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Avalanche transistor operation at extreme currents: physical reasons for low residual voltages
Solid-State Electronics 47 (2003) 1255–1263 www.elsevier.com/locate/sse
Avalanche transistor operation at extreme currents: physical reasons for low residual voltages Sergey N. Vainshtein a
a,*,1
, Valentin S. Yuferev b, Juha T. Kostamovaara
a
Electrical Laboratory, Department of Electrical Engineering, University of Oulu, Linnanmaa SF-90570 Oulu, Finland b A.F. Ioffe Institute of the Russian Academy of Science, 194021, Politechnicheskaya Ul. 26, St. Petersburg, Russia Received 21 August 2001; received in revised form 25 November 2002; accepted 30 December 2002
Abstract Low residual voltages (70–95 V) were observed in our experiments with Si nþ –p–n0 –nþ avalanche transistors at current pulses of a few nanoseconds with an amplitude of 100 A. The voltages are much lower than that predicted by a simple theory of avalanche transistor switching. A physical explanation is suggested and a numerical model is produced which explains the low residual voltages by a strong rebuilding of the electric field domain in the n0 collector. This reconstruction takes place when the current density significantly exceeds a critical value, which is associated with a drift of equilibrium carriers in the collector at a saturated velocity. The final electric field distribution across the collector region was shown to be greatly dependent on both total current density and the ratio of the injection current component to the total current. A voltage drop of less than 50 V was calculated at high total currents (105 A/cm2 ) provided that the ratio of the electron injection current to the total current exceeded 0.7. The maximum possible value of this ratio is determined by the fundamental properties of the semiconductor material and plays an essential role in the phenomenon. By contrast, we did not succeed in obtaining any appreciable reduction in the residual voltage for pþ –n–p0 –pþ transistors either experimentally or numerically. The physical reasons for this behaviour were found to be mainly determined by the difference in the electron and hole mobilities. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Semiconductor switches; Avalanche breakdown; Microwave switches
1. Introduction High amplitude (10–100 A), nanosecond range current pulses are used for various applications, particularly for the pumping of high-power, wide stripe pulsed laser diodes. One of the simplest and most convenient ways to generate such pulses is to use high voltage (200–400 V) bipolar transistors in avalanche mode [1,2]. * Corresponding author. Tel.: +358-8-553-2702; fax: +358-8553-2700. E-mail addresses: [email protected].fi (S.N. Vainshtein), [email protected]ffe.rssi.ru (V.S. Yuferev). 1 S.N. Vainshtein is on leave from the A.F. Ioffe Institute of the Russian Academy of Science.
A theory of the operation of avalanche transistors with relatively low breakdown voltages at the base– collector junction was developed long ago and is applicable to both n–p–n and p–n–p structures [3]. The general idea in the various models is that a transistor switches between the maximum emitter–collector voltage VCB0 for the shorted base–emitter electrodes, and the minimum emitter–collector voltage VCE0 for the open base. The voltage VCB0 actually represents the breakdown voltage for the base–collector junction, while the voltage VCE0 is defined by intrinsic positive feedback between a weak carrier multiplication in the base– collector junction and associated carrier injection from the emitter. An electric field domain of invariable shape defined by the doping concentrations in the base and collector is assumed in models of this kind. Accordingly,
0038-1101/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0038-1101(03)00007-8
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the value VCE0 is the minimum residual voltage, which can be achieved when the transistor is switched on. Let us consider as an example, however, the commercial avalanche transistor FMMT-417 (ZETEX Semiconductors) produced for high-current applications. The values VCB0 and VCE0 , measured with a curve tracer at low frequency (100 Hz), are shown in Fig. 1. The S-like curve is a schematic representation of a characteristic switching curve, as could be expected in the framework of conventional models. One can see that the minimal expected voltage should not be less than 170 V. Meanwhile the residual voltage drop across the transistor, as specified by the manufacturer, is as low as 100 V at a current amplitude of 60 A, which is contradictory to the measured VCE0 value. Let us evaluate the operating current density. For an operating transistor area of 103 cm2 , the current density J is 60 kA/cm2 at the maximum current of 60 A. This value should be compared with the critical current density Jcr ¼ qn0 vns , where q is the electron charge, n0 is the equilibrium electron density, equal to the donor concentration Nd in the collector, and vns is the saturated electron velocity. Given a collector–base breakdown voltage of 340 V, the donor concentration is 1015 cm3 (or lower, depending on the thickness of the n0 collector), and Jcr 1600 A/cm2 . One can see that the operating current density exceeds the critical value by a factor of 40, which should mean that the electrical field domain cannot be determined by the collector doping any longer but must depend mostly on the current density. This was not the case with the low-voltage transistors [3], in which the critical currents significantly exceeded any realistic operating current values. Although high voltage avalanche transistors have been widely used at high currents, we have not found any clear physical description in the literature that does not contain rough assumptions drastically affecting the
Fig. 1. Schematic presentation of FMMT-417 avalanche transistor switching in terms of conventional models.
results. Current mode secondary breakdown associated with avalanche injection has been considered in a number of papers (see [4] and references therein). A reconstruction of the collector field domain with the reduction in the collector voltage has been presented, but the total current was assumed to be equal to its electron component, which cannot be the case when the injection current exceeds the critical value (see Section 3). Then both the electron velocity and hole velocity are considered to be saturated, which appears to be a rough assumption significantly affecting the shape of the field domain in the collector region (Section 3). Numerical simulations of the voltage–current characteristics of nþ –p–n0 –nþ avalanche transistors at high current densities are provided in Ref. [5]. One can see from these characteristics that residual voltage reduction is possible at a very high density of the injected carriers. No electric field distribution is shown, however, and its relation to the electron and hole current components is not analysed. Moreover, as will be seen below, a reliable description of the collector field domain is problematic in this respect, because of the rough approximations used for the carrier velocities and the ionisation rates. Switching of reverse-biased diodes has been considered without the carrier injection [6] and with a week injection (J < qn0 vns ) [7], 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 [8], could be related to the processes in a transistor, but the injection contact approximation used there is not applicable to carrier transport analyses in the n0 collector region. Rebuilding of the electric field domain in the base region of a reverse-biased diode at extreme currents is shown in [9]. The electric field distribution presented in this work 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 was not specified in the paper, which does not allow us to evaluate whether the analyses presented in [9] for a diode structure are related to the processes in the collector of the avalanche transistor. It follows from our further consideration that both the total current and its injection component define the shape of the electric field domain in the collector of a transistor, and no such analysis has yet been presented in the literature. Our goal is to analyse the main conditions under which low residual voltages are achieved across the switched-on avalanche transistor at extreme currents. Commercial device simulators such as MEDICI, ATLAS or DESSIS should allow a dynamic 2D solution of this problem for a detailed comparison of the current and voltage waveforms with the experiment. We will show, however, that simple 1D static consideration of
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the processes in the n0 collector region supplemented with analytical treatment of some aspects can provide a clear vision of the physical reasons for low residual voltage across the switched-on nþ –p–n0 –nþ structure and for the lack of this behaviour in pþ –n–p0 –pþ transistors. Analyses of the results show that even carrier diffusion can be neglected in the steady-state approach, since the drift currents vastly exceed the diffusion ones in any part of the collector region.
