Properties of the transient of avalanche transistor ... - IEEE Xplore

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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 49, NO. 1, JANUARY 2002

Properties of the Transient of Avalanche Transistor Switching at Extreme Current Densities Sergey N. Vainshtein, Valentin S. Yuferev, and Juha T. Kostamovaara, Member, IEEE

Abstract—Avalanche transistor switching at extreme currents is studied under conditions in which the charge of the excess carriers drastically rebuilds the collector field domain, causing fast switching and a low residual voltage across the switched-on device. The dynamic numerical model includes carrier diffusion and considers different dependencies of the velocities and ionization rates for the electrons and holes on the electric field. These dependences determine the principal difference in the switching process between n+ -p-n0 -n+ and p+ -n-p0 -p+ structures. Reasonably good agreement is found between the simulated and measured temporal dependences of the collector current and voltage drop across the device for a particular type of avalanche transistor. Certain differences in the switching delay can partly be attributed to limitations in the one-dimensional (1-D) approach. It is now certain that collector domain reconstruction defines the transient in a n+ -p-n0 -n+ transistor at high currents, but is not very pronounced in a p+ -n-p0 -p+ transistor. Some nontrivial features of the device operation are found, depending on the semiconductor structure. In particular, it is shown that the thickness of the low-doped collector region affects mainly the switching delay, and does not significantly effect the current rise time. Index Terms—Avalanche breakdown, microwave switches, semiconductor switches.

I. INTRODUCTION

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HERE is a need to generate current pulses of a few nanoseconds in length with an amplitude of 10–10 A across a low-ohmic load for certain commercial and laboratory applications. This concerns particularly the pumping of high-power, broad-stripe laser diodes for laser radars and other systems [1], [2]. One of the simplest, cheapest, and most effective ways is to make use of high-voltage 300 V avalanche transistors [3], [4]. Surprisingly, the switching transient for such transistors in a high-current mode has not been analyzed so far, in spite of the fact that these devices have been in use for many years. One specific feature of this mode from a practical point of view is the reduction in the residual

Manuscript received June 1, 2001; revised September 25, 2001. This work was supported in part by the Academy of Finland (Project 50460) and in part by INTAS Project 97-OPEN-1609. The review of this paper was arranged by Editor J. N. Burghartz. S. N. Vainshtein is with the Electronics Laboratory/Department of Electrical Engineering, University of Oulu, Linnanmaa SF-90570, Oulu, Finland, on leave from the A. F. Ioffe Institute of the Russian Academy of Science, St. Petersburg 194021, Russia (e-mail: [email protected]). V. S. Yuferev is with the A. F. Ioffe Institute of the Russian Academy of Science, St. Petersburg 194021, Russia, and also with the Electronics Laboratory/Department of Electrical Engineering, University of Oulu, Linnanmaa SF-90570, Oulu, Finland (e-mail: [email protected]). J. T. Kostamovaara is with the Electronics Laboratory/Department of Electrical Engineering, University of Oulu, Linnanmaa SF-90570, Oulu, Finland. Publisher Item Identifier S 0018-9383(02)00221-6.

voltage across the switched-on device by a factor of two or three, as compared with the prediction given by a simple model for an avalanche transistor [5]. From the physical point of view, the most essential fact is that the carrier concentration in the low-doped collector should exceed the doping concentration by about two orders of magnitude at extreme currents. This means collector of the that the domain of the electric field in the switched-on transistor can no longer be defined by the donor concentration. The analyses in the literature concerning the rebuild of the field domain in an avalanche transistor [6], [7] were performed within a quasi-static approach and the simplifications used in them lead to erroneous conclusions in some important cases, which will be discussed in Sections IV-C and D. The numerical simulations of the voltage–current characteristics for an n -p-n -n structure performed in [8] show the possibility for reducing the residual voltage at high current densities. No electric field distribution is shown, however, and its relation to the electron and hole current components is not discussed. Moreover, the rough approximations used for the carrier velocities and ionization rates presumably do not allow an entirely correct description of the collector field domain to be obtained. Switching of reverse-biased diodes has been considered without carrier injection [9] and with a weak injection [10], but the voltage reduction across the diode was not especially significant, since avalanche multiplication in the final state took place near both sides of the base region. Avalanche injection in a bulk semiconductor, as considered in [11], could be related somehow to the processes in the -collector, but the injection contact approximation used is not applicable to the transistor as such. Rebuilding of the electric field domain in the base region of a reverse-biased diode at extreme currents is shown in [12]. The electric field distribution presented in this paper is qualitatively similar to that expected in the collector of an avalanche transistor with a strong electron injection from the emitter. Unfortunately, the full set of boundary conditions is not specified in the paper, so that we cannot evaluate whether the analyses presented in [12] for a diode structure is related to the processes taking place in the collector of an avalanche transistor. Thus, to the best of our knowledge, not only dynamic, but even a reliable quasi-static consideration of the avalanche transistor at extreme currents is lacking in the literature. The purpose of this paper is to present the results of one–dimensional (1-D) dynamic simulations of an avalanche transistor at extreme current densities, to analyze physical processes in the structure during the switching transient, and to compare the simulation results with the experiment.

