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The book by M. Levinshtein, J. Kostamovaara, S. Vainshtein "Breakdown Phenomena in Semiconductors and Semiconductor Devices" (Review) has been published just now in two versions: 1. As a Review in the Special Issue of International Journal of High Speed Electronics and Systems: M. Levinshtein, J. Kostamovaara, S. Vainshtein "Breakdown Phenomena in Semiconductors and Semiconductor Devices" (Review) Intern. Journ. of High Speed Electron. and Systems (IJHSES):14 (4), 921-1114 (2005) and 2. As a hard cover book M. Levinshtein, J. Kostamovaara, S. Vainshtein "Breakdown Phenomena in Semiconductors and Semiconductor Devices" World Scientific Publishing Company Singapore - New Jersey - London - Hong Kong, (2005), ISBN: 9812563954 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ We dedicated this book to our relatives: To the memory of Julia Titova Michael Levinshtein

To my family Juha Kostamovaara To my parents Serafima and Naum Vainshtein Sergei Vainshtein

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Why we wrote this book?

One of the main reasons for that is the intention to draw the attention to the original and promised results obtained at University of Oulu with silicon, and, especially, with GaAs Avalanche Transistors and at the Ioffe Institute with silicon and GaAs Avalanche Sharpers (Diodes with Delayed Breakdown (DDB)

However, the book has been written not as a description of separate effects and devices but rather as a consecutive course on the breakdown phenomena in semiconductors and semiconductor devices.

We believe that the knowledge of breakdown phenomena is very important and useful for any scientist or engineer dealing with semiconductor devices because, generally speaking, in order to provide maximal speed and maximal power, many semiconductor devices must operate either under breakdown conditions or very close to these.

The general content of the book looks as follows:

Preface

1. Introductory Chapter

2. Avalanche Multiplication

3. Static Avalanche Breakdown

4. Avalanche Injection

5. Dynamic Breakdown

Conclusion

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Preface

It is noted in the preface that one form of avalanche breakdown has been known to mankind from ancient times: lightning is inscribed in the tales and myths of all primitive tribes.

The first known practical application of the avalanche breakdown goes back to the first century of our era. There is a fish in the Mediterranean, the electric ray, or skate, which was called “narcae” by the ancient Greeks, a word which means “paralyzing”. It is known nowadays that the voltage generated by this fish can reach 200 Volts. The Roman physician Scribonius, in his famous writing «De Compositiones Medicamentorum», published in AD 40, described the using of this narcae for the treatment of headaches, gout and some other diseases. The treatment was rather painful. This may be the reason why the term “breakdown” is associated very often with such unpleasant concepts as “failure” and “destruction”.

However, electrical breakdown itself is not connected with any form of destruction.

One widely used microwave device, the IMPATT diode, for example, has a characteristic operation frequency of about 100 GHz (1011 Hz), which means that it goes into a mature avalanche breakdown regime 1011 times a second. Since the guaranteed lifetime of a commercial IMPATT diode is at least 5000 hours, each diode will go into this regime safely no less than ~ 3×1018 times. (Note that the age of the Universe is just ~ 1018 s).

Moreover, impact ionization, avalanche and breakdown phenomena form the basis of many very interesting and very important semiconductor devices, such as avalanche photodiodes,

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avalanche transistors, suppressors, sharpening diodes (diodes with delayed breakdown), and IMPATT and TRAPATT diodes.

As it's mentioned in the Preface, many books contain chapters or sections devoted to the principal features of the avalanche and breakdown phenomena, and there are many good books and outstanding reviews concerning certain special aspects of these phenomena.

The aim of this book is to summarize the main experimental results on avalanche and breakdown phenomena in semiconductors and semiconductor devices and to analyse them from a unified point of view. This book has been written by experimentalists for experimentalists.

We do not deal at all with fundamental theoretical aspects such

as the distribution function of hot electrons, nuances of the band structure at high energies, etc., but instead we focus our attention on the phenomenology of avalanche multiplication and the various kinds of breakdown phenomena and their qualitative analysis.

It is noteworthy that in order to provide maximal speed and

maximal power, many semiconductor devices must operate

either under breakdown conditions or very close to these.