2. Experiment A commercial FMMT-417 avalanche transistor with a surface-mounted package was selected for the experiments. The plastic case of the transistor was removed from one side by polishing in order to use the metal plates closest to the transistor chip for the electrical connection with the external circuit. This allowed serial parasitic inductance LEC paras between the ohmic contacts to the structure and the external circuit connections to be reduced to 1.3 nH (Fig. 2(a)). This value is still significant at high dI=dt rates, and correct measurements of the voltage across the transistor structure required the voltage drop across this inductance to be subtracted from the voltage measured by the oscilloscope. Parasitic inductance values were found in the preliminary experiments from comparison of the experimental waveforms with PSPICE simulations. We evaluate the accuracy of the resulting parasitic inductance values, shown in Fig. 2(a), to be 0.2 nH. Five 4.7 X chip resistors connected in parallel were used as a load, and three 2.2 nF ceramic (NP0) capacitors C0 connected in parallel were preliminarily charged to voltage U0 . The total inductance LP of the external circuit includes the inductance of the capacitor and the electrical connections. Triggering pulses of length 50 ns from a 5 V/50 X pulse generator with a 5 ns rise time and a 1 kHz repetition rate were used. The current pulses and corresponding voltage waveforms across the ohmic contacts to the emitter and collector regions of the transistor structure are shown in Fig. 2(b). Strictly speaking, the voltage drop across the transistor at the instant when the current pulse achieves its maximum value cannot be considered as a steady state value. The further decrease in the voltage is not very significant, however, even if a 3 times higher value for the capacitance C0 is used. Moreover, the voltage drop across the transistor several nanoseconds after the switching starts is a matter of the most essential practical interest. Thus we take this value here as a first-order approach to the quasi-steady state of the switched on transistor. One can see that the residual voltage is lower than that specified by the manufacturer for the same or even higher current amplitudes, and ranges from 70 V (for Im 70 A) to 95 V (for Im 120 A).
Fig. 2. (a) Schematic presentation of the circuit with the parameters of the vital parasitic inductances included, (b) load current (curves 1–3) and the voltage across the emitter–collector ohmic contacts to the structure (curves 10 –30 ) as a function of time for various initial biasing values U0 : 1,10 ––290 V, 2,20 ––240 V, 3,30 ––180 V. The current waveforms IðtÞ are derived from the VLOAD ðtÞ dependence by iteration, accounting for the parasitic load inductance LLOAD , and the voltage waveforms are obtained by subtracting LEC paras dI=dt from the experimentally observed VEC ðtÞ dependences.
3. Numerical simulations 3.1. Model and numerical solution As has already been mentioned, our consideration is restricted to the static process in the collector region, and the home-made code used here is specially oriented towards the simulation of avalanche transistors at high current densities. We neglected the recombination and diffusion of electrons and holes. The first assumption is associated with the fact, that we are interested in the condition of the device a few nanoseconds after switching. The validity of the second assumption was checked by comparing the drift and diffusion current
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components for the simulated electric field and the carrier density profiles. It was found that the diffusion currents are negligibly small even for the regions with the highest gradient of carrier density. The carrier transport and electrical field distribution in the collector region under a steady-state regime are then described by the following equations: oJn ¼ ðan Jn þ ap Jp þ GÞ ox
ð1aÞ
oJp ¼ ðan Jn þ ap Jp þ GÞ Jn ¼ qnvn ; Jp ¼ qpvp ox
ð1bÞ
oE 1 ¼
ox ee0
Jp Jn þ qND vp vn
ð1cÞ
where Jn and Jp are the electron and hole current densities, q is the electron charge, vn and vp are the velocities of the electrons and holes, an and ap are their ionisation rates, G is the rate of thermal generation, e is the dielectric constant, ND is the doping concentration, E is the electrical field and the upper sign on the right-hand side corresponds to the n0 collector and the lower sign to the p0 one. Note also that in the case of the n0 collector the value E in (1c) means that the electric field takes the opposite sign. In such a formulation we have a collector–base junction located at the left boundary x ¼ 0, the injected carriers move from left to right and E, Jn , Jp are positive for both n0 and p0 collectors. It follows from (1a) and (1b) that the total current J ¼ Jn þ Jp is constant. The electron and hole velocities in (1) were determined by expressions taken from Ref. [10]: vn ¼ vns vp ¼ vps
E=En ð1 þ ðE=En Þbn Þ1=bn E=Ep
;
ð1 þ ðE=Ep Þbp Þ1=bp
ð2Þ
where the coefficients vns , vps , En , Ep , bn and bp depend on the temperature. At T ¼ 300 K we have vns ¼ 1:0705 107 cm/s; vps ¼ 0:83447 107 cm/s; En ¼ 6980 V/cm; Ep ¼ 17 988 V/cm; bn ¼ 1:1087; bp ¼ 1:213. The ionisation rates an and ap are usually represented as an ¼ an1 expðbn =EÞ;
ap ¼ ap1 expðbp =EÞ
ð3Þ
where bn , bp , an1 and ap1 were taken from Ref. [9]: bn ¼ 1:75 106 V=cm; 6
1
an1 ¼ 3:8 10 cm ;
bp ¼ 3:26 106 V=cm; ap1 ¼ 2:25 107 cm1
Since the system (1) consists of three equations, it is necessary to give three boundary conditions. Carrier densities or current densities for the transport equations (1a) and (1b) are usually specified at the boundaries.
When applied to transistor simulation, the second way is preferable: for the n0 collector at x ¼ 0 Jn ¼ Jn0 ; at x ¼ w Jp ¼ 0 for the p0 collector at x ¼ 0 Jp ¼ Jp0 ; at x ¼ w Jn ¼ 0 ð4Þ where w is the thickness of the region and Jn0 , Jp0 are the electron and hole injection currents. Either the total current, the voltage drop across the structure or the value of E at one of the boundaries can be set as a boundary condition for PoissonÕs equation (1c). All these conditions are in principle equivalent, but the last one is more convenient since it simplifies the numerical procedure significantly. The third boundary condition was therefore written as at x ¼ 0 E ¼ E0
ð5Þ
The model contains two parameters: E0 and Jn0 ðJp0 Þ. The actual aim is to obtain a relation between the total current J and the collector–base voltage U with the injection current Jn0 ðJp0 Þ as a parameter. It is obvious, however, that every pair of values E0 and Jn0 ðJp0 Þ provide a single pair of values J and U , and this allows us to find the required relation. Eqs. (1)–(5) were solved numerically by the finitedifference method, performing the calculations for a collector thickness of 40 lm and a doping concentration ND ¼ 1015 cm3 . A couple of remarks are needed here. This doping concentration would provide a VCB0 comparable to that of a FMMT-417 transistor, provided the thickness of the collector was >20 lm. The actual thickness of the n0 collector region for this transistor is 13 lm, however, and the doping concentration ND 5 1014 cm3 according to C–V measurements. One reason for selecting the doping and the collector thickness in this way in our simulations was associated with an attempt to compare the results with those presented in [9], where 50 lm and 1015 cm3 were considered. Another aim was to understand how significantly the collector thickness effects the collector voltage at high currents. It will be seen further that the electric field domain near the right-hand boundary determines a dominant fraction of the voltage across the collector. An increase in the collector region thickness of over 15 lm will increase the width of the low-field region, which determines only 30–45% of the total voltage drop even for a collector as thick as 40 lm. The simulated total voltage drop at high currents can be compared with the experimental data for the transistor FMMT-417 by simple subtraction of the voltage drop across the lefthand side of the collector, since the rightmost strong field domain is invariant relative to the total collector thickness (see next section). Special simulations for a thinner collector region would provide no any additional information.
