Switching-on Delay Jitter Caused by Lateral Distribution of Current Channel of Avalanche Transistor

Abstract

The stability of the avalanche transistor’s (AT’s) switching-on process is essential for its extensive application in power semiconductors. The switching-on process was typically described in one-dimensional terms, overlooking the effects of multi-dimensional structural variations on stability. This paper investigated the influence of the lateral distribution of current channels on the switching-on delay jitter in the AT. The lateral size of the current channel affects the transit time by changing the electron path in the base region, resulting in the switching-on delay jitter of the AT. An analytical formula for the lateral size of the current channel and the switching-on delay jitter has been proposed. The two-dimensional simulation model of the AT gave the distribution of current channels. The model’s accuracy was verified by comparing experimental and simulation data. The experimental data proved that the base transit time was the main component of the switching-on delay. The results show that the switching-on delay jitter can be significantly reduced by adjusting the current channel’s lateral size. In addition, the trigger signal’s characteristics also change the current channel’s lateral distribution and then affect the stability of the switching-on delay, which provides a new perspective for the design and application of ATs.

1. Introduction

High voltage pulse power supply had broad applications in food non-thermal pasteurization [1], electro-chemotherapy [2], ground-penetrating radar [3], medicine [4], plasma [5], and ultra-wideband communication systems [6]. It is an effective method to develop such pulse power supply based on power semiconductor devices. Avalanche transistor is a commonly used power semiconductor device [7] whose switching process has to undergo avalanche breakdown and secondary avalanche breakdown [8,9]. This unique breakdown mechanism gives avalanche transistors a fast on-off capability of nanoseconds and even sub-nanoseconds [10,11,12]. Therefore, the avalanche transistor can develop nanosecond pulse generation equipment with kilovoltage amplitude.

The rated pulse current of a silicon-based avalanche transistor is usually less than 100 A. To improve the output power of the pulse-generating equipment, researchers generally adopted methods such as circuit power synthesis [13], spatial power synthesis [14], or avalanche transistor parallel [15]. These methods have taken advantage of the avalanche transistor’s excellent stability, which refers to the almost unchanged delay period between the trigger signal’s start and the transistor’s switching-on instant. Therefore, studying the variation law of delay jitter (switching-on delay jitter in this paper) is vital. Literature [16] combined semiconductor equations and finite element methods to achieve a one-dimensional simulation of avalanche transistor conduction characteristics and thoroughly described the physical process of secondary breakdown but ignored the changes of parameters in the two-dimensional structure (primarily the lateral direction) during the conduction process. In literature [17], a two-dimensional simulation model of avalanche transistors was established using ATLAS (version 5. 0. 10. R) software to simulate the current gathering edge and concentration effects in the avalanche process. Researchers have given the lateral size of the conducting current. Still, the influence of such lateral current distribution on the switching-on characteristics was unclear. Literature [18] studied the secondary breakdown process of avalanche transistors at a two-dimensional scale and put forward the view that the base width will significantly affect the switching-on delay. Literature [19] pointed out that the width of the base region was the main factor determining the switching-on delay of avalanche transistors, and the delay jitter was mainly affected by the impact ionization of electrons in the collector depletion layer. However, researchers only derived the relevant analytical formula under one-dimensional coordinates. According to the literature [20], the switching-on delay of the avalanche transistor triggered by the base injection current is much more significant than the switching-on delay of the avalanche transistor triggered by the collector overvoltage ramp. This difference may provide a new perspective for studying the stability of the pulse sources based on avalanche transistors.

This paper proposes that the lateral size of the current channel will change the path of electrons through the base region, affect the base transit time of the electron, and cause the switching-on delay jitter. This paper has given the analytical formula for the change of switching-on jitter with the lateral size of the current channel without considering the randomness of electron impact ionization in the collector region. A two-dimensional model of an avalanche transistor has been established, and the variation of the lateral size of the current channel with trigger signal was simulated. The results show that increasing the base doping concentration gradient and decreasing the current channel’s lateral size can reduce the avalanche transistor’s switching-on delay jitter.

2. Theoretical Analysis

2.1. Principle Analysis of Switching-on Delay Jitter

Due to a much more significant delay, the avalanche transistor triggered by the base injection current tends to cause much more delay jitter. Therefore, this paper only analyzed the switching-on delay jitter in this case. When the avalanche transistor plays a power semiconductor switch, there is a high voltage between the emitter and the collector in the initial state, and both the emitter and collector junction are in the cut-off state. When the pulse trigger loads to the base, holes flow into the emitter from the base, and then many electrons enter into the base region from the emitter region. Due to the emitter current gatherer effect, the electron current channel mainly appears at the interface between the emitter and the base, as shown in Figure 1. Subsequently, these electrons cross the base region and enter the collector depletion layer due to drift and diffusion motion. Under the high electric field in the depletion, the electrons generate impact ionization, generating an intense avalanche current and making the avalanche transistor switch on [21].