Consequently, an acquaintance with breakdown phenomena is very important and useful for any scientist or engineer dealing with semiconductor devices.

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Chapter 1. Introductory Chapter 1. Introductory Chapter 1.1 Elementary act of impact ionization

1.2 Auger recombination 1.3 Energy of electrons and holes as a function of electric field

1.4 Main approaches for describing ionization phenomena 1.4.1 Approximation of the characteristic breakdown field Fi

1.4.2 Monte-Carlo method 1.4.3 Ionization rates approximation

In the introductory chapter (Chapter 1) we will briefly discuss the main definitions and establish the main approaches to describing breakdown phenomena.

The basis of all ionization, breakdown and avalanche effects, without exception, is the elementary act of ionization

The elementary act of impact ionization. Collision of an energetic

electron (or hole) with an atom produces two new free carriers: an electron and a hole.

The minimal energy necessary to carry out the act of impact ionization (the threshold energy Eth) and its connection with band structure has been discussed.

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It has been demonstrated that the probability of finding an electron (hole) that is able to cause an act of impact ionization in comparatively wide band semiconductors (Si, GaAs) at room temperature is very small at low electric fields.

However, in narrow-gap semiconductors (InSb) with a small

value of the forbidden gap Eg, this probability can be large enough even at equilibrium, in the absence of an electric field. Consequently, the avalanche ionization plays an important role in such semiconductors even at equilibrium.

The probability of ionization increases exponentially with the energy of the carrier E , so that in a strong electric field, when the energy E is large enough, the effects of impact ionization becomes very important in semiconductors of any value of Eg.

Accordingly, the dependencies of the electron and hole energy and electron and hole drift velocities versus electric field have been discussed for different semiconductors.

From three main approaches to describe the breakdown processes, the most rigorous and complicated approach: Monte-Carlo simulation is just mentioned. (However, several main References on this technique have been given) Two other approaches:

Approximation of the characteristic breakdown field Fi and Ionization rates approximation are analyzed in detail.

Approximation of the characteristic breakdown field Fi is obviously a very rough approximation indeed, since the transient problems cannot be even formulated in the framework of this approach, for example. Nevertheless, this approximation is not infrequently used, e.g. to estimate the magnitude of the breakdown voltage at p-n junctions and in Schottky diodes with arbitrary doping

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distribution across the bias, in field-effect and bipolar transistors (FETs and BJTs), in thyristors and in many other semiconductor devices. On the other hand, it can be used for qualitative analysis in very complicated situations connected with breakdown phenomena (dynamic breakdown)

Ionization rates approximation is the "workhorse" of the theory of ionization phenomena. It is a very productive and effective compromise between the “over-simplified” approach of the effective breakdown field Fi and the rigorous but rather complicate Monte-Carlo simulation procedure. The definitions of the ionization rates, their dependencies on the electric field in different semiconductors, the limitations of the local models, and the main equations describing avalanche phenomena in the frame of this approach are established and analyzed. Shortly speaking, in the Introductory Chapter (Chapter 1) we briefly discussed the main definitions and establish the main approaches to describe the breakdown phenomena.

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Chapter 2. Avalanche Multiplication 2. Avalanche Multiplication 2.1 Fundamentals of avalanche multiplication

2.2 Avalanche photodiodes 2.2.1 Spectral sensitivity 2.2.2 Dark current 2.2.3 Quantum efficiency

2.2.4 Time response 2.2.5 Multiplication factor

2.2.6 Avalanche excess noise Let us consider the following simple situation. An electron is injected

into a sample of length L with a homogeneous field F

Let us suppose that at this given field F the electron ionization rate αi

is much larger than the hole ionization rate (αi >> βi), so that αiL >> 1

but βiL << 1. This means that the electron performs numerous

elementary acts of impact ionization as it passes through the sample.

On the other hand, the probability of a hole making even one act of

impact ionization is practically zero.

This is the simplest case of avalanche multiplication.

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An electron is injected into a sample from the left. In travelling a distance li = 1/αi (on average) it will create a new electron and a hole. The hole will move to the left (without ionization), while emergent electrons will move to the right and create further electron-hole pairs, and so on.