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3.2. Simulation results for a n0 collector related to a nþ – p–n0 –nþ transistor structure
VOLTA GE , V
400
COLLECTOR
500
200
7
5
6 5
10
4
3
10
1
3
1
3
100
0.60
0.65
0.70
Jn / J Fig. 4. Dependence of the collector voltage on the ratio of the electron injection current to the total current Jn0 =J . J (kA/cm2 ): 1––40, 2––70 and 3––100.
4.0 3.5 3.0 2.5 2.0
1 8
1.5 1.0
2
0.5
3
4
5
67
0.0 0 4 8 12 16 20 24 28 32 36 40
4 2
10
2
0
5
10
300
E / 10 , V / cm
When simulating this structure we used the upper signs on the left-hand side of Eq. (1). The set of current– voltage characteristics (CVC) presented in Fig. 3 was obtained for various values of the injection current Jn0 , exceeding the critical one (Jcr ). The conventional characteristic (dashed curve), corresponding to breakdown in a diode without an injection current is analogous to that obtained in [6,7]. It is seen that the injection current powerfully influences the branch of the current–voltage characteristic with positive differential resistance, and the voltage across the structure at large Jn0 can become several times lower than the breakdown voltage without injection. Furthermore, an increase in Jn0 at constant total current J causes a marked decrease in the collector voltage, as shown in Fig. 4. This can be attributed to redistribution of the electric field and the formation of the narrow peak E near the right boundary (Fig. 5). The higher the injection current, the narrower is the peak in the electric field domain. This is one reason for the voltage reduction as Jn0 rises. The second reason is associated with the fact that the greater part of a collector on the left side of the peak is a region where ionisation is absent and the electric field is practically constant and very close to the value Eðx¼0Þ ¼ E0 . We will denote this as a quasi-neutral region, despite the fact that a drift (not diffusion) current predominates there. It is clear that the smaller E ¼ E0 is in this domain, the lower is the voltage over the structure. Thus the minimum voltage occurs if a
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X , µm Fig. 5. Electric field distributions along the n0 collector for various combinations of the injection current and total current Jn0 (kA/cm2 )/J (kA/cm2 ): 1––0/0, 2––0.85/0.856, 3––5/7.15, 4–– 12/18.6, 5––26/41.3, 6––40/64, 7––70/110, 8––40/81.1.
2
0
100 200 300 400 500 600 700 COLLECTOR VOLTAGE, V
Fig. 3. Current–voltage characteristics for various electron injection current densities Jn0 (A/cm2 ): 1––1750, 2––3000, 3–– 5000, 4––12 000, 5––26 000, 6––40 000, 7––70 000. The dashed line corresponds to Jn0 ¼ 0. The critical current density Jcr ¼ 1712 A/cm2 . The left (initial) point of each curve corresponds to E0 ¼ 4000 V/cm.
combination of a high injection current and small values of E0 takes place. At a fixed injection current the minimum voltage will be achieved when E0 tends to zero (the carrier concentration on the left side tends to infinity). Note that this case cannot be treated as an ‘‘injecting contact’’ [8] because the electric field proves to be close to zero not only at x ¼ 0, but everywhere apart from the peak at the right boundary.
It is seen that this relation tends towards the limit 1 þ ðvps En =vns Ep Þ ¼ 1:302 when E0 ! 0, which is determined by a fundamental property of the material––the ratio of mobilities b ¼ vns Ep =vps En . Correspondingly, referring to Fig. 4, we can conclude that the minimum collector voltage is reached when Jn0 =J is equal to 1=ðb þ 1Þ at a fixed value of J . At high injection currents it is also possible to obtain an approximate analytical expression for the electrical field Em at the right boundary. Since an ap and Jn > Jp , ionisation by holes can be neglected in (1a) and (1b). The system (1) can then be written as follows Z x an dx ; Jp ¼ J Jn Jn ðxÞ ¼ Jn0 exp 0
Z x oE 1 1 1 J ¼ þ qND Jn0 exp an dx ox ee0 vn vp vp 0 Taking into account that E0 is small with respect to E in the avalanche region, we obtain, after some transformation, Em bn 1 1 1 Jn0 ¼ þ 1 E2 J ee0 ans bn vns vps bn Em J J J þ qND ln ð6bÞ vps Jn0 R1 where E2 ðxÞ ¼ 1 expðxtÞt2 dt. The maximum value of Em is again reached in the limiting case Jn0 =J ¼ 0:768. At J ¼ 100 kA/cm2 , for example, we obtain Em ¼ 3:65 105 V/cm. Formulas (6a) and (6b) and the curves in Fig. 5 show that the distribution of the electric field near the right-hand boundary at high currents ceases to depend on the collector thickness starting from a n0 region a few lm thick. The carrier density distributions, which determine the field domain, are shown in Fig. 6. Three regions can be conditionally segregated, the stretched quasi-neutral domain where the electron and hole concentrations are practically constant and differ by the value ND , the rightmost high-field domain, and a buffer domain where the electric field exceeds E0 value significantly but ionisation is still absent. Electron velocity in the latter domain increases with respect to that in the quasi-neutral
-3
A simple relation between E0 and Jn0 =J can be found when a domain with a constant electrical field is formed. The derivative dE=dx in (1c) can be set at zero for this domain. Neglecting the donor concentration at high Jn0 , we obtain: bn 1=bn 1 þ EEn0 J vps En ¼1þ ð6aÞ bp 1=bp Jn0 vns Ep 1 þ EEp0
4.0
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3.5
CONCENTRATIONS (n , p) / 10 , cm
1260
Electrons
3.0 2.5
Holes
2.0 1.5 1.0 0.5 0
8
16
24
32
40
X , µm Fig. 6. Distributions of carrier density at Jn0 ¼ 40 and J ¼ 64 kA/cm2 .
region, which can lead to a reduction in the electron concentration, as shown in Fig. 6. As soon as the right boundary is approached, the ionisation process becomes essential and the electron concentration begins to increase again. The rise in Jn0 =J (reduction in E0 ) causes an increase in the carrier concentration in the quasineutral domain, and the concentration tends to infinity as Jn0 =J tends to 0.768. This means that the ratio of the injection current to the total current is the most essential parameter determining the residual voltage across the collector region of a nþ –p–n0 –nþ transistor operating at extreme currents. The appearance of a high-field domain at the right boundary is similar to that observed in [9]. The role of the injection current cannot be understood from Ref. [9], however, since it does not specify the full set of boundary conditions. The quasi-neutral and buffer field domains cannot be compared, since they are not presented in [9], and finally the shape of the field distribution at moderate currents differs from that obtained in our simulations. 3.3. Simulation results for a p0 collector related to a pþ – n–p0 –pþ transistor structure The models for p0 and n0 collectors are quite similar, being distinguished only by the signs on the right-hand sides of Eq. (1), and one could assume analogous behaviour for both kinds of collector. The operational characteristics of the two collectors turn out to differ markedly, however. The injection current in the n0 collector can amount to a significant fraction of the total collector current, so that the ratio Jn0 =J can be as high as 0.768 in principle. A relatively small part of the total current must be generated by ionisation in this case. The situation is essentially different for the p0 collector, calculations for which show that the injection current,
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1
4.5
E / 10 , V/c m
550 2
4.0 3.5 3.0
5
C O L L E C T O R VO L T A G E , V
5.0 560
540
530 0.1
2.5 2.0
1
1.5
2
1.0 0.2
0.3
0.4
0
Jp0 / J
4
8 12 16 20 24 28 32 36 40
, m
Fig. 7. Dependence of the collector voltage on the ratio of the hole injection current to the total current. Total current density J (kA/cm2 ): 1––70, 2––100.
