-1}R$ is a characteristic dose of the transformation reaction of VO in Si.

Summing up Eqs. (9) and (12), we at last obtain the defect dynamics model for the negative ISE of the base current in PNP transistors

see the black curves in Fig. 7 (c).

Eq. (13) predicts that, there would be two independent terms in the ISEs of the PNP transistors: one linear decay term depending on the $\gamma$-ray dose, which stems from the ionization-irradiation-induced transformation and annihilation of $\rm V_{2}$ in n-type silicon; and one exponential decay term depending on the $\gamma$-ray dose, which stems from the ionization-irradiation-induced transformation and annihilation of VO in n-type silicon. Eq. (13) also predicts an almost linear dependence of the amplitude of the exponential decay on the initial DD due to VO ($f_{2}$) and a quadratic dependence of the slope of the linear decay term on the initial DD due to $\rm V_{2}$ ($f_{1}$). Eq. (13) also predicts a dose rate dependence of both terms of the ISE.

Following a similar way as the NPN transistor case (Sec. III C), the proposed model, Eq. (13), is testified by the data of PNP devices displayed in Sec. II.B. The fitting model is

Again, the first term in the fitting model contains both the linear decay term in the proposed physical model and the unknown linear ID; the presence of the linear decay term can be identified by the statistical difference between $k_{j}$ and $k_{0}$. The correspondence between the fitting parameters in Eq. (14) and physical parameters in Eq. (13) is $k_{j}-k_{0}=-\eta_{1}f_{1}^{2}/D_{c}^{j}$. The second term in Eq. (14) is exactly the exponential decay term in Eq. (13). The correspondence between the parameters are $A_{j}=(f_{2}+\eta_{2}f_{2}^{2})\sim f_{2}$ and $D_{c}^{j}=D_{c}\textquoteright$. It is noted that $A_{j}$ is exactly the initial base current due to the VO defects. The fitting profiles are plotted in Fig. 5 as solid curves, which are found to work very well for all samples. The dashed curves in Fig. 8 (b) show the linear component and the exponential decay term of a typical fitting curve of the PNP device. A significant decrease is found for the linear component (blue dashed) with respect to the average slope of pure ID (black dashed). These facts verify the predicted linear and exponential decay terms in the ISE of PNP transistors, reflecting the proper defect evolution physics in the prediction of the dose dependence of the ISE in PNP transistors.

The fitting parameters $k_{j}$ and $A_{j}$ are plotted in Figure 10 as a function of the base current, to further testify the initial DD dependence of the proposed model. containing the initial DD ($f_{1}+f_{2}$). The fitting characteristic dose ($D_{c}\textquoteright$) is not sensitive to the initial DD, and is 0.47 krad and 0.50 krad for the low and high dose rates, respectively. Note, these values are larger than those for the $V_{2}$ defects as shown in tab. 1, reflecting a slower transformation process for VO defects in Si. The data of high dose rate with the neutron fluence of $1\times 10^{12}\rm cm^{-2}$ is of low quality and are excluded from the fitting process. It is clear from Fig. 10 (b) that, the amplitude of the exponential decay term, $A=f_{2}$, shows an almost linear dependence on the initial DD, which is started from the pre-irradiation base currents, $\rm I_{B}^{0}\approx 14nA$, see the dashed guide lines. This dependence is consistent with the prediction of the proposed model. It is also found that for the low (high) dose rate $\gamma$-ray irradiation, the value of $A=f_{2}$ is only 1/38 (1/170) of the initial DD, $f_{1}+f_{2}$. This fact implies that the neutron-induced (annealable) VO defects are much fewer than the neutron-induced $\rm V_{2}$ defects. As can be seen in Fig. 10 (a), the slope of the linear term $k_{j}$ decreases as the initial DD increases. A quadratic dependence is found for the overall trend, $k_{j}=k_{0}-\alpha(f_{1}+f_{2})^{2}$. Since $f_{2}$ is much smaller than $f_{1}$, this result means $k_{j}=k_{0}-\alpha f_{1}^{2}$, which is also consistent with the model’s prediction. The value of $\alpha$ is $2\times 10^{-5}\rm nA^{-1}krad^{-1}$ for both low and high dose rates. The above results further verify the predicted initial DD dependence of the ISE in the proposed model, reflecting the proper physics of defect evolution underlying the model.

The significant difference between the red and blue dots in Fig. 10 reflects the dose-rate dependence of the ISE predicted by the model. The reason is that the ionization-induced defect transformation and annihilation becomes stronger for lower dose rate irradiation. This fact readily explains the dose rate dependence of the ISE in PNP transistors, which cannot be explained by the Coulomb interaction mechanism. Different from the NPN transistor case, the physical parameters cannot be calculated from the fitting parameters.