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JOURNAL OF APPLIED PHYSICS 97, 024502 (2005) Ultrahigh field multiple Gunn domains as the physical reason for superfast (picosecond range) switching of a bipolar GaAs transistor S. N. Vainshteina) University of Oulu/Dept. of El. Eng./El. Lab. Linnanmaa SF - 90570, Oulu, Finland V. S. Yuferev Ioffe Physico-Technical Institute, Politehnicheskaya 26, 194021, St.-Petersburg, Russia J. T. Kostamovaara University of Oulu/Dept. of El. Eng./El. Lab. Linnanmaa SF - 090570, Oulu, Finland (Received 8 April 2004; accepted 2 November 2004; published online 22 December 2004) The superfast (picosecond range) high-current switching observed recently in a GaAs junction bipolar transistor is explained by practically homogeneous carrier generation in the volume of the switching channels by a moving train of avalanching Gunn domains of large amplitude. The very fast 共⬃200 ps兲 reduction in the collector voltage is determined by shrinkage of each domain, provided the negative electron mobility in ultrahigh electric fields is taken into account and current filamentation takes place. The results of one-dimensional simulations show good quantitative agreement with experimental voltage and current wave forms when the simulated switching area is equal to the summed areas of the filaments observed in the experiment. © 2005 American Institute of Physics. [DOI: 10.1063/1.1839638] I. INTRODUCTION The design of a high-speed electron device which enables high current 共⬃1 – 100 A兲, high voltage 共⬃100 – 1000 V兲, subnanosecond and picosecond (ps) range switching is a fairly attractive proposition for the pumping of ps laser diodes1 in high-resolution optical radars,2 laser tomography, and three-dimensional imaging systems,3 as also for ultrawide band radars and communications,4 electron and optical shutters, streak-camera sweep modules, etc. The experimental observation of superfast switching in a GaAs junction bipolar transistor operating in the avalanche mode has recently been reported.5 This provides unique turn-on parameters not achievable with any active commercial semiconductor component available at present. The time required for the reduction in collector voltage from the initial biasing of ⬃300 V to a value of ⬃90– 100 V was shorter by a factor of ⬃15 than that in Si avalanche transistors, and an order of magnitude shorter than that predicted by the driftdiffusion model for a Si-like shape of the electron velocity versus electric field dependence. One should thus expect an original physical mechanism for the observed superfast switching associated with some of the specific properties of GaAs. Two obvious possibilities are photon-assisted carrier transport in the direct-band material, or alternatively negative differential electron mobility. The first possibility has been analyzed lately6 using a specially developed simulation code, and it has been shown that no notable acceleration in the switching transient can be obtained from photon transport due to the comparatively moderate rate of photon generation, even when account is taken of the stimulated emission. Alternatively, one could expect negative differential a) Electronic mail: vais@ee.oulu.fi 0021-8979/2005/97(2)/024502/9/$22.50 mobility to cause the generation of moving Gunn domains in the collector, and acceleration in the switching transient could be caused by impact ionisation within these domains. A preliminary evaluation of the effect of negative differential mobility was performed using the “Atlas” device simulator (Silvaco Inc.) with a two-dimensional (2D) drift-diffusion model included adopting a popular approximation7 for the dependence of electron velocity on the electric field e共F兲. Pronounced high-frequency voltage and current oscillations were observed in the simulations,6 associated with fairly complicated 2D dynamic mapping of the Gunn-like field domains, but no acceleration in the switching transient was found. The purpose of this article is to demonstrate that an explanation for the superfast switching is still possible even in the framework of a one-dimensional (1D) model including negative electron mobility, and to analyze the necessary conditions for the phenomenon to exist. II. A BRIEF HISTORY OF THE PROBLEM One of the most popular ways of generating high current pulses (up to ⬃100 A) of a few nanoseconds in duration in practical applications8 is to employ Si bipolar junction transistors (BJT) operating in the avalanche mode. From the physical point of view, the relatively fast switching of the device to a relatively low residual voltage is due to the avalanche injection, with pronounced shrinkage of the collector field domain caused by the formation and spread of a quasineutral (plasma) domain across the n0 collector region.9–11 This quasineutral domain is formed by the electron injection from the emitter, and by the impact generation of the holes near the collector contact and their drift towards the emitter. Therefore the switching time of a Si avalanche 97, 024502-1 © 2005 American Institute of Physics Downloaded 02 Feb 2005 to 130.231.49.186. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 024502-2 Vainshtein, Yuferev, and Kostamovaara transistor cannot be shorter (but rather significantly longer) than the time it takes for the holes to travel across the n0 collector layer at the saturated velocity. When much shorter (subnanosecond and ps range) current/voltage pulses are needed, generators based on dynamic breakdown in Si diodes (see Refs. 12 and 13 and references therein) can be used, even though they are not very compact, nor are they cheap. Finally, superfast switching in a GaAs (or AlGaAs/ GaAs) thyristor structure was reported,14,15 with a switching time that ranged from 200 to 600 ps. These devices are not yet commercially available, and the physical interpretation of this superfast switching remains an open question, despite formulation of the hypotheses of the streamer character of the experimentally observed narrow switching channel16 or multiple channels.17 The main problem in any physical explanation of the phenomenon consists in the fact that very powerful modulation of the conductivity of the n0 region in a GaAs thyristor structure is achieved faster than it takes for the carriers to travel through the layer with maximum possible (saturation) velocity. The streamer hypothesis is indeed able to explain this fact, but we are lacking a model which could provide evidence for how a sufficiently strong electric field can appear in the structure to cause the streamer discharge. The streamer hypothesis is also in contradiction with the requirement of sufficiently low conductivity near the front of the streamer head,18 while in the thyristor structure the superfast switching starts at an instant when the carrier density in the n0 region is relatively high. As in a GaAs thyristor, the recently observed ps range switching in a GaAs transistor5 occurs faster than the carriers can cross the blocking n0 layer. In general this kind of superfast switching can be associated with fast (faster than the carrier velocity) propagation of an ionization wave across the structure (like the streamer) or, alternatively, with more or less homogeneous carrier generation across the whole thickness of the n0 collector layer. We will show in this article by means of a 1D simulation that the second option can be realized in a GaAs transistor, provided that the differential electron mobility remains negative up to ultrahigh electric fields and switching occurs in a sufficiently small area of the device. The train of the moving and avalanching Gunn domains appear in the n0 collector at a certain stage of transistor switching, which causes very intensive carrier generation across the entire switching volume and determines a much shorter switching time than that in a Si avalanche transistor,10,11 where the carriers have to be delivered from the n+ emitter and the n0 − n+ interface in the collector. As for the streamer hypothesis, we think that this mechanism as less probable in a GaAs transistor (see discussion at the end of Sec. IV). III. SIMULATIONS OF THE SWITCHING TRANSIENT IN A GAAS TRANSISTOR A. Model The 1D dynamic drift-diffusion model used here is fairly closely analogous to that used for simulating the switching transient in a Si transistor: all the details can be found in Ref. 10. One essential difference concerns the use of Fermi in- J. Appl. Phys. 97, 024502 (2005) FIG. 1. External circuit used in the simulations: load resistor RL = 1 ⍀, storage capacitor C0 = 2.2 nF, total parasitic inductance of the loop L P = 3 nH, collector voltage UC共t兲, measured between the ohmic contacts to the structure, and collector current IC共t兲, derived from the voltage drop across the load resistor. stead of Boltzmann statistics. In doing so we have used the Einstein relation between the mobility and diffusion coefficient and the parabolic band model. The circuit used in the simulations, which corresponds to the experimental conditions,5 is shown in Fig. 1. The main difference relative to the Si transistor simulations concerns the dependences of carrier velocity and ionization rates on the electric field. Furthermore, the device area used in the simulations was fitted to the experimental data on current filamentation.5 In this experiment a number of light-emitting channels were observed along the perimeter of the emitter-base interface in a single transistor switching. These channels had a more or less random spatial distribution, with their locations varying from one current pulse to another. The characteristic size of a single channel (at half-maximum optical power) typically varied from ⬃4 to ⬃8 m, and the typical number of channels was ⬃10– 12. Thus the characteristic area of the device participating in high-current switching was about ⬃3 ⫻ 10−6 cm2, the value used in our 1D simulations. The following approximations for the ionisation rates in GaAs were used 2 ␣n,p = ␣0n,p ⫻ exp兵n,p − 关n,p + 共E0/E兲2兴0.5其, 共1兲 where ␣n and ␣ p are the ionization coefficients for the electrons and holes, E is the electric field, ␣0n = 1.7⫻ 106 cm−1, ␣0p = 2.45⫻ 105 cm−1, E0 = 6.5⫻ 106 V / cm, n = 25.6, and p = 47.6. The correspondence of these approximations to the experimental data19 and to those used in the Atlas device simulator (Silvaco Inc.) is illustrated in Fig. 2(a). The popular approximation for the dependence of the electron velocity n on the electric field E was used7 (the same approximation is employed in the Atlas device simulator, for example): n = 关n ⫻ E + s ⫻ 共E/Et兲4兴/关1 + 共E/Et兲4兴, 共2兲 where n is the low-field electron mobility and Et = 4 kV/ cm is the critical field at which the velocity reaches its peak value m ⬇ 2 ⫻ 107 cm/ s. The velocity saturates at s = 9.5⫻ 106 cm/ s in high electric fields E 艌 25 kV/ cm [see curve 1 in Fig. 2(b)]. It will be seen from the results of the simulations, however, that the factor that is particularly important for superfast switching is the range of electric fields in which negative differential mobility manifests itself. One can see from a comparison of curve 1 in Fig. 2(b) with the scattered graph corresponding to the experimental data20 that Downloaded 02 Feb 2005 to 130.231.49.186. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 024502-3 J. Appl. Phys. 97, 024502 (2005) Vainshtein, Yuferev, and Kostamovaara FIG. 2. (a) Ionization coefficients for the electrons and holes: the solid and dashed lines correspond to the approximation used in this work [see formulae (1)], the experimental data (see Ref. 19) for the direction 具100典 are shown in the scatter graph, and the dotted lines represent the approximation used in Atlas; (b) dependence of electron velocity on the electric field: curves 1 and 2 are calculated using the relations (2) and (3), respectively, and the scatter graphs correspond to the experimental data (see Ref. 20). negative differential mobility occurs up to very high electric fields, which is not accounted by formula (2). We have therefore used the following fit of the approximation for n共E兲 to the experimental data in most of our simulations: modif = n · 关a + b ⫻ exp共− E/E0兲兴. n 共3兲 Here the dimensionless coefficients a = 0.576, b = 0.49, and E0 = 1.5⫻ 105 V / cm. The electron velocity values calculated from Eq. (3) do not differ in practice from those calculated using Eq. (2) at low electric fields, but the velocity modif n does not saturate any longer at high fields and fits well with the experimental data [compare curves 1, 2, and the scattered graph in Fig. 2(b)]. Despite the very low modulus of the negative differential mobilities in very high fields (curve 2), the difference in simulation results for the two approximations (curves 1 and 2) is drastic (see next section). Since the results of the simulations presented later show a fine spatial structure, certain details of the numerical calculations deserve to be mentioned. The structure was properly meshed (6400 nodes along 46 m of the structure thickness), which means that the average distance between two neighbouring nodes was about 7 nm, and even the domains of minimal width 共⬃0.04 m兲 contained a sufficient number of nodes. (An increase or reduction by a factor of ⬃3 in the total number of nodes did not cause any notable change in the results of the calculations.) Temporal variation of the time step was used, since convergence could be achieved in some cases only with as small a time step as 10−15 – 10−17 s (due to the negative differential mobility and a high carrier FIG. 3. Collector voltage (a) and collector current (b) wave forms: curves 1 represent the simulation results obtained using the “popular” n共E兲 approximation defined by formula (2), curves 2 show those obtained using the n共E兲 approximation defined by formula (3), and curves 3 present the experimental data. density), and the maximum possible time step had to be used to shorten the total computation time. Another measure for improving the convergence was replacement of the Poison equation by the equation for the displacement current (see details in Ref.10). The error in the calculation of the residual between the positive and negative charges was checked in each time step.10 The following speculations speak in favor of the reliability of the calculation results. The same 1D model as employed earlier for a Si avalanche transistor has shown fairly good agreement with the experimental voltage and current wave forms and with the results of 2D simulations using the commercial (Atlas, Silvaco Inc.) code.10,11 The concentration at which the Gunn domains appear in the simulations agrees well with the Kroemer criterion. As in a Si transistor,10 the initial state of the transistor structure was calculated for the applied biasing and the transient calculations involving the external circuit were then performed, after the base current of the same shape and density as in the experiment5 had been applied at instant t = 0. (The same base current density means that the base current in the simulations was scaled according to the difference in total device area between that used in the experiment and that used in the simulations—see data on current filamentation earlier.) B. Effect of the negative differential mobility in high electric fields The simulated-voltage and current wave forms using both approximations (2) and (3) for the electron velocity are shown in Fig. 3, together with the experimental curves. One can see that there is fairly good agreement between the mea- Downloaded 02 Feb 2005 to 130.231.49.186. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 024502-4 Vainshtein, Yuferev, and Kostamovaara J. Appl. Phys. 97, 024502 (2005) FIG. 4. Doping profile across the transistor structure and electric field distributions at different instants t 共ns兲/collector current densities j (in units j c): 0 / 0; 1.5/ 1.4; 2.27/ 2.1 (from left to right). This particular example corresponds to the simulations with the n共E兲 approximation defined by formula (3). sured collector voltage and that simulated with the negative differential mobility in high electric fields [approximation (3)], while the simulations with velocity saturation [approximation (2)] give a much longer switching time. This conclusion was verified by performing the simulations with modified approximations n共E兲 analogous to formulae (2), but with a different saturation velocity for the electrons (higher or lower than that for the holes). Alternatively, the simulations were performed for n共E兲 dependences analogous to that represented by curve 2 in Fig. 2(b), but with a polynomial fit to the experimental data [not exponential, as in formula (3)]. This verification has convincingly confirmed that superfast switching is associated only with the presence of negative mobility in ultrahigh electric fields, and is not sensitive to the values of the saturated velocities or to any particular (exponential or polynomial) approximation for n共E兲. A significant difference can be seen in Fig. 3(b) between the simulated (curve 2) and measured (curve 3) current wave forms at low currents. Otherwise the plotting of these curves on a linear current scale provides fairly good agreement between the experimental and simulated dependences, the rise time of the current tr ⬇ 2.6 ns (from 0.1 to 0.9 of Imax) in both cases being determined by L P / RL of the circuit (Fig. 1). The pronounced difference at low currents can be understood by taking the current filamentation into account, a point discussed in detail in Sec. IV. The strong oscillations in the collector voltage [curves 1 and 2 in Fig. 3(a)] are caused by the generation of multiple Gunn domains, their temporal evolution and their absorption by the collector contant. The electric field profiles across the structure are shown in Figs. 4–6. Figure 4 presents the transistor structure and the electric field profiles in the initially biased structure, together with typical filed profiles at the beginning of the transient, when the current density slightly exceeds the critical level jc ⬃ q ⫻ ND ⫻ s ⬃ 1.1⫻ 103 A / cm2 [ND is the donor density in the n0 region, saturated velocity s ⬃ 107 cm/ s], so that the electric field across the structure is reconstructed but the moving Gunn domains have not yet FIG. 5. Electric field profiles corresponding to the transient presented in curves 1 in Figs. 3(a) and 3(b). The four instants t 共ns兲 correspond to the following current densities j (in units j c), t / j: 4 / 9.7, 4.8/ 81, 5.6/ 560, 7.3/ 1010. appeared. The field profiles at different instants of the switching transient for the approximation n共E兲 given by the formulae (2) are shown in Fig. 5, and those corresponding to formulae (3) in Fig. 6. The formation of moving (from p base to n+ collector) Gunn domains of relatively high amplitude starts in both cases (curve t = 4 ns in Fig. 5 and curve t = 2.63 ns in Fig. 6) after the current density has exceeded jc by a factor of ⬃5 (see Figs. 5 and 6) and the average density of both electrons and holes in the quasineutral regions of the n0 collector has exceeded ⬃3 ⫻ 1015 cm−3. From that point FIG. 6. Electric field profiles related to the transient presented in curves 2 in Figs. 3(a) and 3(b). The four instants t 共ns兲 correspond to the following current densities j (in units j c), t / j: 2.63/ 13.0, 2.98/ 120, 3.08/ 600, 3.25/ 2900. Downloaded 02 Feb 2005 to 130.231.49.186. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 024502-5 J. Appl. Phys. 97, 024502 (2005) Vainshtein, Yuferev, and Kostamovaara onwards a drastic difference in domain behavior has been observed for the two electron velocity approximations. Fairly broad domains of a “moderate” amplitude 共艋400 kV/ cm兲 determine relatively slow switching in the one case (Fig. 5), and the rapidly narrowing domains of growing amplitude (up to ⬃600 kV/ cm at some instants) lead to fast switching in the other case (Fig. 6). A very fast reduction in the collector voltage [curve 2 in Fig. 3(a)] is caused by drastic narrowing in the width of the running Gunn domains, from full width at half maximum (FWHM) ⬃1.5 m (curve t = 2.98 ns in Fig. 6) to FWHM ⬃0.05 m (curve t = 3.25 ns). Despite a simultaneous increase in the number of domains from ⬃6 to ⬃20 (see Fig. 6), the average voltage per domain is reduced, from ⬃50 V 共t = 2.98 ns兲 to ⬃2.5 V 共3.25 ns兲, thus reducing the total collector voltage from ⬃300 to ⬃50 V. (The voltage across the “plasma region” between the domains, ⬃103 V / cm⫻ 30 m ⬃ 3 V, is negligible.) The very fast growth in carrier density in the n0 region is caused by a huge rate of carrier generation. Indeed, the very high amplitude of the Gunn domains 共4 – 6 ⫻ 105 V / cm兲 gives rise to values as high as 1.6⫻ 104 – 105 cm−1 for the ionization coefficients [relations (1)] for both electrons and holes. These values are quite comparable to the reverse domain width, thus providing a considerable probability of an ionisation act within a single domain. Carrier generation is then distributed practically homogeneously across the structure, since the characteristic distance between the domains (ranging from ⬃6 to ⬃0.5 m during the transient) is