Fig. 8. Electrical field distributions along the p0 collector for two values of the hole injection current Jp0 (kA/cm2 ): 1––30, 2–– 43.
which in this case is the hole current, cannot exceed 43% of total current J . The major part of the collector current will thus be determined by the avalanche process, resulting in a very high collector voltage. The collector voltage dependences on the injection current fraction Jp0 =J for two values of the total current are shown in Fig. 7. The computed voltage values (500 V) are actually unphysical from the point of view of transistor switching from an initial voltage value of about 300 V. This simply means that no switching to the current value of 100 kA/cm2 can be obtained for this transistor at any realistic initial biasing. The calculated collector voltage varies only slightly with the injection current. The distributions of the electric field in the p0 collector at J ¼ 100 kA/cm2 are presented in Fig. 8. It is seen that they differ from that obtained for the n0 collector at high injection currents and resemble instead the field domain in a diode when the injection current is absent [6]. The only effect of the injection current to be seen in Fig. 8 is the reduced peak in the electric field near the left boundary, which causes an insignificant reduction in the collector voltage (Fig. 7). The maximum value of E at the right boundary turns out to be as high as 4:93 105 V/cm (compare with Em ¼ 3:65 105 V/cm for the n0 collector at the same collector current). Finally, although we see that the electric field varies insignificantly in most of the collector, its magnitude remains fairly large and sufficient for appreciable ionisation. Thus no quasi-neutral domain is formed. At first glance one could assume that the reason for the formation of a quasi-neutral domain lies in the difference in ionisation rate between the electrons and holes. The simulations performed for the p0 collector
with the interchanged ionisation coefficients for the electrons and holes did not show any appreciable reduction in the residual collector voltage, however. On the contrary, the simulations of the n0 region with interchanged ionisation coefficients provided only a slightly higher residual voltage than that shown in Fig. 4. This result can be qualitatively understood as follows: the formation of a quasi-neutral domain requires intensive hole generation by the ‘‘fast’’ electrons and ‘‘slow’’ sweeping of the holes out of the quasi-neutral region. The peak value of the electric field on the right-hand boundary is adjusted automatically in order to maintain a sufficient rate of hole generation at a certain collector current. (The lower coefficients an ðEÞ are used in the simulations, the higher one being the peak value for the electric field.) The corresponding increment in the residual voltage at a collector current of 105 A/cm2 with interchanged ionisation coefficients was found to be only moderate (10–20 V). In addition, we will show here by the reduction ad absurdum method that no quasi-neutral domain can exist in the p0 collector, due to details of the field dependence of the hole and electron velocities in Si (relations (2)). Let us assume that a quasi-neutral domain does exist. Then, denoting the right-hand side of Poisson equation (1c) by F ðEÞ and taking into account that ionisation is absent in this domain and carrier currents are constant, we can represent the electrical field in the form e E ¼ E þ E where E is constant and satisfies the equation F ðE Þ ¼ e is a small addition satisfying the linearized 0, while E equation
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ee0
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e dE dF e E ¼ dx dE E¼E
The solution of this equation is written in the form e ¼ E þ ðE0 E Þ expðcxÞ; where c ¼ 1 dF E ee0 dE E¼E Neglecting ND we obtain the following expression for c: c¼
1 Jp ee0 E vp ðE Þ
1þ
E En
E En
bn ! bn
E Ep
1þ
bp
E Ep
bp !
Here, as before, the upper and lower signs correspond to n0 and p0 collectors, respectively. It is seen that c < 0 for a p0 collector. This means that the electrical field will diminish on removal from the base–collector junction. This is impossible since the quasi-neutral solution cannot be matched with the solution in an avalanche region near the right boundary in this case. Thus, no domain in which ionisation is absent can exist in the p0 collector at high currents, which would inevitably lead to a high voltage over the structure. Vice versa, c is positive for the n0 collector, so that the analogous problem does not arise. This conclusion is in qualitative agreement with the experimental results of Ref. [11], where it is shown that n–p–n Si transistors are more addicted to nonthermal secondary breakdown than p–n–p transistors. The authors of Ref. [11] assume that this behaviour must be caused by the difference in ionization coefficients between electrons and holes, while our results clearly show that the main reason lies in the difference in carrier mobilities.
4. Discussion We failed to observe high-current switching experimentally in various types of pþ –n–p0 –pþ transistor with VCB0 ranging from 200 to 350 V when using the circuit shown in Fig. 2(a). No collector current pulse with an amplitude higher than a few kA/cm2 and a rise time shorter than 10 ns was observed, apparently confirming the results described in the previous section. Then, as shown in Section 3.2, the ratio of the injection current to the total current is the most essential parameter determining the residual voltage across the collector region of a nþ –p–n0 –nþ transistor operating at extreme currents. One can see from Fig. 2 that the experimentally observed residual voltage ranges from 70 V (for I 70 A) to 95 V (for I 120 A). These values can be compared with the data shown in Fig. 4 in order to evaluate the ratio Jn0 =J as achieved in the
experiment. Allowance should be made, of course, for the difference between the collector thickness in the FMMT-417 transistor (Wexp ¼ 13 lm) and one used in the simulations (W ¼ 40 lm). Simple subtraction of the product E0 ðW Wexp Þ from the collector voltage in Fig. 4 gives a range 0.60–0.63 for the ratio Jn0 =J , which would correspond to the experimentally observed voltages. Here E0 was calculated from expression (6). The theoretical prediction for Jn0 =J should be dependent on a specific (1D or even 2D dynamic) operating mode in the transistor. The processes in the emitter and base regions that are dependent on the structure and external circuit will affect the actual Jn0 =J ratio, and the current 1D static model is not proficient for independent prediction of this value. In spite of this, the steady-state solution presented here appears to be useful, since it shows in demonstrative form the physical reason and requirements for low residual voltage to be achieved across the switched-on device. 5. Conclusions A relatively low residual voltage (<100 V) across an avalanche nþ –p–n0 –nþ transistor at extreme currents (100 kA/cm2 ) was demonstrated both experimentally and by means of quasi-steady state numerical modelling. The voltage, which drops mostly across the right-hand boundary of the n0 collector region, is mainly defined by the ratio of the injection (electron) current to the total current. An increase in both this ratio Jn0 =J and the total current J will reduce the residual voltage, which can be as low as 20 V, provided that Jn0 =J tends towards its upper limit 1=ðb þ 1Þ ¼ 0:768 (b is the ratio of mobilities in Si). Details of the carrier velocities and the ionisation rates dependences on the electric field affect the collector voltage drastically. It was shown in particular that no high current switching is possible for the pþ –n–p0 –pþ Si transistor structure, mainly due to the difference in electron and hole mobilities.
Acknowledgements This work was supported by the Academy of Finland (project #50460) and by INTAS (project INTAS-010364). The authors would like to express their deep gratitude to M.E. Levinshtein for the fruitful and instructive discussions.
References [1] Herden WB. Application of avalanche transistors to circuits with a long mean time to failure. IEEE Trans Instrum Meas 1976;25:152–60.
S.N. Vainshtein et al. / Solid-State Electronics 47 (2003) 1255–1263 [2] Kilpela A, Kostamovaara J. Laser pulser for a time-offlight laser radar. Rev Sci Instr 1997;68:2253–8. [3] Streetman BQ. Solid state electronic devices. Englewood Cliffs, Inc, NJ: Prentice-Hall Inc; 1972. [4] Mallik K. Nonuniform doping of the collector in avalanche transistors to improve the performance of Marx bank circuits. Rev Sci Instr 2000;71:1853–61. [5] Mizushima Y, Okamoto Y. Properties of avalanche injection and its application to fast pulse generation and switching. IEEE Trans Electr Dev 1967;14:146–57. [6] Bowers HC. Space-charge-induced negative resistance in avalanche diodes. IEEE Trans Elecr Dev 1968;15:343–50.
