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
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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
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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-
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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.
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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.)
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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
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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
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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.
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024502-9
1
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