much less than the thickness on the n0 collector 共⬃30 m兲. This is precisely the reason for the superfast switching. Namely, the carriers in a Si transistor10,11 have to be delivered to the blocking n0 layer from the n+ emitter (the electrons) and from the high-field domain located near the n+ collector (the holes), and thus carrier transport across the n0 layer limits the switching speed. In a GaAs transistor both the electrons and the holes are generated by the train of avalanching domains across the whole switching volume, and this reduces the switching time drastically. As we have seen, negative differential mobility in very strong electric fields is of major importance for the phenomenon of superfast switching, since it determines the possibility for a drastic reduction in the width of the Gunn domains and for growth in its amplitude. (This important point will be discussed in more detail in Sec. III.) density. This means that an increase in area A will reduce the voltage across the structure at each instant (for given values of L p , RL , jc and djc / dt), thus reducing the domain amplitude and the impact ionization rate. A comparison of the simulated voltage wave forms for three values of the parameter A is presented in Fig. 7(a). One can see that the switching time increases from ⬃200 ps to ⬃1 ns when the switching area is increased (and the current density reduced) by two orders of magnitude. The corresponding broadening in the Gunn domains for the lower current (and carrier densities) at about the same collector voltage is illustrated in Figs. 7(b)–7(d). In other words, the reduction in the area of the device (at a given voltage and current) causes an increase in the current density and carrier concentration. As we will see in the next sections, the higher carrier density causes stronger impact ionisation, and consequently a faster reduction in the collector voltage. D. Some properties of avalanching Gunn domains of very high amplitude C. Effect of switching area Another very important factor is the operational area of structure A. It is obvious that the switching rate should be determined by the instant current density and voltage across the structure, since each of these values will affect the parameters of the Gunn domains. The voltage Uc across the structure at each instant can be expressed as Uc = U0 − L p ⫻ 共dIc/dt兲 − Ic ⫻ RL = U0 − 关L p ⫻ 共djc/dt兲 − jc ⫻ RL兴 ⫻ A, FIG. 7. Simulated voltage wave forms (a) for three operational areas A 共cm2兲: 1 – 3 ⫻ 10−6, 2 – 3 ⫻ 10−5, and 3 – 3 ⫻ 10−4. The dashed lines represent the simulation results, but a more illustrative picture is given by the solid lines, which “generalize” the very marked voltage oscillations. Electric field profiles for the three operational areas at instants when the collector voltage is about 200 V are shown in (b)–(d). 共4兲 where U0 is the voltage across a voltage source (or storage capacitor) and Ic and jc are the collector current and current We have seen that superfast switching is determined by very fast reduction in the width of the moving Gunn domains with a simultaneous increase in the domain amplitude for a given collector voltage. Higher domain amplitude allows the sustaining of a high rate of avalanche multiplication despite the reduction in the voltage across the transistor during the transient. The properties of these domains deserve to be discussed, since there is no description of domains of ultrahigh amplitude in the literature. (Neither have we found any model in which the parameters of the Gunn domains are similar to those found in our case.) Downloaded 02 Feb 2005 to 130.231.49.186. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 024502-6 J. Appl. Phys. 97, 024502 (2005) Vainshtein, Yuferev, and Kostamovaara FIG. 8. The electric field, on a linear (a) and logarithmic (b) scale, carrier density (c), and current component profiles (d) around a single domain corresponding to instant t = 3.25 ns (compare with Fig. 6). Jn, J p, and Jd are the electron, hole and the displacement current densities, and the dotted line in (d) represents the total current, i.e., the sum of the three current components. The velocity of the domain d is about equal to the saturated velocity of the holes s ⬃ 107 cm/ s. The vectors of the electron velocities d − n shown in (a) illustrate the spatial gradient of the electron velocity within the “backwall” of the domain in a coordinate system moving together with the domain. An example of field and carrier concentration profiles, together with profiles for the various current components around one of the domains in the train (see the total profile t = 3.25 in Fig. 6) is shown in Fig. 8. The difference between the electron and hole densities (c) determines the domain shape 共dE / dx ⬃ n − p兲, and the peaks in the displacement current Jd (d) characterize the dynamics of the local electric field 共dE / dt ⬃ Jd兲. The electron current Je determines in practice the conductivity in the electron-hole plasma on the right and left sides of the domain, while the contribution of the hole current J p is highly essential within the domain. The shapes of the profiles shown in Fig. 8 are qualitatively similar to those corresponding to different domains presented in Fig. 6 within the interval 2.98– 3.25 ns, while the