0018–9383/02$17.00 © 2002 IEEE

VAINSHTEIN et al.: TRANSIENT OF AVALANCHE TRANSISTOR SWITCHING

II. MODEL Carrier transport in a bipolar transistor is, in principle, multidimensional. It will be shown, however, that the switching transient of an avalanche transistor at extreme current, when described in the framework of a 1-D dynamic approach, provides reasonably good agreement with the experimental data. An essential problem is how to induce an external (base) current in a 1-D model. The approach used here implies a source of the majority carriers that is uniformly distributed across the base. The areas of the collector, base, and emitter are considered cm for the selected transistor type), to be equal (each 10 and the base current is treated as a generation of the majority carriers in the base region. The base current value in the external circuit is comprehended as the generation rate, integrated over the base volume. The equations governing an – – – structure with the emitter on the left-hand side can be presented as follows with the carrier recombination neglected: (1a)

(1b)

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ment current (2a) was used to calculate the electric field instead of the Poisson (1e). Such a modification was very important at extreme current densities, when a quasi-neutral domain was formed, since the right-hand side of the Poisson equation presents a small difference of large magnitudes in this case (2a) if if if

belongs to the collector belongs to the base belongs to the emitter.

(2b)

[This expression can easily be obtained by differentiating (1e) with respect to the time, substituting (1a) and 1(b) into it and integrating the resulting equation with respect to .] The last term in (2a) corresponds to the hole injection due to the base current. The total current density in the collector is equal to that through the collector contact, and the current density in the base increases from its value in the collector to that passing through the emitter contact. It is worth noting that the donor and acceptor concentrations do not enter directly to (2a). Thus, the use of this equation instead of (1e) requires the following conservation condition to be satisfied for each instant at each value of :

(1c) (2c) (1d) (1e)

The same carrier velocities as in [13] and the same ionization rates as in [12] were used

where and

electron and hole concentrations, respectively; and fluxes; electron charge; base current; base thickness; transistor area; , and , velocities and diffusion coefficients of the electrons and the holes; and electron and hole ionization rates; dielectric constant in Si; vacuum dielectric constant; and densities of the donors and acceptors, respectively; electric field taken with the opposite sign. Here, the positive direction for electron (hole) motion is from in (1b) is left to right (from right to left). The function equal to unity if belongs to the base, and zero otherwise. The neglecting of carrier recombination seems to be a reasonable simplification for a switching transient of a few nanoseconds’ duration. A significant improvement in the convergence of the iterative process was achieved when an equation for the displace-

(3)

(4) cm/s; cm/s; V/cm; V/cm; ; ; correspondingly the ratio of the mobilities ; V/cm; V/cm; /cm; /cm. Equations (1a)–(1e) and (2a)–(2c) requires initial and boundary conditions. The carrier and electric field distributions established under constant reverse biasing with the leakage current neglected were used as initial conditions, and the equilibrium values for the electron and hole densities at the contacts as the boundary conditions for (1a)–(1e). When a p -n-p -p transistor was simulated, the semiconductor structure was “inverted” and (1a)–(1e) and (2a)–(2b) adapted accordingly. The problem can now be defined if one prescribes the values for the collector and base currents, which are provided by an external circuit in our simulations.