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[7] Hane K, Suzuki T. Jpn J Appl Phys 1975;14:1961–8. [8] Caruso A, Spirito P, Vitale G. Negative resistance induced by avalanche injection in bulk semiconductors. IEEE Trans Elecron Dev 1974;21:578–86. [9] Neudeck GW. Reverse biased pþ –n –nþ junction at extreme currents. Electron Lett 1975;11:397–8. [10] Jacoboni C, Canali C, Ottaviani G, Quaranta AA. A review of some charge transport properties of silicon. Solid-State Electron 1977;20:77–89. [11] Howel PL, Reddi GK. Avalanche injection and second breakdown in transistors. IEEE Trans Electron Dev 1970;17:320–35.
Avalanche transistor operation at extreme currents: physical reasons for low residual voltages Sergey N. Vainshtein a
a,*,1
, Valentin S. Yuferev b, Juha T. Kostamovaara
a
Electrical Laboratory, Department of Electrical Engineering, University of Oulu, Linnanmaa SF-90570 Oulu, Finland b A.F. Ioffe Institute of the Russian Academy of Science, 194021, Politechnicheskaya Ul. 26, St. Petersburg, Russia Received 21 August 2001; received in revised form 25 November 2002; accepted 30 December 2002
Abstract Low residual voltages (70–95 V) were observed in our experiments with Si nþ –p–n0 –nþ avalanche transistors at current pulses of a few nanoseconds with an amplitude of 100 A. The voltages are much lower than that predicted by a simple theory of avalanche transistor switching. A physical explanation is suggested and a numerical model is produced which explains the low residual voltages by a strong rebuilding of the electric field domain in the n0 collector. This reconstruction takes place when the current density significantly exceeds a critical value, which is associated with a drift of equilibrium carriers in the collector at a saturated velocity. The final electric field distribution across the collector region was shown to be greatly dependent on both total current density and the ratio of the injection current component to the total current. A voltage drop of less than 50 V was calculated at high total currents (105 A/cm2 ) provided that the ratio of the electron injection current to the total current exceeded 0.7. The maximum possible value of this ratio is determined by the fundamental properties of the semiconductor material and plays an essential role in the phenomenon. By contrast, we did not succeed in obtaining any appreciable reduction in the residual voltage for pþ –n–p0 –pþ transistors either experimentally or numerically. The physical reasons for this behaviour were found to be mainly determined by the difference in the electron and hole mobilities. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Semiconductor switches; Avalanche breakdown; Microwave switches
1. Introduction High amplitude (10–100 A), nanosecond range current pulses are used for various applications, particularly for the pumping of high-power, wide stripe pulsed laser diodes. One of the simplest and most convenient ways to generate such pulses is to use high voltage (200–400 V) bipolar transistors in avalanche mode [1,2]. * Corresponding author. Tel.: +358-8-553-2702; fax: +358-8553-2700. E-mail addresses: [email protected].fi (S.N. Vainshtein), [email protected]ffe.rssi.ru (V.S. Yuferev). 1 S.N. Vainshtein is on leave from the A.F. Ioffe Institute of the Russian Academy of Science.
A theory of the operation of avalanche transistors with relatively low breakdown voltages at the base– collector junction was developed long ago and is applicable to both n–p–n and p–n–p structures [3]. The general idea in the various models is that a transistor switches between the maximum emitter–collector voltage VCB0 for the shorted base–emitter electrodes, and the minimum emitter–collector voltage VCE0 for the open base. The voltage VCB0 actually represents the breakdown voltage for the base–collector junction, while the voltage VCE0 is defined by intrinsic positive feedback between a weak carrier multiplication in the base– collector junction and associated carrier injection from the emitter. An electric field domain of invariable shape defined by the doping concentrations in the base and collector is assumed in models of this kind. Accordingly,
0038-1101/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0038-1101(03)00007-8
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the value VCE0 is the minimum residual voltage, which can be achieved when the transistor is switched on. Let us consider as an example, however, the commercial avalanche transistor FMMT-417 (ZETEX Semiconductors) produced for high-current applications. The values VCB0 and VCE0 , measured with a curve tracer at low frequency (100 Hz), are shown in Fig. 1. The S-like curve is a schematic representation of a characteristic switching curve, as could be expected in the framework of conventional models. One can see that the minimal expected voltage should not be less than 170 V. Meanwhile the residual voltage drop across the transistor, as specified by the manufacturer, is as low as 100 V at a current amplitude of 60 A, which is contradictory to the measured VCE0 value. Let us evaluate the operating current density. For an operating transistor area of 103 cm2 , the current density J is 60 kA/cm2 at the maximum current of 60 A. This value should be compared with the critical current density Jcr ¼ qn0 vns , where q is the electron charge, n0 is the equilibrium electron density, equal to the donor concentration Nd in the collector, and vns is the saturated electron velocity. Given a collector–base breakdown voltage of 340 V, the donor concentration is 1015 cm3 (or lower, depending on the thickness of the n0 collector), and Jcr 1600 A/cm2 . One can see that the operating current density exceeds the critical value by a factor of 40, which should mean that the electrical field domain cannot be determined by the collector doping any longer but must depend mostly on the current density. This was not the case with the low-voltage transistors [3], in which the critical currents significantly exceeded any realistic operating current values. Although high voltage avalanche transistors have been widely used at high currents, we have not found any clear physical description in the literature that does not contain rough assumptions drastically affecting the
Fig. 1. Schematic presentation of FMMT-417 avalanche transistor switching in terms of conventional models.
results. Current mode secondary breakdown associated with avalanche injection has been considered in a number of papers (see [4] and references therein). A reconstruction of the collector field domain with the reduction in the collector voltage has been presented, but the total current was assumed to be equal to its electron component, which cannot be the case when the injection current exceeds the critical value (see Section 3). Then both the electron velocity and hole velocity are considered to be saturated, which appears to be a rough assumption significantly affecting the shape of the field domain in the collector region (Section 3). Numerical simulations of the voltage–current characteristics of nþ –p–n0 –nþ avalanche transistors at high current densities are provided in Ref. [5]. One can see from these characteristics that residual voltage reduction is possible at a very high density of the injected carriers. No electric field distribution is shown, however, and its relation to the electron and hole current components is not analysed. Moreover, as will be seen below, a reliable description of the collector field domain is problematic in this respect, because of the rough approximations used for the carrier velocities and the ionisation rates. Switching of reverse-biased diodes has been considered without the carrier injection [6] and with a week injection (J < qn0 vns ) [7], 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 [8], could be related to the processes in a transistor, but the injection contact approximation used there is not applicable to carrier transport analyses in the n0 collector region. Rebuilding of the electric field domain in the base region of a reverse-biased diode at extreme currents is shown in [9]. The electric field distribution presented in this work 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 was not specified in the paper, which does not allow us to evaluate whether the analyses presented in [9] for a diode structure are related to the processes in the collector of the avalanche transistor. It follows from our further consideration that both the total current and its injection component define the shape of the electric field domain in the collector of a transistor, and no such analysis has yet been presented in the literature. Our goal is to analyse the main conditions under which low residual voltages are achieved across the switched-on avalanche transistor at extreme currents. Commercial device simulators such as MEDICI, ATLAS or DESSIS should allow a dynamic 2D solution of this problem for a detailed comparison of the current and voltage waveforms with the experiment. We will show, however, that simple 1D static consideration of
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the processes in the n0 collector region supplemented with analytical treatment of some aspects can provide a clear vision of the physical reasons for low residual voltage across the switched-on nþ –p–n0 –nþ structure and for the lack of this behaviour in pþ –n–p0 –pþ transistors. Analyses of the results show that even carrier diffusion can be neglected in the steady-state approach, since the drift currents vastly exceed the diffusion ones in any part of the collector region.