In the switching-on process of the avalanche transistor, electrons from emitter to collector mainly go through four time periods of t_e, t_b, t_c, and t_cc [22], as shown in Figure 1. The variable t_e represents the charge and discharge time of the emitter junction barrier capacitor. The variable t_b represents the base transit time of electrons. The variable t_c represents the delay time for electrons to pass through the collector junction depletion layer. The variable t_cc represents the charge and discharge time of the junction barrier capacitor. The longitudinal size of avalanche transistors is generally in the order of microns or more, and the width of the base region of the conventional avalanche transistor is about 2 μm. The cross-sectional area of the p-n junction is about 0.01 cm². According to relevant formulas in references [21,22], the base transit time of the electron in the avalanche transistor is about 250 ps. The charge and discharge time of the PN junction barrier capacitor is about 10 ps. Therefore, the variable t_e and the variable t_cc are negligible compared to the variable t_b. The strong electric field in the collector junction depletion layer makes the drift velocity of electrons much larger than the average drift velocity of electrons in the base region. The longitudinal size of the n₀ region is much larger than the longitudinal size of the p region. Hence, the difference between the longest path and the shortest path of the electron current channel in the n₀ region is not significant, so the jitter of the variable t_c will be small. However, the lateral dimension of the electron current L_lateral is comparable to or even larger than the longitudinal width of the thin base region W_base. The longest and shortest paths of the electron current channel in the base region are very different, resulting in a significant jitter of the variable t_b. Therefore, the lateral distribution of the electron current channel becomes the main factor affecting the switching-on delay jitter of the avalanche transistor (mainly refers to the variable t_b) when the noise of the peripheral bias circuit is negligible and the trigger signal is changeless.

2.2. Base Transit Time in One-Dimensional Case

According to [23,24], the base transit time t_b of an electron is equal to the non-equilibrium minority electron charge divided by the minority electron current in the base region, which can be calculated as follows.

$$t_{\mathrm{b}}=q{\displaystyle {\int }_0^{W_{\mathrm{base}}}\frac{n_{\mathrm{B}}(x)}{J_{\mathrm{n}}(x)}\mathrm{d}x}$$

(1)

where q is the electron charge. n_B(x) is the electron concentration distribution in the base region. J_n(x) is the electron current density. The origin of the x coordinate is at the base region near the emitter junction, and the direction is longitudinal downward.

The impurity concentration distribution in the base region can be approximated by

$$N_{\mathrm{B}}(x)=N_{\mathrm{B}}(0)\exp (-\frac{\mathsf{\eta }x}{W_{\mathrm{base}}})$$

(2)

The constant coefficient η represents the doping gradient of impurity concentration.

According to [25,26], the electron current density is

$$J_{\mathrm{n}}(x)=-q{\mathsf{\mu }}_{\mathrm{n}}{n}_{\mathrm{B}}(x)E(x)-q{D}_{\mathrm{B}}\frac{\mathrm{d}{n}_{\mathrm{B}}(x)}{\mathrm{d}x}$$

(3)

where μ_n is the electron drift coefficient and D_B is the electron diffusion coefficient. E(x) is the electric field built in the base region.

In the equilibrium state, J_n(x) is zero. The expression of the base built-in electric field can be obtained using Einstein relation as follows.

$$E(x)=\frac{kT}{q}\frac{1}{p_{\mathrm{B}}(x)}\frac{\mathrm{d}{p}_{\mathrm{B}}(x)}{\mathrm{d}x}=\frac{kT}{q}\frac{1}{N_{\mathrm{B}}(x)}\frac{\mathrm{d}{N}_{\mathrm{B}}(x)}{\mathrm{d}x}$$

(4)

where k is the Boltzmann constant, T is the Kelvin temperature, and p_B(x) is the hole concentration distribution in the base region.