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For this case, multiplication factor Mn for electrons injected at the cathode Mn = jn(L)/jn(0) (i.e. ratio of the number of electrons at the "output" to the number of the electrons at "output is evidently equal to

L

n

nn

ie)(j)L(jM α==

0

In general case, both types of carrier are able to create electron-hole pairs: An electron is injected into the sample from the left. In travelling the distance

1/in il α≈ it creates an electron-hole pair. The hole will then move to the left and, in travelling the distance iip /l β1≈ (αi ~ βi), will also creates an electron-hole pair. The new electron created by the hole will now move to the right and create a new electron-hole pair, and so on. For this general case multiplication factor M is given by:

∫ ∫ ′−−−= L x

iii

n

dx]xd)(exp[M

0 01

1

βαα

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At αi = βi = α,

∫−=== Lpn

dxMMM

01

1

α

It is seen from (2-7) that at ,dxL

i 10

→∫α ∞→nM .

The condition ∞→nM corresponds to avalanche breakdown.

For homogeneous field distribution along the avalanche region, the condition of breakdown has the simplest form: 1=Lα . This means that avalanche breakdown occurs (at ii βα = ) when the electron (and hole) creates just one electron-hole pair on average while travelling through the avalanche region L. Three important points are worth noting

1. With ∞→nM , it is not necessary to have any external carriers to support the avalanche breakdown process. Breakdown is a self-supporting process.

The condition )( dx ii

Lβαα ==∫ 1

0

must be satisfied at any bias voltage 0V (for thVV >0 ) and at any current density j.

2. With a change in current density, the field distribution along the avalanche region is reconstructed to support the main condition of a

breakdown 10

=∫L

dxα

3. The larger the ratio ii / βα (or ii /αβ ) is, the larger the multiplication factor M that can be reached before breakdown occurs. The most favourable condition for reaching maximum values of M is

ii βα >> (or ii βα << ), and the most unfavourable condition is ii βα =

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The phenomenon of avalanche multiplication is used directly in one of the most important types of photodetector: the avalanche photodiode. There are many types of photodetectors: photoconductors, reverse-biased p-i-n diodes, Schottky diodes, phototransistors etc. Avalanche photodiodes (APD), which combine fairly high internal gain (up to several hundred times) with a high limiting frequency of operation (up to several Gigahertzes), are widely used in fibre-optic communication systems. From a physical point of view, an APD is simply a conventional photodiode (reverse-biased p-n or p-i-n structure, or a reverse-biased Schottky diode) that is under a bias which provides a substantial multiplication factor M in the reverse-biased barrier. In such a regime, the carriers generated by light create other carriers that lead to internal gain in the APD. Special precautions must be taken, however, to guard against surface breakdown in APDs (by means of guard rings, junction termination extensions formed by implantation, etc.. There are a rich variety of APDs constructions and modifications, and we discuss here only the principal features of APDs. All main parameters of APDs: spectral sensitivity, dark current, quantum efficiency, time response, multiplication factor, and avalanche excess noise have been briefly considered. It has been emphasized that

in a well-designed APD, the optimal compromise must be achieved between acceptable quantum efficiency (which requires a relatively large value for the space-charge width W), a small time response (which requires a relatively small value for W), not too great magnitudes of the multiplication factor M, high gain, and low noise.

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Chapter 3. Static Avalanche Breakdown

3. Static Avalanche Breakdown 3.1 Introduction 3.2 General form of the static “breakdown” current-voltage characteristic 3.2.1 Microplasma breakdown 3.2.2 Homogeneous (“mature”) breakdown 3.2.2.1 Contact resistivity 3.2.2.2 Thermal resistance 3.2.2.3 Space-charge resistance 3.2.3 Negative differential resistance 3.2.3.1 Qualitative consideration 3.2.3.2 The zero doping ( p i n− − ) structure 3.2.2.3 Computer simulation

3.2.4 Second part of the current-voltage characteristic, with positive differential resistance at very high current densities 3.3 Avalanche suppressor diodes 3.3.1 Principle of operation 3.3.2 Main parameters

3.4 IMPATT diodes 3.4.1 Principle of operation 3.4.2 Some physical problems that arise at very high frequencies

Systematic studies of breakdown phenomena in solids began

more than 80 years ago, in the early 1920s.