where

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III. TRANSISTOR STRUCTURE AND EXTERNAL CIRCUIT A detailed comparison of the simulation results with the experiment requires precise information concerning the transistor structure and the external circuit. Neither problem is entirely trivial, and both need certain comments. Unfortunately, we do not know all the details of the structure from the transistor manufacturer, and most of the parameters had to be evaluated from special measurements. Then, it is essential to have precise information on the parasitic inductances, since these drastically affect the transient with a characteristic time of 1 ns in a circuit with an impedance of 1 . A commercial transistor (FMMT417, ZETEX SEMICONDUCTORS) was used in our experiments and simulations. The donor profile in the collector region was found from the capac) characteristics. The acceptor concentraitance–voltage ( cm in the -base was evaluated from tion the breakdown voltage of the emitter–base p-n junction. Measurement of the electron lifetime in -base by the Lax method and its comparison with the current gain allowed the thickness 3 m of the quasi-neutral part of the base region to be evaluated (the sign “ ” is used here since the injection coefficient is not known exactly). It was assumed in the simulations that the quasi-neutral part of the base region was of a thickness of 2.5 m, and that exponential reduction in the acceptor concentration occurred within a 1.5 m transition region toward the collector. Finally, the donor concentrations in both the emitter and the n collector layer were voluntarily assumed to cm . be The doping profile across the transistor structure, used in the simulations, and the external circuit with parasitics, used both in the simulations and in the experiment, are shown in Fig. 1. The values of the parasitic inductances indicated in the figure were carefully verified in the independent experiments. The transistor in these experiments was replaced by practically identical waxwork with a short connection between the emitter and the collector, and the current oscillations were excited by a mercury relay (200 ps pulse rise time) in the whole circuit and in various of its fragments with additional capacitances induced. Finally, the parasitic inductances for the total circuit and its fragments were estimated from the oscillation period in each of the experiments. We evaluate the accuracy of the inductance values shown in Fig. 1 as being better than 0.1 nH. The following equations for the external circuit were used in the simulations: (5a) (5b) where the total inductance of the collector circuit includes the parasitic inductances of the , the parasitic inducemitter and collector leads and the soldering connections , tance of the capacitor (see and the parasitic inductance of the load resistor is the collector current; is the load resistor; Fig. 1); is the capacitor in the collector circuit; and are the

Fig. 1. Doping profile assumed for the transistor FMMT417, and the external circuit. The scatter graph shows the donor profile in the collector calculated from the C V measurements.

0

voltages between the collector–emitter and the base–emitter ohmic contacts, respectively (5c) and are the base current and the resistor in the base ciris cuit, respectively. The effect of the emitter inductance neglected in the base circuit because of the high value of the re100 . An analytical approximation for the trigsistor used in the simulations is similar to that in gering pulse the experimental setup. for (5a)–(5c) are The initial conditions (6) Three waveforms were measured in the experiment for comparison with the results of the simulations (see Fig. 1): (Ch. 1 of the oscilloscope); 1) (differential probes Ch. 2, 3); 2) (Ch. 4). 3) was then extracted by an iteration The collector current , with taking account of the values procedure from and . The collector–emitter voltage in the experiment was calculated from the measured waveforms, neglecting (7) The finite volume method was used to obtain a numerical solution in combination with exponential approximation of the carrier fluxes and quasi-linearization of the nonlinear terms related to the dependences of the carrier velocities on the electric field. Highly nonuniform grids were used, and each grid was


Fig. 2. Comparison of the measured (solid lines) and simulated (dashed lines) current and voltage waveforms for two values of the initial bias voltage U (V): (a) 200 and (b) 290.

subdivided in the vicinity of the collector and emitter junctions, and also in the right-hand part of the -collector, where the impact ionization is most intensive. Then, when using (2a) for calculating the electric field, it is necessary to control the residual in the Poisson equation (2c), which did not exceed a value of Coulomb/cm in our simulations. The numerical scheme 10 was elaborately verified and tested. IV. SIMULATIONS AND EXPERIMENTAL RESULTS

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Fig. 3. (a), (c) Electric field and (b), (d) carrier density distributions across the p-base (x = 4÷0 m) and in the n-collector (x = 0÷16 m) regions at various instances (t = 0÷9 ns). Electron (solid lines) and hole (dotted lines) concentration are presented in (b) and (d), respectively. The profiles correspond to the simulated current and voltage waveforms shown in Fig. 2(a).

0

voltages across the transistor in the simulation and experiment during the transient do not differ by more than 6 V if the simulated waveforms are shifted to the right by 1.7 ns. This agreement is surprisingly good, bearing in mind the accuracy of our knowledge on the doping level in the emitter and the details of the doping profile near the base–collector junction.