The expression of J_n(x) is obtained by combining (3) and (4).

$$\begin{array}{l}{J}_{\mathrm{n}}(x)=-\frac{q{D}_{\mathrm{B}}}{N_{\mathrm{B}}(x)}[n_{\mathrm{B}}(x)\frac{\mathrm{d}{N}_{\mathrm{B}}(x)}{\mathrm{d}x}+N_{\mathrm{B}}(x)\frac{\mathrm{d}{n}_{\mathrm{B}}(x)}{\mathrm{d}x}]\\ =-\frac{q{D}_{\mathrm{B}}}{N_{\mathrm{B}}(x)}\frac{\mathrm{d}[n_{\mathrm{B}}(x)N_{\mathrm{B}}(x)]}{\mathrm{d}x}\end{array}$$

(5)

Equation (5) is integrated from x to W_base. Due to the thin base region of the avalanche transistor, the recombination of electrons in the base region is very small, so J_n(x) can be regarded as a constant when integrating. This results in the following expression

$$J_{\mathrm{n}}(x)=\frac{q{D}_{\mathrm{B}}{n}_{\mathrm{B}}(x)N_{\mathrm{B}}(x)}{{\displaystyle {\int }_x^{W_{\mathrm{base}}}{N}_{\mathrm{B}}(x^{\prime })\mathrm{d}{x}^{\prime }}}$$

(6)

$$n_{\mathrm{B}}(x)=\frac{J_{\mathrm{n}}(x)}{q{D}_{\mathrm{B}}{N}_{\mathrm{B}}(x)}{\displaystyle {\int }_x^{W_{\mathrm{base}}}{N}_{\mathrm{B}}(x^{\prime })\mathrm{d}{x}^{\prime }}$$

(7)

By combining (2) and (7), the electron concentration distribution in the base region can be obtained as follows

$$n_{\mathrm{B}}(x)=\frac{J_{\mathrm{n}}(x)W_{\mathrm{base}}}{q{D}_{\mathrm{B}}}\frac{1-\exp [-\mathsf{\eta }(1-\frac{x}{W_{\mathrm{base}}})]}{\mathsf{\eta }}$$

(8)

By substituting (8) into (1), the expression of the base transit time of the electron is obtained as follows

$$t_{\mathrm{b}}=\frac{W_{\mathrm{base}}^2}{D_{\mathrm{B}}}\frac{1}{\mathsf{\eta }}[1-\frac{1}{\mathsf{\eta }}+\frac{\exp (-\mathsf{\eta })}{\mathsf{\eta }}]$$

(9)

When the base region is uniformly doped, η approaches 0. According to L’Hôpital’s rule, the expression of base transit time is as follows

$$\underset{\mathsf{\eta }\to 0}{\lim }{t}_{\mathrm{b}}=\frac{W_{\mathrm{base}}^2}{2D_{\mathrm{B}}}$$

(10)

According to (9) and (10), it can be seen that when the base region is uniformly doped, there is no built-in electric field in the base region that accelerates the electron drift, so the base region transit time of the electron is the largest in this case.

2.3. Base Transit Time in Two-Dimensional Case

In the two-dimensional case, due to the lateral distribution of the current, the path of electrons across the base region becomes L_electron, and its size ranges are as follows

$$W_{\mathrm{base}}\le L_{\mathrm{electron}}\le \sqrt{W_{\mathrm{base}}^2+{(0.5L_{\mathrm{lateral}})}^2}$$

(11)

Therefore, according to (9)–(11), the base transit time of the electron in the two-dimensional case becomes as follows

$$\{\begin{array}{c}{t}_{\mathrm{b}}=\frac{L_{\mathrm{electron}}^2}{D_{\mathrm{B}}}\frac{1}{\mathsf{\eta }}[1-\frac{1}{\mathsf{\eta }}+\frac{\exp (-\mathsf{\eta })}{\mathsf{\eta }}]\cdots \cdots \mathsf{\eta }>0\\ t_{\mathrm{b}}=\frac{L_{\mathrm{electron}}^2}{2D_{\mathrm{B}}}\cdots \cdots \mathsf{\eta }=0\end{array}$$

(12)

The base transit time range of the electron can be obtained from (11) and (12)

$$\{\begin{array}{c}\frac{W_{\mathrm{base}}^2}{D_{\mathrm{B}}}M\le t_{\mathrm{b}}\le \frac{W_{\mathrm{base}}^2+{(0.5L_{\mathrm{lateral}})}^2}{D_{\mathrm{B}}}M\cdots \cdots \mathsf{\eta }>0\\ \frac{W_{\mathrm{base}}^2}{2D_{\mathrm{B}}}\le t_{\mathrm{b}}\le \frac{W_{\mathrm{base}}^2+{(0.5L_{\mathrm{lateral}})}^2}{2D_{\mathrm{B}}}\cdots \cdots \mathsf{\eta }=0\end{array}$$

(13)

The variable M is shown below

$$M=\frac{1}{\mathsf{\eta }}[1-\frac{1}{\mathsf{\eta }}+\frac{\exp (-\mathsf{\eta })}{\mathsf{\eta }}]$$

(14)

The variation of t_b and its jitter with the lateral distribution of the avalanche transistor current channel is shown in Figure 2. When η remains constant, t_b and its jitter increase with the increase in L_lateral. When the L_lateral remains unchanged, t_b and its jitter decrease with the increase in η, which is caused by the increase in the base built-in electric field that accelerates the electron drift process.

