1. Introduction

A component's reliability is associated with the stability of its electrical parameters in different environments, among which the transistor's common-emitter current amplification factor, the current gain $h_{\rm FE}$, varies with the temperature and influences the performance and reliability of transistors. Thus it is significant to study and analyze the relationship between the current gain and the temperature. Research has shown that the temperature features of the current gain are mainly related to the following factors: the band gap narrowing caused by being heavily doped in the emitter region, the relationships between the mobility, other physical quantities and the temperature, the ideal factor of the base current is not 1^{[1, 2]}, the high current injection effect and other factors^[3]. Based on the studies above, in this paper, the ideal factor, $V_{\rm BE}$ and band gap changes with the temperature are studied and analyzed, and the current gain expression is corrected to make the results closer to the actual test.

In addition, the accelerating lifetime study method in the constant temperature-humidity stress is used to estimate the reliability of the same batch transistors. Applying the revised findings from the expression, the current gains before and after the test are compared and analyzed. Then, according to the degradation data of the current gain, the transistor lifetimes in the test stress are respectively extrapolated in the different failure criteria.

2. Correction of the current gain expression

2.1 Current gain theory model

It is known that the temperature model of current gain $h_{\rm FE}$ can be represented by the following formula^{[4, 5]}:

$\begin{equation} h_{\rm FE} =\frac{N_{\rm E} \mu_{\rm B} W_{\rm E} }{N_{\rm B} \mu_{\rm E} W_{\rm B} }\exp \left(\frac{-\Delta E_{\rm g} }{kT}\right). \end{equation}$

(1)

In Eq. (1), $N_{\rm E}$, $\mu_{\rm E}$, $W_{\rm E}$ are respectively the impurity concentration in the emission region, the minority carrier mobility and emission region width, $N_{\rm B}$, $\mu_{\rm B}$ and $W_{\rm B}$ are respectively the impurity concentration in the base region, the minority carrier mobility and the base region width, and $\Delta E_{\rm g}$ are the relative band gap narrowing values in the emission and base regions caused by the heavy doping^[5]. As the base region generally uses light doping, its band gap narrowing value can be negligible. Thus $\Delta E_{\rm g}$ is determined by the band gap narrowing value in the emission region.

It is known that the ideality factor n contains the non-ideal element impacts from the barrier region, the surface recombination current and the base current. The subtle difference between n and 1 may influence the transistor's electrical characteristics a little, so it is also not noticed and it is generally thought that $n=$ 1. However, the tests have verified the n's impact on the temperature characteristics $h_{\rm FE}$ cannot be ignored, even in the medium current range, the objective fact n $>$ 1 should be considered^[1]. Therefore, the impact of the ideal factor of the base current $n \neq$ 1 on the current gain needs to be considered, and its relationship between the current gain and the temperature can be converted into:

$\begin{equation} h_{\rm FE} =\frac{N_{\rm E} \mu_{\rm B} W_{\rm E} }{N_{\rm B} \mu_{\rm E} W_{\rm B} } \exp \left(\frac{-\Delta E_{\rm g} }{kT}\right) \exp \left[\left(1-\frac{1}{n}\right)\frac{qV_{\rm BE} }{kT}\right]. \end{equation}$

(2)

Equation (2) comprehensively reflects the impacts from both the heavy doping $\Delta E_{\rm g}$ and base current ideality factor n on the current gain. The band gap narrowing effect caused by the heavy doping in the emission makes $h_{\rm FE}$ have the significant positive-temperature coefficient characteristics. The base current ideality factor n is not equal to 1, which makes the $\exp [(1-\frac{1}{n})qV_{\rm BE} /kT]$ also have a negative-temperature coefficient and further weakens the impact of the band gap narrowing effect on $h_{\rm FE}$.

2.2 Test

By the test, the current gain features are analyzed. 3DK105B transistors are selected as the test samples, taking a temperature point at 10 K intervals (totally 19 temperature points) between 218 K and 398 K ($-65$ to 125 ℃) to test the current gain changes with the $I_{\rm C}$ and BE junction forward characteristics. The test conditions are: $V_{\rm CE}$ = 1 V and $I_{\rm C}$ = 10 mA. The test equipment is the semiconductor parameter analyzer KEITHLEY 4200-SCS. The test environment temperature is controlled within 25 $\pm$ 3 ℃, and the humidity within 50 $\pm$ 5% RH. The whole test is conducted in a shielded room in order to reduce the external interference on the test results. The temperature control equipment is the high precision incubator DESPATCH, whose temperature control accuracy can reach $\pm$0.5 ℃, using the liquid nitrogen refrigeration to realize the experimental low-temperature control.

2.3 Analyses

The parts of the $h_{\rm FE}$-$I_{\rm C}$ test results at the various temperature points are shown in Fig. 1.

Choosing $h_{\rm FE}$ when $I_{\rm C}$ = 10 mA and the temperature point T = 298 K (room temperature 25 ℃) as the datum point, the test results of the current gain are normalized, which is shown in Fig. 2.