Approximately at the same time it was found that the scenario

for breakdown depends critically on the magnitude of the ratio dV0/dt,

where V0 is the bias applied to the structure and t is time. Over a very

wide range of magnitudes of dV0/dt, from very small (quasi-static)

to fairly large, just the same "conventional" scenario is followed,

that usually known simply as “breakdown”.

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Considering the static breakdown we followed to the current-voltage (I-V) characteristic of a reverse biased barrier: p-n junction, or Schottky diode, or heterojunction over a very wide range of current densities j

Qualitative current-voltage characteristic of a reverse-biased p-n junction (or

Schottky diode) in a condition of static breakdown.

Seven characteristic parts can be distinguished in this curve. Part 1 is associated with conventional leakage current considered in Chapter 2 Part 2 is associated with the avalanche multiplication phenomenon which is also considered in Chapter 2.

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Part 3, an area of "microplasma breakdown", occurs usually in any semiconductor diode structures of large area and even in diodes of small area if fabricated on the basis of relatively new semiconductor materials). In this part of the curve, avalanche breakdown occurs only at local points in the reverse-biased junction (the microplasma channels).

Part 4, representing homogeneous ("mature”) breakdown, is the most important and best-studied part of the current-voltage characteristic. Sometimes just this part is regarded as a "breakdown" current-voltage Part 4 is characterized by a very sharp increase in j with growth in V0

and a small positive differential resistance dj

dVRd0= , which is the

main parameter of this part of the I-V characteristic.

Part 5 is characterized by a very sharp increase in dR with further growth in V0. It is very often difficult to observe this part of the curve, because the appropriate range of V0 can be rather narrow.

Part 6 is the section of Negative Differential Resistance (NDR). To observe this part experimentally “point by point” it is necessary to use a circuit with a large load resistance lR . In the circuit with a low load resistance the current density will “jump” from the point 5 to point 7 As the amplitude of such a jump can be very large (several orders of magnitude), “overheating” of the device is possible, leading to its destruction. As in other systems with an NDR of the S-type (current increases as the bias decreases), the current filaments in devices can appear in this NDR part of the I-V characteristic. This filamentation increases the local current density, and consequently the “overheating” of the “hot points” in the structure.

In Part 7, with extremely high current densities, the differential resistance of the In Part 7, with extremely high current densities, the differential resistance of the structure becomes positive again. This effect appears mainly as a result of saturation of the ionization rates

ii βα and in very high electric fields. Electron- hole scattering and the recombination of carriers can also contribute to this effect.

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Microplasma breakdown. The phenomenon of microplasma breakdown was observed and studied for the first time in Si p-n junctions (W. Shockley, “Problems related to P-N junctions in silicon” Sol. State. Electron, 2 (1), 35-60 (1961). Later, however, it was demonstrated that it is observed in reverse-biased junctions in any semiconductor materials: Ge, GaAs, GaP, ternary A3B5 compounds, SiC, GaN, etc.

This type of breakdown reveals usually the presence of imperfections in the space-charge region of a reverse-biased junction, as an imperfection, especially when located close to a p-n junction, will causes a local increase in the field at some point in the junction

As a result, with the reverse bias increasing, the breakdown occurs not across the whole area of the junction but only at one local point where the field is at its maximum and the breakdown condition is satisfied at the smallest magnitude of the reverse bias V0 (known as the “first microplasma”). The presence of dislocations, micro-occurrences of "second phase", metallic or dielectric particles, etc., will cause breakdowns at local points.

As a rule, the beginning of a microplasma breakdown manifests itself in a series of the current pulses.

In the case of large imperfections, microplasma breakdown can begin at a much lower bias than calculated magnitude of the homogeneous breakdown voltage Vi. In this case the device can be destroyed at a relatively low current, because the density of the current flowing through the first microplasma can reach the critical value even at small average current magnitudes.

On the other hand, in materials of good, modern quality (Si, Ge), the magnitude of the microplasma breakdown bias can be only a few percent smaller than Vi even in devices with a large operation area. In this case the microplasma processes are important only at relatively low current densities (“at the beginning” of breakdown), while at high current densities homogeneous (mature) breakdown is dominant.