3. Simulation and Experiment

To quantitatively study the size of the lateral distribution of the current channel and estimate the switching-on delay jitter of the avalanche transistor, the mix-mode model of a five-stage Marx circuit built in ATLAS software, as shown in Figure 3. Parameters of the avalanche transistor Q_1–5 are from FMMT417 of DIODES Inc. (Plano, TX, USA), whose doping and structure references [9,17]. The doping concentration of the emitter n⁺ region and collector n⁺ region is 1 × 10¹⁸ cm⁻³. The doping concentration of the base region is 2 × 10¹⁷ cm⁻³. The width of the base region is 2 μm. The doping concentration of the n₀ region of the collector is 5 × 10¹⁴ cm⁻³. The width of the n₀ region of the collector is about 13 μm. Other circuit components adopt PSPICE models. DC voltage source V_DC is 300 V, charge resistance R_Un, and ground resistance R_Ln is 100 kΩ, R_BE is 750 Ω, C_n is 1 nF, and load R_load is 50 Ω. l_B, l_E, and l_C are lead inductors of each electrode. The PSPICE model of the avalanche transistor gives the reference value of lead inductance. l_B, l_E, and l_C are 2 nH, 2 nH, and 1 nH, respectively. The rectangular trigger pulse introduced from both ends of the R_BE has a rising edge of 1 ns, an amplitude of 10 V, and a pulse width of 50 ns. The simulation applied the impact ionization model proposed by Selberherr [27]. A pulse source based on the circuit shown in Figure 3 was developed during the experiment. As shown in Figure 4, the output characteristics of the pulse source are tested. The blue box in the circuit board shown in Figure 4 identifies the grounding panel. Resistances R_L1~5 can be directly grounded nearby. As for the part marked by the red box, it is a microstrip line of about 5 cm in length. Of course, we also compared the output pulse waveform when the microstrip line length was only 1 cm. Compared to the high-frequency microwave PCB (printed circuit board) with a low-loss tangent, the results show that the output pulse amplitude of the PCB with a longer microstrip line decreased by less than 1%, and the rising edge increased by no more than 10 ps. In addition, the most important reason for using a longer microstrip line here is to optimize the output pulse waveform through impedance transformation.

The circuit simulation and experimental results are shown in Figure 5. The comparison of the load waveform shows that the simulation error of the pulse width is about 6.5%, and the simulation error of the waveform peak value is about 1.4%. So, the model has a high simulation accuracy. It is highly reliable to extract the data of semiconductor parameters in the model and explore the conduction mechanism of each avalanche transistor. Figure 6 shows the differential voltage V_CE variation curve with time for each avalanche transistor. The delay of the base trigger to the switching-on instant of Q₁ is about 1.5 ns, corresponding to the period between point 1 and point 2. In the above period, the V_CE of avalanche transistor Q_2–5 increases with time, and no violent avalanche occurs. Then, at almost the same moment, the V_CE of Q_2–5 decreases (as shown in points 3 to 6), and the avalanche process occurs. Therefore, the switching-on delay jitter of Q₁ has the most significant influence on the stability of the pulse source.

Figure 7 shows the electron current distribution simulation data when avalanche transistor Q₁ switches on. As

Table 1. The variations of L_lateral with the trigger signals of different amplitudes and rising edges.

Trigger Signal	Llateral	Switching-on Delay Jitter
#1	2.1 μm	30~71 ps
#2	2.4 μm	39~92 ps
#3	2.7 μm	49~117 ps

shows, the L_lateral is 2.4 μm here. According to the base doping concentration curve given in [9,18], η is calculated to be about 10, and the base width W_base is 2 μm. According to ATLAS’s concentration-dependent low-field mobility model and the doping concentration variation range of the base region, the variation range of electron diffusion coefficient D_B is 14 cm²/s~34 cm²/s. According to (13), the transistor switching-on jitter ranges from 38 ps to 92 ps. In contrast, the jitter of FMMT417 is about 50 ps by counting tens of thousands of waveforms in the experiment as in Figure 4. Because the diffusion coefficient varies with the base doping concentration, the jitter calculated theoretically is an interval value. Nevertheless, the above results confirm that the lateral distribution of current channels in the base region is essential to cause the switching-on delay jitter of the avalanche transistor.