2. Experiment A commercial FMMT-417 avalanche transistor with a surface-mounted package was selected for the experiments. The plastic case of the transistor was removed from one side by polishing in order to use the metal plates closest to the transistor chip for the electrical connection with the external circuit. This allowed serial parasitic inductance LEC paras between the ohmic contacts to the structure and the external circuit connections to be reduced to 1.3 nH (Fig. 2(a)). This value is still significant at high dI=dt rates, and correct measurements of the voltage across the transistor structure required the voltage drop across this inductance to be subtracted from the voltage measured by the oscilloscope. Parasitic inductance values were found in the preliminary experiments from comparison of the experimental waveforms with PSPICE simulations. We evaluate the accuracy of the resulting parasitic inductance values, shown in Fig. 2(a), to be 0.2 nH. Five 4.7 X chip resistors connected in parallel were used as a load, and three 2.2 nF ceramic (NP0) capacitors C0 connected in parallel were preliminarily charged to voltage U0 . The total inductance LP of the external circuit includes the inductance of the capacitor and the electrical connections. Triggering pulses of length 50 ns from a 5 V/50 X pulse generator with a 5 ns rise time and a 1 kHz repetition rate were used. The current pulses and corresponding voltage waveforms across the ohmic contacts to the emitter and collector regions of the transistor structure are shown in Fig. 2(b). Strictly speaking, the voltage drop across the transistor at the instant when the current pulse achieves its maximum value cannot be considered as a steady state value. The further decrease in the voltage is not very significant, however, even if a 3 times higher value for the capacitance C0 is used. Moreover, the voltage drop across the transistor several nanoseconds after the switching starts is a matter of the most essential practical interest. Thus we take this value here as a first-order approach to the quasi-steady state of the switched on transistor. One can see that the residual voltage is lower than that specified by the manufacturer for the same or even higher current amplitudes, and ranges from 70 V (for Im 70 A) to 95 V (for Im 120 A).
Fig. 2. (a) Schematic presentation of the circuit with the parameters of the vital parasitic inductances included, (b) load current (curves 1–3) and the voltage across the emitter–collector ohmic contacts to the structure (curves 10 –30 ) as a function of time for various initial biasing values U0 : 1,10 ––290 V, 2,20 ––240 V, 3,30 ––180 V. The current waveforms IðtÞ are derived from the VLOAD ðtÞ dependence by iteration, accounting for the parasitic load inductance LLOAD , and the voltage waveforms are obtained by subtracting LEC paras dI=dt from the experimentally observed VEC ðtÞ dependences.
3. Numerical simulations 3.1. Model and numerical solution As has already been mentioned, our consideration is restricted to the static process in the collector region, and the home-made code used here is specially oriented towards the simulation of avalanche transistors at high current densities. We neglected the recombination and diffusion of electrons and holes. The first assumption is associated with the fact, that we are interested in the condition of the device a few nanoseconds after switching. The validity of the second assumption was checked by comparing the drift and diffusion current
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components for the simulated electric field and the carrier density profiles. It was found that the diffusion currents are negligibly small even for the regions with the highest gradient of carrier density. The carrier transport and electrical field distribution in the collector region under a steady-state regime are then described by the following equations: oJn ¼ ðan Jn þ ap Jp þ GÞ ox
ð1aÞ
oJp ¼ ðan Jn þ ap Jp þ GÞ Jn ¼ qnvn ; Jp ¼ qpvp ox
ð1bÞ
oE 1 ¼
ox ee0
Jp Jn þ qND vp vn
ð1cÞ
where Jn and Jp are the electron and hole current densities, q is the electron charge, vn and vp are the velocities of the electrons and holes, an and ap are their ionisation rates, G is the rate of thermal generation, e is the dielectric constant, ND is the doping concentration, E is the electrical field and the upper sign on the right-hand side corresponds to the n0 collector and the lower sign to the p0 one. Note also that in the case of the n0 collector the value E in (1c) means that the electric field takes the opposite sign. In such a formulation we have a collector–base junction located at the left boundary x ¼ 0, the injected carriers move from left to right and E, Jn , Jp are positive for both n0 and p0 collectors. It follows from (1a) and (1b) that the total current J ¼ Jn þ Jp is constant. The electron and hole velocities in (1) were determined by expressions taken from Ref. [10]: vn ¼ vns vp ¼ vps
E=En ð1 þ ðE=En Þbn Þ1=bn E=Ep
;
ð1 þ ðE=Ep Þbp Þ1=bp
ð2Þ
where the coefficients vns , vps , En , Ep , bn and bp depend on the temperature. At T ¼ 300 K we have vns ¼ 1:0705 107 cm/s; vps ¼ 0:83447 107 cm/s; En ¼ 6980 V/cm; Ep ¼ 17 988 V/cm; bn ¼ 1:1087; bp ¼ 1:213. The ionisation rates an and ap are usually represented as an ¼ an1 expðbn =EÞ;
ap ¼ ap1 expðbp =EÞ
ð3Þ
where bn , bp , an1 and ap1 were taken from Ref. [9]: bn ¼ 1:75 106 V=cm; 6
1
an1 ¼ 3:8 10 cm ;
bp ¼ 3:26 106 V=cm; ap1 ¼ 2:25 107 cm1
Since the system (1) consists of three equations, it is necessary to give three boundary conditions. Carrier densities or current densities for the transport equations (1a) and (1b) are usually specified at the boundaries.
When applied to transistor simulation, the second way is preferable: for the n0 collector at x ¼ 0 Jn ¼ Jn0 ; at x ¼ w Jp ¼ 0 for the p0 collector at x ¼ 0 Jp ¼ Jp0 ; at x ¼ w Jn ¼ 0 ð4Þ where w is the thickness of the region and Jn0 , Jp0 are the electron and hole injection currents. Either the total current, the voltage drop across the structure or the value of E at one of the boundaries can be set as a boundary condition for PoissonÕs equation (1c). All these conditions are in principle equivalent, but the last one is more convenient since it simplifies the numerical procedure significantly. The third boundary condition was therefore written as at x ¼ 0 E ¼ E0
ð5Þ
The model contains two parameters: E0 and Jn0 ðJp0 Þ. The actual aim is to obtain a relation between the total current J and the collector–base voltage U with the injection current Jn0 ðJp0 Þ as a parameter. It is obvious, however, that every pair of values E0 and Jn0 ðJp0 Þ provide a single pair of values J and U , and this allows us to find the required relation. Eqs. (1)–(5) were solved numerically by the finitedifference method, performing the calculations for a collector thickness of 40 lm and a doping concentration ND ¼ 1015 cm3 . A couple of remarks are needed here. This doping concentration would provide a VCB0 comparable to that of a FMMT-417 transistor, provided the thickness of the collector was >20 lm. The actual thickness of the n0 collector region for this transistor is 13 lm, however, and the doping concentration ND 5 1014 cm3 according to C–V measurements. One reason for selecting the doping and the collector thickness in this way in our simulations was associated with an attempt to compare the results with those presented in [9], where 50 lm and 1015 cm3 were considered. Another aim was to understand how significantly the collector thickness effects the collector voltage at high currents. It will be seen further that the electric field domain near the right-hand boundary determines a dominant fraction of the voltage across the collector. An increase in the collector region thickness of over 15 lm will increase the width of the low-field region, which determines only 30–45% of the total voltage drop even for a collector as thick as 40 lm. The simulated total voltage drop at high currents can be compared with the experimental data for the transistor FMMT-417 by simple subtraction of the voltage drop across the lefthand side of the collector, since the rightmost strong field domain is invariant relative to the total collector thickness (see next section). Special simulations for a thinner collector region would provide no any additional information.