domain width and amplitude changes depend to a great extent on the moment in time and the instant carrier density in the plasma region (between the domains). The Gunn domain width wd and amplitude Em are plotted against the carrier density 共n ⬇ p兲 in the plasma region surrounding the domain in Fig. 9 (a graph based on simulation data corresponding to various instants and spatial domain locations). Despite the fairly chaotic behavior of Gunn domains during the transient, one can see a pronounced correlation in the case of negative differential mobility [formula (3)]. The domain width wd is reduced significantly in this case as the plasma density increases [Fig. 9(a)], while the analogous dependence in the case of velocity saturation [formula (2)] is not pronounced and the width of the domains is not reduced below ⬃1 m (related data are not presented in the graph). Then, despite the fact that the amplitude of the domains corresponding to a FIG. 9. Effect of average electron-hole density in the plasma surrounding a Gunn domain on the domain width wd (a) and amplitude Em (b) in the case of nonsaturated electron velocity [formula (3)]. No regular correlations of this kind were found in the alternative case of velocity saturation [formula (3)] where the domain width always exceeded 1 m and the amplitude did not exceed ⬃4 ⫻ 105 V / cm at any carrier density. particular plasma density may lie in a fairly broad range of electric fields, one can mark a sector [see Fig. 9(b)] in which the domain amplitudes are typically confined, so that the maximum achievable (and average) domain amplitudes correlate to the plasma density. Again, no pronounced correlation with plasma density has been found for the case of velocity saturation, and the amplitude of the moving Gunn domains never exceeds ⬃4 ⫻ 105 V / cm in such a case. IV. DISCUSSION A. Multiple Gunn domains: Generation and spread The generation and spread of a single Gunn domain in a slab of a semiconductor with negative differential mobility is very well known.21 Various aspects of the problem, including domains with both electrons and holes present,22 with avalanche multiplication,23 etc., have been considered in the numerous publications. The domains of ultrahigh amplitude, however, analogous to those observed in our simulations have not been discussed so far. The generation of multiple Gunn domains has been observed and this typically occurs under fairly nonsteady conditions, e.g., when a steep voltage front 共dU / dt ⬃ 1012 V / s兲 is applied to a semiconductor slab (see Ref. 24). The transient in the avalanche GaAs BJT is extremely fast, so that the generation of multiple domains is not surprising as such. Fairly specific features, however, are a relatively regular domain structure and a systematic change in the intervals between the domains (see the snapshots t = 3.08, 3.25 ns in Fig. 6). What happens is that new domains are continually generated near the base-collector junction (in some cases also in the base, and even in the emitter), from where they spread towards the collector contact and are absorbed there. The average velocity d of the domain spread is Downloaded 02 Feb 2005 to 130.231.49.186. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 024502-7 J. Appl. Phys. 97, 024502 (2005) Vainshtein, Yuferev, and Kostamovaara fairly close to the magnitude of the saturated velocity for the holes 共⬇107 cm/ s兲, but d may differ from this typical value by ⬃共−35% – + 50%兲 at various instants and spatial positions in each particular case. There is no regular correlation of this velocity with the instant current density across the structure, so that the same velocity values can be observed in principle at the beginning and end of the switching transient. Furthermore, as one can see from the profiles t = 3.08 and t = 3.25 ns in Fig. 6, the spatial density of the domains decreases from left to right. This is not caused by a regular change in the spread velocity as one might assume when looking at a particular snapshot, but reflects mainly the “history” of domain generation, in that the domains are generated more frequently with increasing time (and current), and at a later instant (when these domains are approaching the collector contact) the domains “remember” the longer spatial intervals between them at the instant of their generation. This point is correct in general, but in some cases (not very frequently) the disappearance of a moving domain has also been observed within the n0 region before the domain reaches the collector contact. This also affects the intervals between the particular domains in some cases, and finally the intervals are affected by the history of certain variations in velocity during the spreading of the domains. B. Avalanche multiplication in a single domain Let us compare the intensity of impact ionization in the middle of the transient for two types of domain: one represented by the profile at t = 4.8 ns in Fig. 5 and the other by that at t = 3.08 ns in Fig. 6. The number of electron-hole pairs generated by a single carrier (electron or hole) during its run through a high field domain can be evaluated as follows: Gn = n ⫻ d − n 冕 x2 x1 ␣ndx, Gp = p ⫻ d + p 冕 x2 ␣ pdx, x1 共5兲 where n and p are the electron and hole velocities in a high field region of the