4. Discussion

4.1. Trigger Signal Changes Switching-on Delay Jitter

Previous studies have pointed out that trigger signal rise slope dV/dt will affect the avalanche intensity of transistors [19], thus changing the switching-on delay jitter. Here, the trigger signal rise slope dV/dt is set unchanged, and whether trigger signals with different amplitudes and rising edges will change the lateral distribution of electron current channels is studied. Amplitudes of trigger signal #1~3 are 5 V, 10 V, and 15 V, respectively, and the rising edge is 0.5 ns, 1.0 ns, and 1.5 ns, respectively. With the lateral distribution of electron current density obtained by simulations when the avalanche transistor switches on, the curves of the variation of current density with the lateral coordinate at the longitudinal coordinate 4.6 μm are shown in Figure 8. The statistics are in Table 1.

As Table 1 shows, after the trigger signal #1~3 is applied, the current lateral widths L_lateral are 2.1 μm, 2.4 μm, and 2.7 μm, respectively. Even if the rise slope dV/dt holds constant, the change of the trigger signal’s amplitude and the rising edge will change the lateral distribution of the current channel, thus affecting the switching-on delay jitter of the avalanche transistor. Since D_B changes with the doping concentration of the base region, the calculated value of t_b here should be a value in a range. So, the value of the switching-on delay jitter is also in a range. According to the previous analysis and the data in Table 1, the switching-on delay jitter of the avalanche transistor FMMT417 may increase by 30~76 ps for every 1 μm increase in the lateral size of current density.

4.2. Potential Distribution Makes Lateral Distribution of Electron Current

The lateral distribution of the current channels can be explained by the potential distribution when the avalanche transistor switches on. As can be seen from the distribution of the potential when the avalanche transistor switches on in Figure 9. The distribution of the potential when the avalanche transistor switches on., there is a lateral electric field E_b in the base region, which causes electrons to diffuse to the right when passing through the base region, and the repulsion between electrons causes the current channel to diffuse to the left. The formation of electron current channels is the result of the two diffusion actions mentioned above. The higher the electron charge density in the base region, the more obvious the repulsion effect between charges. The larger the amplitude of the base trigger signal, the greater the diffusion effect of E_b. Therefore, trigger signals with different amplitudes and rising edges will change the lateral distribution of electron current.

As shown in Figure 9. The distribution of the potential when the avalanche transistor switches on., a lateral electric field E_b exists in the avalanche transistor’s on-base region. This electric field causes electrons to spread to the right as they pass through the base region, and the repulsion between electrons causes the current channel to spread to the left. The formation of electron current channels results from the two diffusion actions mentioned above. The higher the electron charge density in the base region, the more pronounced the repulsion effect between charges. The larger the base trigger signal’s amplitude, the more significant the diffusion effect by E_b. Therefore, as the simulation results shown in Table 1, trigger signals with different amplitudes and rising edges will change the lateral distribution of electron current.

5. Conclusions

This paper uses theoretical analysis, simulation, and experiments to study the influence of current channel lateral distribution on the switching-on delay jitter of avalanche transistors. It is found that the switching-on delay jitter of the avalanche transistor is mainly affected by the transit time of electrons in the base region. The transit time is closely related to the lateral size of the current channel. An analytical formula is presented to describe the relationship between the switching-on delay jitter and the lateral size of the current channel, and a two-dimensional model is established to simulate this phenomenon.

The simulation results show that the switching-on delay jitter of the avalanche transistor can be effectively reduced by increasing the base doping concentration gradient and decreasing the current channel’s lateral size. The experiment verifies the accuracy of the simulation model and further proves the importance of lateral distribution to the switching-on delay jitter. In addition, the influence of trigger signal amplitude and rising edge on the lateral distribution of the current channel is also discussed. It is pointed out that trigger signals with different amplitudes and rising edges will cause the lateral distribution of the current channel to change even if the rising slope of the trigger signal remains constant and then affects the switching-on delay jitter.

In conclusion, it can be concluded that the base transit time of electrons is an essential factor of the switching-on delay jitter of the avalanche transistor, and the lateral distribution of the current channel in the two-dimensional structure should be noticed. The research in this paper provides a necessary theoretical basis and practical guidance for understanding and controlling the switching-on delay jitter of avalanche transistors in high-power applications. It is also beneficial to improve the stability and performance of related electron equipment.