By the craft of 3DK105B transistors, it is known $N_{\rm E}$/$N_{\rm B}$ = 100, and $W_{\rm E}$/$W_{\rm B}$ = 7/15. For the silicon devices, the relationship between its mobility and the temperature^[6] is: $\mu_{\rm B}$ $\propto$ $T-2.3 \pm 0.1$, $\mu_{\rm E}\propto T-2.2 \pm 0.1$. Thus it can be seen that $\mu_{\rm B}/\mu_{\rm E}=T-0.1$.

For the silicon material transistors, the $\Delta E_{\rm g}$ empirical formula^[7] is:

$\begin{equation} \Delta E_{\rm g} =3.4\times 10^{-8}\left(N_{\rm D} ^{\frac{1}{3}}-N_{\rm d} ^{\frac{1}{3}}\right). \end{equation}$

(3)

In Eq. (3), $N_{\rm d}$ $=$1.85 × l0$^{15}$ cm$^{-3}$, $N_{\rm D}$ means the donor impurity concentration.

In terms of the craft of 3DK105B transistors, we can get: $\Delta E_{\rm g}$ = 0.068 eV.

Through the transistor samples test, when choosing $I_{\rm C}$ = 10 mA, the corresponding n = 1.033 and $V_{\rm BE}$ = 0.575 V, which is brought into Eq. (2). The curve of the formula results is normalized (Fig. 3(b)), and is compared with that of the actual test results (Fig. 3(a)), as shown in Fig. 3.

When T = 398 K, the actual test normalized result $h_{\rm FE}(T)$/$h_{\rm FE}(T_{0})$ = 1.57, and Equation (2) normalized result $h_{\rm FE}(T)$/$h_{\rm FE}(T_{0})$ = 1.43.

When T = 218 K, the actual test normalized result $h_{\rm FE}(T)$/$h_{\rm FE}(T_{0})$ = 0.53, and Equation (2) normalized result $h_{\rm FE}(T)$/$h_{\rm FE}(T_{0})$ = 0.59.

From Fig. 3, it is seen that the higher the temperature, the larger the differences between the Eq. (2) results (n = 1.033, $V_{\rm BE}$ = 0.575 V) and the actual test ones. In order to more accurately reflect the current gain changes with the temperature, the effects of temperature changes on n and $V_{\rm BE}$ need to be considered.

With the current relationship of the PN junction $I=I_0 \exp \frac{qV_{\rm BE} }{nkT}$, the ideal factor $n=\frac{{\rm d}V}{{\rm d} \ln I}\frac{qV_{\rm BE} }{kT}$ can be achieved. According to the BE junction forward I-V feature curve, $I_{\rm B}$ and the corresponding $V_{\rm BE}$ are obtained, so the ideality factor n at different temperature points can be calculated. The changes of $V_{\rm BE}$ and n with the temperature are shown in Fig. 4.

Figure 5 respectively shows three normalized curves: one of Equation (2) considering $V_{\rm BE}$ and n changes with the temperature (Fig. 5(a)), one of the actual test results (Fig. 5(b)), and one of Eq. (2) not considering $V_{\rm BE}$ and n changes with the temperature (Fig. 5(c)).

At the different temperature points, the test conditions of current gain are $V_{\rm CE}$ = 1 V and $I_{\rm C}$ = 10 mA. For different temperature points, the corresponding base current $I_{\rm B}$ also changes. Therefore, n and $V_{\rm BE}$ also change with the temperature. It can be seen from Fig. 5, Eq. (2) results are much closer to the actual test (Fig. 5(a)), for it considers $V_{\rm BE}$ and n changes with the temperature (Fig. 5(b)).

2.4 Expression correction of current gain

In order to more accurately calculate the current gain changes with the temperature, the impact of the band gap changes with the temperature on the current gain needs to be considered. Equation (2) does not take this into account, so it cannot fully reflect the current gain changes with the temperature. Thus Equation (2) needs to be corrected. Reference ^[6] points out its empirical formula is:

$\begin{equation} \Delta E_{\rm g}'=\frac{\alpha T^2}{T+\beta }, \end{equation}$

(4)

where $\alpha$ = (4.73 $\pm$ 0.25) × 10$^{-4}$ eV/K, and $\beta$ = 636 $\pm$ 50 K. $\Delta E_{\rm g}'$ is the change amount caused by band gap narrowing changes with the temperature.

Considering the band gap changes with the temperature, the corrected Eq. (2) can be inverted into:

$\begin{equation} \begin{split} h_{\rm FE} = {} & \frac{N_{\rm E} \mu_{\rm B} W_{\rm E} }{N_{\rm B} \mu_{\rm E} W_{\rm B} }\exp \left(\frac{-\Delta E_{\rm g} }{kT}\right) \\ & \times \exp \left[\left(1-\frac{1}{n}\right)\frac{qV_{\rm BE} }{kT}\right] \exp \left(\frac{-\Delta E'_{\rm g}}{kT}\right). \end{split} \end{equation}$

(5)

Figure 6 respectively gives three normalized results: the actual test one (Fig. 6(a)), an Eq. (2) one considering $V_{\rm BE}$ and n changes with the temperature (Fig. 6 (b)), an Eq. (2) one not considering that (Fig. 6(c)) and an Eq. (4) one (Fig. 6(d)).