It is worth noting that only microplasma-free reverse bias structures can be used with devices operating at very high current densities (IMPATT and TRAPATT diodes).

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Homogeneous breakdown As far as homogeneous breakdown (part 4 in the Figure is concerned, the differential resistance dj/dVRd 0= is determined mainly by three components:

scthcd RRRdj/dVR ++== 0 ,

where cR is the contact resistivity, thR is the thermal component, scR is the resistivity of the space-charge region.

The nature of all three components has been analyzed in details.

It has been demonstrated that as soon as a current density j is close to the critical dsN Nevj ⋅= (Nd is the doping level of the base), the width of the space charge region increases rather sharp.

Indeed, when an appreciable avalanche current flows through the diode the Poisson equation should be written as

)ev

jN(edxdF

sd −=

0εε

(free carriers compensate the charge of the ionized donors) In the situation where j ∼ jN, it is necessary to take into account the second boundary of the sample. At dF/dx → 0, for any length of sample, the region of the strong field reaches the second contact

Field and carrier distributions under conditions of breakdown at j ∼ jN.

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Accordingly, the differential resistance sharply increases (Part 5 in the Figure). With further j increase (j > jN), when avalanche ionization takes place in the whole base, the holes inevitably contribute to the field distribution along the base. The Poisson equation must now be written as follows:

)pnN(edxdF

d +−=0εε

One can show that the contribution of the holes to the field distribution along the base causes the emergence of a negative differential resistance (NDR) in the current-voltage characteristic (Part 6 in the Figure).

If both electrons and holes ionize, the electron concentration will increases

from left to right and the hole concentration from right to left, on account of

impact ionization (b).

Field distribution along the base in the case where F > Fi. NDR appears on

account of partially compensation of the electron space charge by holes (c).

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Due to the very strong (exponential) dependences of the ionization rates versus electric field:

]F/Fexp[ ni 00 −=αα

and

]F/Fexp[ pi 00 −= ββ ,

the fields Fm close to the boundaries of the sample increase only slightly with increasing current, whereas the field in the middle part of the sample decreases markedly. As a result, the NDR appears.

However, at very high current densities, when the fields Fm close to the boundaries of the sample are of order of fields Fn0 and Fpo, respectively, the ionization rates αi and βi are rather weakly dependent on F. Hence, to achieve any noticeable increase in the current density j, it is necessary to increase Fmax very markedly at the boundaries of the base. In this case, the decrease in the field in the middle of the base cannot compensate for the increase in Fmax, and NDR disappears

It is easy to estimate that the threshold value for NDR appearance, sdN veNj~j = , is

~ 160 A/cm2 for Nd = 1014 cm-3, and

~ 1.6×105 A/cm2 for Nd = 1017 cm-3.

The characteristic current density at which NDR becomes PDR. This density was found to be 106 – 108 A/cm2.

Analytical results are thoroughly compared with appropriate computer simulations.

The principles of operation and main parameters of the Avalanche suppressor diodes and IMPATT diodes which operate in the part 4 of the current-voltage characteristics of the reverse-biased p-n junction are analyzed. Some physical problems that arise at very high operation frequencies have been discussed.

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Chapter 4. Avalanche Injection

4.1 Introduction 4.2 Avalanche injection in n+-n-n+ (p+-p-p+) structures 4.3 Avalanche injection in bipolar transistors 4.3.1 Introduction

4.3.2 Conventional regime of operation 4.3.2.1 Difference in breakdown voltages of a BJT

between the common-base and common-emitter configurations 4.3.2.2 Dependence of the bipolar transistor gain coefficient

α0 on current density 4.3.2.3 Main features of ABT operation in a conventional

regime

4.4 Operation regime of a Si avalanche transistor at very high current densities

4.4.1. Introduction 4.4.4.2 Steady-state collector field distribution. Residual collector voltage 4.4.3 Transient properties of Si avalanche transistor at extreme current densities.