A. Comparison of the Simulated and Experimental Waveforms

B. Electric Field and Carrier Density Profiles

and emitter The experimental and simulated base currents and the collector–emitter voltage during the switching are shown in Fig. 2. The comparison is made here for the turn-on process only, since the model does not profess to be a reliable description of the later relaxation stage. Indeed, the carrier recombination and a shunting of the emitter–base junction should be included for analyses of this kind. The most pronounced difference concerns the switching delay, which is 2 ns larger in the experiment than in the simulations. The actual characteristic time for electron diffusion ns, across a base region of thickness 2 m is a value which is comparable with the observed difference in the switching delay (the electron diffusion coefficient in the cm /s). An obvious source of inaccuracy in the base switching delay is the limitation to a 1-D approach, which implies hole generation in the volume of the base (see Section III), so that lateral carrier transport in the base region is ignored. The switching process was investigated experimentally and also simulated for various amplitudes of the base current. The results are not presented here, since an increase in the triggering pulse does not cause any significant changes other than a reduction in the switching delay. The best agreement was achieved at a relatively low bias V [see Fig. 2(a)], where the simulated and voltage measured amplitudes of the collector current pulse differ by only 4 A (58 A in the simulations and 54 A in the experiment). The

The electric field and carrier density profiles in the base and collector regions during the switching transient are shown in Fig. 3. The process can be conditionally separated into three stages. The first stage corresponds to an increase in the collector current within the time range 0–1.5 ns [see Fig. 3(a) and (b)] A/cm from 0 to a critical value (here, the donor density in the part of the collector cm and the saturated velocity of the electrons cm/s). There is no appreciable rebuild in the collector field domain [see Fig. 3(a), 0 and 1 ns], since the electron and hole densities remain below the donor concentration [see Fig. 3(b), 1 ns]. In the second stage (1.5–3 ns), the electron density exchanges its sign, ceeds that of the donors, the derivation and its skew rate is defined in practice by the electron density. The third stage [see Fig. 3(c) and (d), 4–9 ns] is responsible rate of the collector current, which is caused for the high by a rapid reduction in the emitter–collector voltage determined by the spread of a “quasi-neutral” domain of the electron–hole plasma from the base to the collector contact [see Fig. 3(d)], with a corresponding transformation in the electric field distribution [see Fig. 3(c)]. (Here and below we put “quasi-neutral” into quotes, because a significant electric field of – kV/cm manifests itself within this domain, even though the electron and hole concentrations differ by only 10 –10 of their values. Even in the base region, the electric field varies from 0.2 to 1 kV/cm at high currents.) The physical mechanism of

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Fig. 4. Electron J (solid lines) and hole J (dashed lines) current profiles across the base and collector regions at various instances.

this plasma spread is worth discussing, since the rate of the collector current is one of the most important parameters of the avalanche switch (if not the main one). The “quasi-neutral” domain is formed by an accumulation of electrons injected from the emitter, and by an accumulation of holes generated by boundary. A certain proporimpact ionization near the – tion of these electrons (holes) will accumulate in the “quasi-neutral” domain, and others will penetrate the -contact ( -base). Both the electron and hole velocities and the ionization rate of the electrons in the high-field domain should thus determine the velocity of plasma spread. Indeed, an increase in ionization rate in the high-field domain and a reduction in the hole velocity in the “quasi-neutral” domain should facilitate hole accumulation, thus increasing the spread velocity. On the other hand, a relatively low electron velocity in the “quasi-neutral” domain can limit the spread velocity. (We neglect the reverse diffusion flux of the holes in the collector in these speculations, since its value is less than the hole drift current by a factor of 30, even in the regions with the highest gradient in hole density. Diffusion is vital before the third switching stage starts.) The dominant limiting factor for plasma spread can be found by considering the electron and hole fluxes in the collector. The electron and hole current profiles are shown in Fig. 4 for various instances within the third switching stage. The gradual reduction in the electron current (from left to right) and that in the hole current (from right to left) correspond to the carrier accumulation which causes the formation of the “quasi-neutral” region and its spread toward the collector contact. One can see that the electron current is much higher than the hole current at any instant in time, which means that carrier accumulation is limited by the rate of hole generation rather than by electron transport. A reduction in the switching time would thus require an increase in the rate of electron ionization, a reduction in the hole velocity within the “quasi-neutral” region, and a reduction in the collector space charge region (SCR), which should be filled by the “quasi-neutral” domain. C. Comparison With the Results of Steady-State Models Certain links can be found between the three stages of the transient mentioned above and steady-state considerations presented in the literature. The first stage should correspond to trivial avalanche transistor switching at low currents, without any rebuild in the collector domain. In our experiments, this stage has the appearance of a switching delay (see Fig. 2). The