domain, d is the domain velocity, and the integration is performed over the domain width. According to Sec. IV A, d ⬇ p ⬇ 107 cm/ s, the electron velocity in the high-field region n ⬇ 5.5– 7 ⫻ 106 cm/ s [see Fig. 2(b)], and the ionization coefficients are defined by the formula (1). The signs + or − in the denominators of the formulae (5) are determined by the fact that the holes are moving from right to left, while both the domain and the electrons are moving in the opposite direction. [It should be noted that the domain will typically overtake the electrons moving in the plasma region with a velocity of ⬇0.8⫻ 107 cm/ s in an electric field of ⬇1100 V / cm—see formula (2) and the electric field profile in Fig. 8.] The estimates arrived at using the formulae (5) and the simulated electric field profiles show that ionization is much stronger for the narrow domains of high amplitude shown in Fig. 6 than for broad domains of lower amplitude (Fig. 5). In the first case (superfast switching), an electron creates on average ⬃ one electron-hole pair while passing the high-field part of the domain and a hole creates ⬃0.2 electron-hole pairs (ionization by the electron is stronger, since ␣n ⬎ ␣ p at E ⬎ 400 kV/ cm and the electron spends a FIG. 10. The electric field on a linear (a) and logarithmic (b) scale, carrier density (c), and current components profiles (d) around a single domain corresponding to instant t = 4.8 ns (compare with Fig. 5). Jn, J p, and Jd are the electron, hole, and displacement current densities, and the dotted line in (d) represents the total current, i.e., the sum of the three current components. The results correspond to n共E兲 dependence with velocity saturation in ultrahigh fields [formula (2)]. longer time in the high-field region than the hole). In the second case (“slow” switching), both an electron and a hole create on average ⬃0.1 electron-hole pair, for despite the domains being broader, the reduction in ionization coefficients in a lower electric field predominates in this case. These estimates illustrate the much higher impact ionization rate achieved by narrow domains of ultrahigh amplitude relative to broad domains of “moderate” amplitude. As we have seen earlier, superfast switching in the case of negative differential mobility in ultrahigh fields is caused by significant domain narrowing and by the growth in the average domain amplitude (see Fig. 9). Unfortunately we do not at present have an illustrative physical explanation for this important result, but simple qualitative speculations show a tendency for the domain to narrow due to an increase in the steepness of its “backwall.” Indeed, in a coordinate system aligned with a moving domain one can see a gradient in electron velocity within the back wall of the domain [see Fig. 8(a)] that causes growth in the spatial gradient of electron density [at x ⬇ 31.62 m in Fig. 8(c)] and should thus cause a sharpening with time of the backwall of the domain. No analogous gradient in electron density exists in the case of velocity saturation in ultrahigh fields [see the corresponding profile at x = 19− 20.5 m in Fig. 10(c)]. A qualitative explanation for the sharpening of the front domain wall would require more sophisticated speculations. C. Filamentation It was shown in Sec. III that two principal conditions must be satisfied for superfast switching to occur. The first (a continuous reduction in electron velocity up to extremely high electric fields) was discussed earlier, and the second is Downloaded 02 Feb 2005 to 130.231.49.186. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 024502-8 J. Appl. Phys. 97, 024502 (2005) Vainshtein, Yuferev, and Kostamovaara the requirement of a very high current density 共⬃106 A / cm2兲, which allows the multiple moving domains to be of very high amplitude (up to ⬃6 ⫻ 106 V / cm) and small width 共⬍0.1 m兲. This condition is automatically satisfied by the device itself by means of current filamentation,5 which appears to be an intrinsic property of avalanching Gunn domains.25 An interesting conclusion can be reached by comparing the simulated and measured current wave forms [curves 2 and 3 in Fig. 3(b)]. These exhibit a very significant difference at the beginning of the transient 共t 艋 3.1 ns兲, which can be understood as follows. The collector current in the experiment (curve 1) increases first along the whole perimeter of the emitter-base interface (2 mm in length) after the triggering base current has been applied. This homogeneous current flux along the perimeter takes place until the current density exceeds the critical value jc ⬃ q ⫻ ND ⫻ s ⬃ 1.1⫻ 103 A / cm2 (ND is the donor density in the n0 region). From that point onwards, a drastic reconstruction of the field domain causes a significant acceleration in switching. Any spatial current fluctuation should tend towards current filamentation, because the higher the local current density is, the higher will be the ionization rate in the Gunn domains, which will cause further growth in the local current density. We assume in our interpretation of the difference between curves 2 and 3 in Fig. 3(b) that the “shelf” manifested in curve 3 around the instant t ⬃ 2 ns corresponds to the beginning of filament formation. The current density in the simulations (curve 