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3.2. Simulation results for a n0 collector related to a nþ – p–n0 –nþ transistor structure
VOLTA GE , V
400
COLLECTOR
500
200
7
5
6 5
10
4
3
10
1
3
1
3
100
0.60
0.65
0.70
Jn / J Fig. 4. Dependence of the collector voltage on the ratio of the electron injection current to the total current Jn0 =J . J (kA/cm2 ): 1––40, 2––70 and 3––100.
4.0 3.5 3.0 2.5 2.0
1 8
1.5 1.0
2
0.5
3
4
5
67
0.0 0 4 8 12 16 20 24 28 32 36 40
4 2
10
2
0
5
10
300
E / 10 , V / cm
When simulating this structure we used the upper signs on the left-hand side of Eq. (1). The set of current– voltage characteristics (CVC) presented in Fig. 3 was obtained for various values of the injection current Jn0 , exceeding the critical one (Jcr ). The conventional characteristic (dashed curve), corresponding to breakdown in a diode without an injection current is analogous to that obtained in [6,7]. It is seen that the injection current powerfully influences the branch of the current–voltage characteristic with positive differential resistance, and the voltage across the structure at large Jn0 can become several times lower than the breakdown voltage without injection. Furthermore, an increase in Jn0 at constant total current J causes a marked decrease in the collector voltage, as shown in Fig. 4. This can be attributed to redistribution of the electric field and the formation of the narrow peak E near the right boundary (Fig. 5). The higher the injection current, the narrower is the peak in the electric field domain. This is one reason for the voltage reduction as Jn0 rises. The second reason is associated with the fact that the greater part of a collector on the left side of the peak is a region where ionisation is absent and the electric field is practically constant and very close to the value Eðx¼0Þ ¼ E0 . We will denote this as a quasi-neutral region, despite the fact that a drift (not diffusion) current predominates there. It is clear that the smaller E ¼ E0 is in this domain, the lower is the voltage over the structure. Thus the minimum voltage occurs if a
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X , µm Fig. 5. Electric field distributions along the n0 collector for various combinations of the injection current and total current Jn0 (kA/cm2 )/J (kA/cm2 ): 1––0/0, 2––0.85/0.856, 3––5/7.15, 4–– 12/18.6, 5––26/41.3, 6––40/64, 7––70/110, 8––40/81.1.
2
0
100 200 300 400 500 600 700 COLLECTOR VOLTAGE, V
Fig. 3. Current–voltage characteristics for various electron injection current densities Jn0 (A/cm2 ): 1––1750, 2––3000, 3–– 5000, 4––12 000, 5––26 000, 6––40 000, 7––70 000. The dashed line corresponds to Jn0 ¼ 0. The critical current density Jcr ¼ 1712 A/cm2 . The left (initial) point of each curve corresponds to E0 ¼ 4000 V/cm.
combination of a high injection current and small values of E0 takes place. At a fixed injection current the minimum voltage will be achieved when E0 tends to zero (the carrier concentration on the left side tends to infinity). Note that this case cannot be treated as an ‘‘injecting contact’’ [8] because the electric field proves to be close to zero not only at x ¼ 0, but everywhere apart from the peak at the right boundary.
It is seen that this relation tends towards the limit 1 þ ðvps En =vns Ep Þ ¼ 1:302 when E0 ! 0, which is determined by a fundamental property of the material––the ratio of mobilities b ¼ vns Ep =vps En . Correspondingly, referring to Fig. 4, we can conclude that the minimum collector voltage is reached when Jn0 =J is equal to 1=ðb þ 1Þ at a fixed value of J . At high injection currents it is also possible to obtain an approximate analytical expression for the electrical field Em at the right boundary. Since an ap and Jn > Jp , ionisation by holes can be neglected in (1a) and (1b). The system (1) can then be written as follows Z x an dx ; Jp ¼ J Jn Jn ðxÞ ¼ Jn0 exp 0
Z x oE 1 1 1 J ¼ þ qND Jn0 exp an dx ox ee0 vn vp vp 0 Taking into account that E0 is small with respect to E in the avalanche region, we obtain, after some transformation, Em bn 1 1 1 Jn0 ¼ þ 1 E2 J ee0 ans bn vns vps bn Em J J J þ qND ln ð6bÞ vps Jn0 R1 where E2 ðxÞ ¼ 1 expðxtÞt2 dt. The maximum value of Em is again reached in the limiting case Jn0 =J ¼ 0:768. At J ¼ 100 kA/cm2 , for example, we obtain Em ¼ 3:65 105 V/cm. Formulas (6a) and (6b) and the curves in Fig. 5 show that the distribution of the electric field near the right-hand boundary at high currents ceases to depend on the collector thickness starting from a n0 region a few lm thick. The carrier density distributions, which determine the field domain, are shown in Fig. 6. Three regions can be conditionally segregated, the stretched quasi-neutral domain where the electron and hole concentrations are practically constant and differ by the value ND , the rightmost high-field domain, and a buffer domain where the electric field exceeds E0 value significantly but ionisation is still absent. Electron velocity in the latter domain increases with respect to that in the quasi-neutral
-3
A simple relation between E0 and Jn0 =J can be found when a domain with a constant electrical field is formed. The derivative dE=dx in (1c) can be set at zero for this domain. Neglecting the donor concentration at high Jn0 , we obtain: bn 1=bn 1 þ EEn0 J vps En ¼1þ ð6aÞ bp 1=bp Jn0 vns Ep 1 þ EEp0
4.0
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3.5
CONCENTRATIONS (n , p) / 10 , cm
1260
Electrons
3.0 2.5
Holes
2.0 1.5 1.0 0.5 0
8
16
24
32
40
X , µm Fig. 6. Distributions of carrier density at Jn0 ¼ 40 and J ¼ 64 kA/cm2 .
region, which can lead to a reduction in the electron concentration, as shown in Fig. 6. As soon as the right boundary is approached, the ionisation process becomes essential and the electron concentration begins to increase again. The rise in Jn0 =J (reduction in E0 ) causes an increase in the carrier concentration in the quasineutral domain, and the concentration tends to infinity as Jn0 =J tends to 0.768. This means that the ratio of the injection current to the total current is the most essential parameter determining the residual voltage across the collector region of a nþ –p–n0 –nþ transistor operating at extreme currents. The appearance of a high-field domain at the right boundary is similar to that observed in [9]. The role of the injection current cannot be understood from Ref. [9], however, since it does not specify the full set of boundary conditions. The quasi-neutral and buffer field domains cannot be compared, since they are not presented in [9], and finally the shape of the field distribution at moderate currents differs from that obtained in our simulations. 3.3. Simulation results for a p0 collector related to a pþ – n–p0 –pþ transistor structure The models for p0 and n0 collectors are quite similar, being distinguished only by the signs on the right-hand sides of Eq. (1), and one could assume analogous behaviour for both kinds of collector. The operational characteristics of the two collectors turn out to differ markedly, however. The injection current in the n0 collector can amount to a significant fraction of the total collector current, so that the ratio Jn0 =J can be as high as 0.768 in principle. A relatively small part of the total current must be generated by ionisation in this case. The situation is essentially different for the p0 collector, calculations for which show that the injection current,
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1
4.5
E / 10 , V/c m
550 2
4.0 3.5 3.0
5
C O L L E C T O R VO L T A G E , V
5.0 560
540
530 0.1
2.5 2.0
1
1.5
2
1.0 0.2
0.3
0.4
0
Jp0 / J
4
8 12 16 20 24 28 32 36 40
, m
Fig. 7. Dependence of the collector voltage on the ratio of the hole injection current to the total current. Total current density J (kA/cm2 ): 1––70, 2––100.