second stage corresponds to the very beginning of the growth in the collector current. Qualitatively analogous analytical steadystate solutions were obtained earlier [7] by considering the Poisson equation with the electron injection induced and the impact ionization neglected. Indeed, the hole density is much lower than the electron density at this stage [see Fig. 3(b)], which makes it possible to ignore the impact ionization when considering the electric field domain. Furthermore, even a steady-state analog of the third stage has been discussed earlier and termed “current-mode secondary breakdown” [7]. However, there exists a difference in principle between these considerations and the results obtained in our paper. The Poisson equation in earlier papers was not solved together with the complete set of transport and continuity equations in the presence of impact ionization. It is obvious, in principle, that inducing a high electron current should create a steep slope in the collector domain [see Fig. 3(c)], but the possibility of a quasi-neutral region existing can be assumed rather than justified by reference to the models that adopt simplification approaches. The model presented in [7], for example, cannot recognize any difference between n-p-n and p-n-p structures, while there is a qualitative difference between their behaviors at extreme currents. The “quasi-neutral” domain in the collector cannot exist in a Si p -n-p -p structure, as will be discussed in Section IV-D. This means that a solution of the type corresponding to the third switching stage can be realized only when certain relations prevail between the velocities and the rates of electron and hole ionization. D. Simulation Results for a p -n-p -p Structure The simulation results for a p -n-p -p transistor are shown in Fig. 5. The biasing voltage, the base current, and the circuit are completely analogous to those used for n -p-n -n , and the results should be compared with the switching shown in Fig. 2(a). One can see that the maximum current is much lower for a p -n-p -p transistor, and the switching process is significantly slower. Only the first and second switching stages manifest themselves in the transient [see Fig. 5(b)], and the third stage is not realized at all. No “quasi-neutral” region is formed, since the electron density remains much lower than that for the holes [see Fig. 5(c)], thus preventing the electric field reduction or any significant carrier accumulation [compare with Fig. 3(d)]. There are two physical reasons that determine such a behavior: 1) the relatively low rate of hole ionization (electron generation) on the right-hand border and 2) high electron velocity. The low generation rate and prompt sweeping of eleccollector region prevents carrier accumulatrons from the tion and makes the solution shown in Fig. 3(c) and (d) impossible for a Si p -n-p -p structure. Current pulses similar to that shown in Fig. 5(a) were observed in our experiments with various 200–350 V p -n-p -p transistors, and the corresponding reduction in the voltage across the transistor was as small as a few volts. E. Transient in a Transistor With a Thick Collector Simulations were performed in order to check the effect of various parameters on the transient process in a n -p-n -n avalanche transistor. Some trivial tendencies were found, such


Fig. 5. Simulated transient in a p -n-p -p transistor at the biasing voltage U 200 V with load resistor R = 0:94 : (a) dependence of the collector current on time and (b), (c) electric field, hole (solid lines) and electron (dotted lines) density profiles across the n-base ( 4÷0 m) and p-collector (0÷16 m) regions at various instances. The “inverted” structure is completely analogous to that presented in Fig. 1.

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=

0

as a reduction in the switching delay with increase in the base current and initial collector-emitter voltage, and it was shown in particular that the base (triggering) current scarcely affects the third stage of the transient. Then, a super-linear increase in the current amplitude was observed when the biasing voltage was increased or the load resistor reduced. Both of the latter are caused by a reduction in the residual voltage across the switched-on device when the collector current density is increasing. For the same reason, the parasitic inductance in the emitter–collector circuit may slow down the switching process more significantly than could be expected for a “linear” switch. A nontrivial result was obtained from the investigation of the collector thickness, in that this thickness was found effect of to affect the switching delay much more significantly than the rate within collector current rise time or the maximum the third switching stage. Simulated current and voltage waveforms for various collector thicknesses, and the electric filed and carrier profiles for a “thick” collector (35 m) are shown in Fig. 6. An increase in the collector thickness from 15 to 35 m causes a rise in the switching delay (second stage) to 20 ns. A qualitative difference was observed within this stage relative to the case of a “thin” collector, in that hole accumulation in a “thick” collector occurs at moderate currents when the impact ionization effect is negligible. This accumulation [see Fig. 6(c), – ns] is determined by hole diffusion from the base, the diffusion current predominating over the drift hole current from the collector contact (because of weak impact ionization), so within that the resulting hole current remains negative at ns. This mechanism causes very the time range ns slow carrier accumulation in the left-hand part of the collector and can be attributed to the Kirk effect [14], or “base push-out” [7]. The third switching stage begins after the peak in the electric