Because $\frac{N_{\rm E} u_{\rm B} W_{\rm E} }{N_{\rm B} u_{\rm E} W_{\rm B}}$ in Eq. (4) has a negative temperature coefficient, it partially compensates the heavily doped effects on the temperature features of bipolar transistors. The index item $\exp [(1-\frac{1}{n})\frac{qV_{\rm BE} }{kT}]$ considers two influences on the current gain: one is the ideal parameter of the base current n which is not 1, and the other is n and $V_{\rm BE}$ change with the temperature. The item $\exp (\frac{-\Delta E'_{\rm g}}{kT})$ considers the effect of band gap changes with the temperature on the current gain. Therefore, it can be seen from Fig. 6, that the Eq. (4) normalized curve (Fig. 6(d)) is closer to the actual test one (Fig. 6(a)).

When analyzing the temperature features of bipolar transistor current gain, such factors should be considered: the band gap narrowing caused by being heavily doped in the emitter region, the relationships between the mobility, the other physical quantities and the temperature, the impact of ideal factor n which is not 1, and the effects of the ideal factor, $V_{\rm BE}$ and the band gap changes with the temperature. Only when all these factors above are considered comprehensively, can the temperature features of the current gain be analyzed more accurately. In addition, the base region is much smaller than its minority-carrier diffusion length, whose impact on the current gain $h_{\rm FE}$ is also very important. However, this needs further study in the future, so it will not be illustrated here.

3. Application examples of current gain correction

Selecting 10 3DK105B bipolar transistors as the study object, which are all from the same batch as the previous tests, Equation (4) is tested. The findings show the normalized curve by Eq. (4) is much closer to the actual test one, which indicates the preceding conclusions have universality.

The test conditions are divided into three groups where: the constant temperature is 100 ℃ and the constant humidity is 100% RH; the constant temperature is 90 ℃ and the constant humidity is 90% RH; the constant temperature is 65 ℃ and the constant humidity is 95% RH. Every group uses 10 samples, and their cumulative test times are respectively 2800, 3800 and 4500 h. At a certain test interval, the electrical parameters of samples are tested. Combining the corrected current gain theory discussed above, after the analyses of current gain before and after the test, it is found that it tends to decline. Table 1 states the current gain values of 10 transistor samples before and after the test in the conditions where the constant temperature is 100 ℃ and the humidity is 100% RH.

Changes of transistor parameters are almost related to the temperature. In the function of the temperature, movable ionic charges in the oxide layer of interface Si-SiO$_{2}$ can both vertically and horizontally move, so that the number and location of negative charges in interface Si also change, which causes the drift of the current gain. Besides, interface traps can also simultaneously capture an electron and hole, playing the role of a recombination center, which results in the reduction of the current gain of the device^{[8, 9]}.

Choosing transistor 10 as the example, Table 2 shows the corresponding $h_{\rm FE}$ degradation data at various test times. Its degradation data is fitted (as shown in Fig. 7), and its lifetime in the temperature of 100 ℃ and humidity of 100% is extrapolated in different criteria.

Taking transistor 10 as the example, the corresponding degradation data of $h_{\rm FE}$ at the different test times are stated in Table 2, which are fitted (as shown in Fig. 7), and the samples' lifetimes at the various criteria are extrapolated in 100 ℃ and comparative 100% RH.

Taking a 25% decrease of the current amplification coefficient $h_{\rm FE}$ as the failure criteria, the lifetime of transistor 10 in the 100 ℃ and 100% RH stress is extrapolated. Table 3 states the extrapolated lifetimes results in the three tests.

The current amplification coefficient $h_{\rm FE}$ reduces by 20%, 25% and 30%, which is taken as the failure criteria to extrapolate the lifetime of Transistor 10 in 100 ℃ and comparative 100% RH stress, as shown in Table 3.

The maximum likelihood estimation method is used to estimate the parameters; the K-S goodness of fit test is applied to analyze the extrapolated lifetimes in Table 4. As shown in Table 4, the goodness of fit tests and parameter estimation in several common distributions are presented.

From Tables 3 and 4, it is seen that, in the test condition and failure criteria, the lifetimes of transistors are extrapolated from their sensitive parameters, which are within 10$^{3}$ to 10$^{4}$. Besides, in terms of the above, the current gain degradation conforms to the power law, which is similar to the results given by Chen, Bravo, Lloyd and others^[9-12].

4. Conclusion

In this paper, the current gain expression is corrected, which is also verified by the test, to make the results closer to the actual test. It is found in the research, the analyses of temperature features of bipolar transistor current gain should consider the following factors: the band gap narrowing caused by being heavily doped in the emitter region, the relationships between the mobility, the other physical quantities and the temperature, the impact of ideal factor n which is not 1, and the effects of the ideal factor, the $V_{\rm BE}$ and the band gap changes with the temperature. Besides, applying the revised findings from the current gain expression, the accelerating lifetime study method in the constant temperature-humidity stress is used to estimate the reliability of the same batch of transistors.