2) at the instant t ⬃ 2 ns is jsim ⬃ 3 ⫻ 103 A / cm2, and that in the experiment will be of the same value 共jexp ⬃ 1 A / Aexp ⬃ 3 ⫻ 103 A / cm2兲, provided that the switching area at the beginning of the transient is Aexp ⬇ 2 mm⫻ 15 m= (perimeter ⫻ lateral current spread). The characteristic size of 15 m assumed in this estimate for the lateral current spread agrees well with that observed in 2D simulations6 for comparable current densities. Thus the current density in the experiment at t ⬍ 2 ns should be the same as that in the simulations, but the current flows homogeneously along the whole perimeter of the structure, and thus its total magnitude is ⬃ two orders of magnitude larger than that in the simulations. Filament formation is accompanied by a powerful acceleration in current growth inside the filaments, which does not happen across the rest of the perimeter. Thus the rapid increase in the current across the filaments (curve 2, t ⬍ 3 ns) is “hidden” in the total current at the shelf on curve 3. After the channels have formed (t 艌 3 ns), the current that is flowing outside them in the experiment becomes negligible relative to the total current magnitude and the simulated current fits well with the measured one. It is worth noting that such a good agreement between the simulated and measured voltage and current wave forms is somewhat surprising, since the diameter of the filaments (switching area) in the simulations is assumed to be time independent, which is hardly likely to be the case in a real structure. The subterrahertz range 共⬃1011 Hz兲 voltage oscillations observed in the simulations (but not in the experiment) are associated with the generation and absorption of the multiple Gunn domains, and their excitation is facilitated by the inductance of the external circuit. One should remember that our model does not take into account the damping of the oscillations by the barrier capacitance of the collector p-n junction of the nonswitched part of the transistor structure. Taking account of this barrier capacitance (which is dependent on the total device area) should smooth out the oscillations in the simulated collector voltage, but will not in principle change the switching mechanism. Finally, it is worth noting that the superfast switching in the narrow channels associated with multiple avalanching Gunn domains is not equivalent to streamer discharge. Indeed, the fast switching a GaAs transistor begins when the carrier density in the n0 region surrounding the switching channel exceeds ⬃1016 cm−3 (see t = 2.98 ns in Fig. 6), while a streamer can hardly remain stable at such a high carrier density near the front of the streamer head.18 Then, streamer discharge means carrier generation near the streamer head, while the solution observed in our 1D simulation demonstrates practically homogeneous carrier generation along the whole channel length. Besides, there is a physical reason for current filamentation in the presence of avalanching Gunn domains (see earlier and Ref. 25), which differs from the streamer mechanism. And finally the characteristic diameter of streamer channels is typically smaller (⬃1 – 3 m)18 than that observed for a GaAs transistor.5 V. CONCLUSIONS The superfast switching observed recently in a GaAs transistor operating in avalanche mode has been explained. Good quantitative agreement was found between the simulated and measured collector voltage wave forms. It is shown that the intrinsic positive feedback in a GaAs BJT operating in the avalanche mode causes the generation of multiple avalanching Gunn domains, which move across the n0 region towards the collector contact and form an electron-hole plasma between the domains that is analogous to practically homogeneous carrier generation in the whole switching volume. The presence of negative differential electron mobility in ultrahigh electric fields causes drastic domain narrowing and growth in domain amplitude, a process which becomes much more pronounced at a higher carrier density. Avalanche multiplication in the domains causes current filamentation, which reduces the operational area, thus increasing the carrier density, and the increase in carrier density in turn causes growth in the domain amplitude and further increases the rate of avalanche generation. These processes allow the time required for voltage reduction across the device to be less than the time it takes for a carrier to cross the n0 collector layer at the maximum possible (saturated) velocity. ACKNOWLEDGMENTS This work was supported by the Academy of Finland (Project No. 50460) and INTAS (Project No. INTAS-010364). The authors are grateful to all the participants in the project for their assistance and to M. E. Levinshtein for stimulating discussions. Downloaded 02 Feb 2005 to 130.231.49.186. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 024502-9 1 J. Appl. Phys. 97, 024502 (2005) Vainshtein, Yuferev, and Kostamovaara S. Vainshtein, J. Kostamovaara, M. Sverdlov, L. Shestak, and V. Tretyakov, Appl. Phys. Lett. 80, 4483 (2002). 2 A. Kilpela, R. Pennala, and J. Kostamovaara, Rev. Sci. Instrum. 72, 2197 (2001). 3 A. Biernat and G. Kompa, J. Opt. 29, 225 (1998). 4 T. W. Barret, Microwave J. , 22 (2001). 5 S. Vainshtein, J. 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