Fig. 6. (a) Simulated collector current (solid lines) and emitter–collector voltage (dashed lines) for a n -p-n -n transistor with various n collector thicknesses w . The capacitor C was replaced in the simulations by the constant voltage source U = 250 V, the parasitic inductances were excluded, and load resistor R = 1 . (b) Electric field and (c) carrier density profiles across the transistor structure with a collector 35 m in thickness [rightmost waveforms in (a)] at various instances. Solid lines represent electron concentration. Dotted lines represent hole concentrations.

field near the collector contact has provided a high rate of hole generation. The thickness of the high-field domain at this instant is comparable to that of a “thin” collector [compare Fig. 6(b), ns and Fig. 3(c), ns], resulting in a comparable duration of the rising current edge in both cases [see Fig. 6(a)]. The somewhat higher voltage drop across the “quasi-neutral” region in a “thick” collector correspondingly leads to the lower amplitude of the collector current. F. Effect of the Doping Profile Near the Collector Junction An interesting phenomenon at the step-like base–collector junction was observed in the simulations at high biasing voltage V . Oscillations in the collector–emitter voltage [see Fig. 7(a)] appeared within the first switching stage, they stop after 2.3 ns, and a further transient was observed that was similar to that shown in Fig. 2(b). The oscillations disappear if , or near the base–collector the biasing voltage junction, or the inductance in the external circuit is reduced. The impurity profile shown by a solid line in Fig. 1, for example, does not cause these oscillations [see Fig. 2(b)]. The oscillation period 120 ps does not depend on the inductance value, thus excluding a trivial interpretation such as oscillations in the loop with the collector–base barrier capacitance and inductance included. Furthermore, the observed period is shorter by a factor of 3. than We suggest the following interpretation for the oscillations. The electric field profiles [see Fig. 7(b)] show that this field has

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ps. This value agrees well with the osps (a factor of three is typical of cillation period the difference between the rise time and the period of the oscillations). An initial reason for the oscillations originates from the peak , which causes intensive multiplicain the electric field at tion of a small initial number of electrons and gives rise to the hole density in the left-hand half of the collector region. The example considered here may provide an explanation for certain differences between the simulated and measured current and voltage waveforms at the very beginning of the transient. There are no oscillations in the collector–emitter voltage [see Fig. 2(b)], since doping profile near the base–collector junction used in the simulations is not step-like but somewhat “softened” (see Fig. 1). This causes a reduction in the peak value of the [see Fig. 7(b)], with dual consequences. electric field at On the one hand, this suppresses the oscillations; on the other (growth hand, it slows down the reduction in the voltage in the current ) during the first switching stage. We would thus ascribe a certain difference between the experiment and the simulations at the very beginning of switching to a difference in the doping profiles used in the simulations and in the real structure. Fig. 7. (a) Collector–emitter voltage oscillations obtained in the simulations for a “thin” collector with a step-like doping profile. The electric field, electron (solid lines) profiles, and hole (dotted lines) profiles related to the instances when the voltage had its maximum (t ) and minimum (t ) values are shown in (b) and (c), respectively.