Fig. 8. Electrical field distributions along the p0 collector for two values of the hole injection current Jp0 (kA/cm2 ): 1––30, 2–– 43.
which in this case is the hole current, cannot exceed 43% of total current J . The major part of the collector current will thus be determined by the avalanche process, resulting in a very high collector voltage. The collector voltage dependences on the injection current fraction Jp0 =J for two values of the total current are shown in Fig. 7. The computed voltage values (500 V) are actually unphysical from the point of view of transistor switching from an initial voltage value of about 300 V. This simply means that no switching to the current value of 100 kA/cm2 can be obtained for this transistor at any realistic initial biasing. The calculated collector voltage varies only slightly with the injection current. The distributions of the electric field in the p0 collector at J ¼ 100 kA/cm2 are presented in Fig. 8. It is seen that they differ from that obtained for the n0 collector at high injection currents and resemble instead the field domain in a diode when the injection current is absent [6]. The only effect of the injection current to be seen in Fig. 8 is the reduced peak in the electric field near the left boundary, which causes an insignificant reduction in the collector voltage (Fig. 7). The maximum value of E at the right boundary turns out to be as high as 4:93 105 V/cm (compare with Em ¼ 3:65 105 V/cm for the n0 collector at the same collector current). Finally, although we see that the electric field varies insignificantly in most of the collector, its magnitude remains fairly large and sufficient for appreciable ionisation. Thus no quasi-neutral domain is formed. At first glance one could assume that the reason for the formation of a quasi-neutral domain lies in the difference in ionisation rate between the electrons and holes. The simulations performed for the p0 collector
with the interchanged ionisation coefficients for the electrons and holes did not show any appreciable reduction in the residual collector voltage, however. On the contrary, the simulations of the n0 region with interchanged ionisation coefficients provided only a slightly higher residual voltage than that shown in Fig. 4. This result can be qualitatively understood as follows: the formation of a quasi-neutral domain requires intensive hole generation by the ‘‘fast’’ electrons and ‘‘slow’’ sweeping of the holes out of the quasi-neutral region. The peak value of the electric field on the right-hand boundary is adjusted automatically in order to maintain a sufficient rate of hole generation at a certain collector current. (The lower coefficients an ðEÞ are used in the simulations, the higher one being the peak value for the electric field.) The corresponding increment in the residual voltage at a collector current of 105 A/cm2 with interchanged ionisation coefficients was found to be only moderate (10–20 V). In addition, we will show here by the reduction ad absurdum method that no quasi-neutral domain can exist in the p0 collector, due to details of the field dependence of the hole and electron velocities in Si (relations (2)). Let us assume that a quasi-neutral domain does exist. Then, denoting the right-hand side of Poisson equation (1c) by F ðEÞ and taking into account that ionisation is absent in this domain and carrier currents are constant, we can represent the electrical field in the form e E ¼ E þ E where E is constant and satisfies the equation F ðE Þ ¼ e is a small addition satisfying the linearized 0, while E equation
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ee0
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e dE dF e E ¼ dx dE E¼E
The solution of this equation is written in the form e ¼ E þ ðE0 E Þ expðcxÞ; where c ¼ 1 dF E ee0 dE E¼E Neglecting ND we obtain the following expression for c: c¼
1 Jp ee0 E vp ðE Þ
1þ
E En
E En
bn ! bn
E Ep
1þ
bp
E Ep
bp !
Here, as before, the upper and lower signs correspond to n0 and p0 collectors, respectively. It is seen that c < 0 for a p0 collector. This means that the electrical field will diminish on removal from the base–collector junction. This is impossible since the quasi-neutral solution cannot be matched with the solution in an avalanche region near the right boundary in this case. Thus, no domain in which ionisation is absent can exist in the p0 collector at high currents, which would inevitably lead to a high voltage over the structure. Vice versa, c is positive for the n0 collector, so that the analogous problem does not arise. This conclusion is in qualitative agreement with the experimental results of Ref. [11], where it is shown that n–p–n Si transistors are more addicted to nonthermal secondary breakdown than p–n–p transistors. The authors of Ref. [11] assume that this behaviour must be caused by the difference in ionization coefficients between electrons and holes, while our results clearly show that the main reason lies in the difference in carrier mobilities.
4. Discussion We failed to observe high-current switching experimentally in various types of pþ –n–p0 –pþ transistor with VCB0 ranging from 200 to 350 V when using the circuit shown in Fig. 2(a). No collector current pulse with an amplitude higher than a few kA/cm2 and a rise time shorter than 10 ns was observed, apparently confirming the results described in the previous section. Then, as shown in Section 3.2, the ratio of the injection current to the total current is the most essential parameter determining the residual voltage across the collector region of a nþ –p–n0 –nþ transistor operating at extreme currents. One can see from Fig. 2 that the experimentally observed residual voltage ranges from 70 V (for I 70 A) to 95 V (for I 120 A). These values can be compared with the data shown in Fig. 4 in order to evaluate the ratio Jn0 =J as achieved in the
experiment. Allowance should be made, of course, for the difference between the collector thickness in the FMMT-417 transistor (Wexp ¼ 13 lm) and one used in the simulations (W ¼ 40 lm). Simple subtraction of the product E0 ðW Wexp Þ from the collector voltage in Fig. 4 gives a range 0.60–0.63 for the ratio Jn0 =J , which would correspond to the experimentally observed voltages. Here E0 was calculated from expression (6). The theoretical prediction for Jn0 =J should be dependent on a specific (1D or even 2D dynamic) operating mode in the transistor. The processes in the emitter and base regions that are dependent on the structure and external circuit will affect the actual Jn0 =J ratio, and the current 1D static model is not proficient for independent prediction of this value. In spite of this, the steady-state solution presented here appears to be useful, since it shows in demonstrative form the physical reason and requirements for low residual voltage to be achieved across the switched-on device. 5. Conclusions A relatively low residual voltage (<100 V) across an avalanche nþ –p–n0 –nþ transistor at extreme currents (100 kA/cm2 ) was demonstrated both experimentally and by means of quasi-steady state numerical modelling. The voltage, which drops mostly across the right-hand boundary of the n0 collector region, is mainly defined by the ratio of the injection (electron) current to the total current. An increase in both this ratio Jn0 =J and the total current J will reduce the residual voltage, which can be as low as 20 V, provided that Jn0 =J tends towards its upper limit 1=ðb þ 1Þ ¼ 0:768 (b is the ratio of mobilities in Si). Details of the carrier velocities and the ionisation rates dependences on the electric field affect the collector voltage drastically. It was shown in particular that no high current switching is possible for the pþ –n–p0 –pþ Si transistor structure, mainly due to the difference in electron and hole mobilities.
Acknowledgements This work was supported by the Academy of Finland (project #50460) and by INTAS (project INTAS-010364). The authors would like to express their deep gratitude to M.E. Levinshtein for the fruitful and instructive discussions.
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