4.5 Operation regime of GaAs avalanche transistor at very high current densities

4.5.1 Experimental results 4.5.2 Breakdown in moving Gunn domain in GaAs: qualitative analysis. 4.5.3 Computer simulations of superfast switching in GaAs avalanche transistor

The term “avalanche injection” was introduced by John Gunn in 1957 to refer to a situation in which “the avalanche behaves as a copious source of electrons (holes) which are injected into the material…” The term “copious” implies that the concentration of the free carriers created by avalanche breakdown exceeds the doping level in the semiconductor sample. In this case, as we have seen in Chapter 3, the space charge and electric field distribution are determined not by the fixed charge of the impurities, but by the charge of the free carriers (electrons and holes).

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In reverse-biased diodes, free carriers can appear at a notable concentration only due to avalanche breakdown (Situation which was considered in Chapter 3). In Chapter 4 we consider the avalanche injection phenomenon in “semiconductor resistances” (i.e. slabs of the semiconductor with two ohmic contacts (n+-n-n+ or p+-p-p+)) and semiconductor transistors and thyristors, i.e. in the structures in which an appreciable concentration of the free carrier can be caused by injection.

It can be injection from the contacts in n+-n-n+ or p+-p-p structures, injection from the emitter in transistor structures, and injection from the gate or one of the emitters in thyristors.

Free carriers (electrons or holes) injected from the outside into a high field region provide a reconstruction of the field domain such that breakdown occurs at relatively small applied voltages, notably smaller than those at which no injection exists. Hence, in the cases under discussion, breakdown appears under the influence of free carriers. Avalanche injection in “semiconductor resistances” (i.e. in the slabs of the semiconductor with two ohmic contacts (n+-n-n+ or p+-p-p+)) has been analyzed in detail in fundamental monograph M. A. Lampert, and P. Mark, “Current Injection in Solids” Academic Press, NY &London, 1970 and in detailed Review by A. M. Barnett, "Current Filament Formation" in Semiconductors and Semimetals, ed. by R.K. Willardson and A.C. Beer, v. 6, 2–172, Academic Press, NY, (1970)

Some important aspects of avalanche injection in Bipolar Junction Transistors (BJTs) have been considered in papers by P. L Hower, and V. G. K. Reddi, “Avalanche Injection and Second Breakdown in Transistors”, IEEE Trans. Electron Devices, ED-17 (4), 320-335 (1970) and A. Herlet and R. Raithel, “The forward characteristic of silicon power rectifiers at high current densities” Solid State Electron., 11 (8), 717 - 742 (1968)

However, the most detailed and comprehensive analysis of the avalanche injection has been given in papers by S. N. Vainshtein, V. S. Yuferev, and J. T. Kostamovaara:

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S. N. Vainshtein, V. S. Yuferev, and J. T. Kostamovaara, "Avalanche transistor operation at extreme currents: physical reasons for low residual voltages", Solid State Electron., 47 (8), 1255-1263 (2003) S. N. Vainshtein, V. S. Yuferev, and J. T. Kostamovaara, "Properties of the Transient of Avalanche Transistor Switching at Extreme Current Densities" IEEE Trans. Electron Devices, ED-49 (1), 142 -149 (2002) S. N. Vainshtein, V. S. Yuferev, and J. T. Kostamovaara, "Nondestructive current localization upon high-current nanosecond switching of an avalanche transistor", IEEE Trans. Electron Devices, ED-50 (9), 1988 -1990 (2003) First of all, the authors were able to observe experimentally a very effective high current switching in the n-collector Si avalanche transistors (n+-p-n0-n+ structure):

Current (curves 1-3) and voltage (curves 1'-3') across an avalanche transistor with n0-collector (FMMT-417, ZETEX SEMICONDUCTORS) as a function of time for various initial biasing values V0: 1,1' - 290 V; 2,2' - 240 V; 3,3' - 180 V. The current waveforms I(t) are derived taking into account the parasitic load inductances. The instant t = 0 corresponds to the input of the base pulse.