not been completely reconstructed, despite the fact that the carrier density exceeds that of the donors by a factor of 3 or more. The holes generated by impact ionization keep the same sign of in the left-hand side of the collector as was defined by the donors [see Fig. 7(b) and (c)]. The electron flux at the instant ns from the base is not strong enough for inversion, and the electron density in the collector is mainly determined by the multiplication of both electrons and holes, which causes an increase in the current, an increase in the voltage across the parasitic inductance in the emitter–collector circuit, and a corresponding reduction in the collector–emitter voltage ns]. This causes a reduction in the [see Fig. 7(a) and (b), ionization rates, saturation of the collector current, a reduction in the voltage drop across the inductance, and an increase in the voltage drop across the transistor. This marks the full period of the oscillation. These oscillations will be suppressed after a certain time, once the electron flux becomes strong enough for the m to be inverted. It is obvious electric field slope at that a reduction in the inductance will reduce the feedback, and the oscillations will be suppressed. The period of the oscillations can now be evaluated. Detailed analyses of the electron and hole show that the most sigcurrent components at the instant nificant contribution to the total current enlargement is provided m. (The electron by the electron impact ionization at current has its maximum at this point, and its value exceeds the .) The electric hole current at any across the structure for V/cm [see Fig. 7(b), ], and field at this point cm the corresponding electron ionization rate (4). Thus, the characteristic time of electron–hole pair generm and is ation caused by the electrons at

G. Estimate for the Effect of Joule Heating The current amplitude in the simulations at high initial voltage exceeds that in the experiment by approximately 13% [102 A against 89 A, Fig. 2(b)]. This difference could be caused in part by the neglecting of the Joule heating and its effect on the ionization rates in the simulations. The maximum possible when the increase in the lattice temperature at the instant current reaches its maximum value can be evaluated as follows:

where and are the material density and heat capacity, revalue of 32 K is reached spectively. The maximum possible near the n - n boundary for the conditions, corresponding to Fig. 2(b), but the actual increase in temperature should be somewhat lower ( 10%) due to the thermal conductivity. The reduction of 18% in the ionization rate for the electrons corresponds to the increase in lattice temperature by 30 K, and can be compensated for by an increase of only 3% in the electric field . These estimates are reached by making use of the relations in [15]. Thus, the Joule heating should cause only a 3% increase in the residual voltage and an analogous reduction in the peak value for the current. The observed 7–13% difference (at – V) between the current pulse amplitude in the simulations and in the experiment should be attributed to uncertainty in the doping profile, rather than to the effect of Joule heating. V. CONCLUSION The results of the dynamic 1-D simulations for an avalanche transistor provide reasonably good agreement with the experimental waveforms at extreme currents. It is specifically shown


that fundamental parameters of the semiconductor material such as the detailed dependences of the electron and hole velocities and ionization rates on the electric field determine the possibility for achieving high current switching. Three consequences of practical importance follow from the dynamic consideration. First, neglecting the effect of the rebuild in the collector field domain, as is typical of simplified models, does not allow fast, high-current switching to be understood. Only moderate current densities of 10 A/cm with a long 10 ns could be expected in this case, as current rise time has been found for a p -n-p -p Si transistor. The fact is nontrivial that the collector thickness in excess of 15 m does not significantly effect the current rise time and the current pulse rate amplitude. Dynamic analyses have shown that the of the collector current in an n -p-n -n Si transistor is mostly limited by the rate of hole generation near the n -n boundary, and to a lesser degree by the electron and hole velocities in the “quasi-neutral” collector region. Finally, one important conclusion is that the details of the doping profile in the collector region significantly affect the shape of the leading edge of the current pulse. The model suggested here seems to be a reliable tool for investigating the switching transient in various transistor structures based on Si and other semiconductor materials. REFERENCES [1] S. N. Vainshtein and J. T. Kostamovaara, “Spectral filtering for time isolation of intensive picosecond optical pulses from a Q-switched laser diode,” J. Appl. Phys., vol. 84, no. 4, pp. 1843–1847, Aug. 1998. [2] A. Biernat and G. Kompa, “Powerful picosecond laser pulses enabling high-resolution pulsed laser radar,” J. Opt., vol. 29, pp. 225–228, 1998. [3] A. Kilpela and J. Kostamovaara, “Laser pulser for a time-of-flight laser radar,” Rev. Sci. Instrum., vol. 68, pp. 2253–2258, 1997. [4] W. B. Herden, “Application of avalanche transistors to circuits with a long mean time to failure,” IEEE Trans. Instrum. Meas., vol. IM-25, pp. 152–160, 1976. [5] B. Q. Streetman, Solid State Electronic Devices. Englewood Cliffs, N.J.: Prentice-Hall, 1972, pp. 344–346. [6] Mallik, “Nonuniform doping of the collector in avalanche transistors to improve the performance of Marx bank circuits,” Rev. Sci. Instrum., vol. 71, no. 4, pp. 1853–1861, 2000. [7] K. Mallik, “The theory of operation of transistorized Marx bank circuits,” Rev. Sci. Instrum., vol. 70, no. 4, pp. 2155–2160, 1999. [8] Y. Mizushima and Y. Okamoto, “Properties of avalanche injection and its application to fast pulse generation and switching,” IEEE Trans. Electron Devices, vol. ED-14, pp. 146–157, 1967. [9] H. C. Bowers, “Space-charge-induced negative resistance in avalanche diodes,” IEEE Trans. Electron Devices, vol. ED-15, pp. 343–350, 1968. [10] K. Hane and T. Suzuki, “Effect of injected current on current-mode second breakdown in silicon PNN structure,” Jpn. J. Appl. Phys., vol. 14, pp. 1961–1968, 1975. [11] A. Caruso, P. Spirito, and G. Vitale, “Negative resistance induced by avalanche injection in bulk semiconductors,” IEEE Trans. Electron Devices, vol. ED-21, pp. 578–586, 1974. [12] G. W. Neudeck, “Reverse biased p –n –n junction at extreme currents,” Electron. Lett., vol. 11, pp. 397–398, 1975. [13] C. Jacoboni, C. Canali, G. Ottaviani, and A. A. Quaranta, “A review of some charge transport properties of silicon,” Solid-State Electron., vol. 20, pp. 77–89, 1977.

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[14] C. T. Kirk, “A theory of transistor cutoff frequency (f ) falloff at high current densities,” IRE Trans. Electron Devices, vol. ED-9, pp. 164–174, 1962. [15] C. R. Crowel and S. M. Sze, “Temperature dependence of avalanche multiplication in semiconductors,” Appl. Phys. Lett., vol. 8, pp. 242–244, 1966.

Sergey N. Vainshtein was born in 1955 in Kostroma, Russia. He received the M.Sc. degree (with honors) in physics from Leningrad Polytechnic Institute, Leningrad, Russia, in 1978, and the Ph.D. degree in semiconductor physics from A. F. Ioffe Institute of Physics and Technology, St. Petersburg, Russia, in 1987. In 1980, he joined the A. F. Ioffe Institute of Physics and Technology, where he has held the position of Senior Scientist since 1995. He is currently a Visiting Senior Researcher with the Department of Electrical Engineering, University of Oulu, Linnanmaa, Oulu, Finland. He has authored more than 50 scientific papers. His research activities in applied physics are in the areas of high-speed, high-power, and high-temperature semiconductor devices. His current research interests include subnanosecond and picosecond kinetics of semiconductor lasers, switching microwave devices, and pulsing circuits.

Valentin S. Yuferev was born in Leningrad, Russia, in 1937. He received the M.Sc. degree in mechanics from the Leningrad Shipbuilding Institute and the Ph.D. and Doctor of Science (physics and mathematics) degrees from the A. F. Ioffe Institute of Physics and Technology, St. Petersburg, Russia, in 1967 and 1984, respectively. He is currently a Leading Researcher and Head of the Research Group at the Laboratory of Applied Mathematics and Mathematical Physics, A. F. Ioffe Institute of Physics and Technology. He has authored more than 170 scientific publications. His main interest is in numerical and mathematical simulation of physical problems described by partial differential equations. His research activities have included semiconductor device simulation, magnetohydrodynamics, heat and mass transfer in crystal growth, and electromagnetic processes in railguns. His current research interests are the simulation of avalanche transistors at high currents, the solution of radiant transport equations, and the study of fluid convection under microgravity.

Juha T. Kostamovaara (S’84–M’85) was born in Tervola, Finland, on December 30, 1955. He received the Dipl. Eng., Lic. Tech., and Dr. Tech. degrees in electrical engineering in 1980, 1982, and 1987, respectively. He was Acting Associate Professor of Electronics in the Department of Electrical Engineering at the University of Oulu, Linnanmaa, Oulu, Finland, from 1987 to 1993, and was appointed Associate Professor of Electronics in 1993. In 1994, he worked as an Alexander von Humboldt Scholar at the Technical University of Darmstadt, Darmstadt, Germany. In 1995, he was invited to become Full Professor of Electronics at the University of Oulu, where he is currently also Head of Electronics Laboratory. His main interest is in high-speed electronic circuits and systems and their applications to pulsed time-of-flight laser radars and radio mobile telecommunications.