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All details of the characteristics observe were explained by the authors in the frame of original semi-analytical approaches and effective computer simulation:

1. In full agreement with the experimental data, the simulations show that the effective switching with high current-low voltage residual state is impossible for transistors with p0 collectors. A simple qualitative explanation has been presented. For avalanche transistors with n0 collectors (n+-p-n0-n+ structures), a very effective high current-low voltage switching mode can be realized. To provide an appreciable decrease in the residual voltage, the injection current jn0 must account for a notable proportion of the total current j (jn0 / j ≥ 0.6).

Dependences 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 [100].

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An optimal jn0 / / j ratio is determined by the ratio of the effective mobilities of electrons and holes peffneffeffb μμ /= , (where sn

neffn

vF

μ = ≈

1530 cm2/Vs and sppeff

p

vF

μ = ≈ 463 cm2/Vs for Si) and has been found

analytically.

Electric field distributions along the n0 collector for various combinations of the injection current jn0 and total current j. 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

This ratio tends towards the limit 1 sp n

sn p

v Fv F

+ ≈ 1.3 as F0 → 0, and

the minimum collector voltage is reached when jn0 / j is equal to 1/(beff +1) ≈ 0.77at a fixed value of j.

The electric field profiles in the base and collector regions

during switching can be conditionally separated into three

stages:

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Electric field distributions across the p-base (x = - 4 ÷ 0 μm) and in the n-collector (x = 0 ÷ 16 μm) regions at various instants (t = 0 ÷ 9 ns). (a) t = 0 ÷ 3 ns; (a) t = 4 ÷ 9 ns The profiles correspond to the simulated current and voltage waveforms shown in Fig. 92(a).

The first stage (0- 1.5 ns) corresponds to an increase in the collector current to a critical value jN= evsNd ≈ 4.8×102 A/cm2, where Nd = 3×1014 cm-3 There is no appreciable rebuilding in the collector field distribution until the electron and hole densities remain below the donor concentration. At the second stage (t ≈ 1.5 - 3 ns), the electron density exceeds that of the donors, and the derivation dF/dx changes its sign. The third stage is responsible for the high dI/dt rate of the collector current, which is caused by a rapid reduction in the emitter-collector voltage due to reduce of the collector field domain. The latter is determined by the spread of a quasi-neutral region from the base towards the collector contact. The quasi-neutral region is formed by an accumulation of the electrons injected from the emitter, and by compensation of their charge by the holes generated due to impact ionization near the n0-n+ boundary.

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One-dimensional simulation provides a fairly good description of the switching transient in an Si avalanche transistor. There is one very important problem, however, that cannot be solved in the framework of a 1D simulation. A very high current density j ~ 105 A/cm2 (a collector current Ic~ 100 A across a device of area ~10-3 cm2) is achieved simultaneously with a high electric field (F ~ 2.5×105 V/cm. Accordingly, the local high power density j× F will generally cause severe local heating and destroy the device within a single pulse. Consequently, time-dependent local heating is an extremely important problem for ATs.

Two-dimensional simulations of the switching transient for the same type of avalanche transistor were performed by the authors the ATLAS device simulator (Silvaco Inc.)

Temperature map of the structure at the end of the current pulse and distribution of the temperature at y = 19 μm at various instants

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Although the local temperature at the end of the pulse is rather high, it does not cause any harm to the device in single pulse operation. Indeed, formation of the thermal filament would require thermal generation across the entire channel. Moreover, the intrinsic carrier concentration in Si at a lattice temperature of 750°K (ni ~ 7.1016 cm-3) is still less than the doping in the n+ collector and comparable to that in the quasi-neutral region. This should mean that the pronounced local heating observed in the simulations does not cause destruction of the device in the single pulse mode, which is in full agreement with experimental results.

Even more interesting and promising results were observed by the same authors with GaAs BJTs: S. Vainshtein, J. Kostamovaara, Y. Sveshnikov, S. Gurevich, M. Kulagina, V. Yuferev, L. Shestak and M. Sverdlov "Superfast high-current switching of GaAs avalanche transistor" Electron. Lett., 40, (1) 85-86 (2004)

S. Vainshtein, V. Yuferev, J. Kostamovaara “Picosecond range switching of GaAs avalanche transistor due to bulk carrier generation by avalanching Gunn domains.” Proc. of SPIE, v. 5352, 382-393 (2004) The GaAs n+-p-n0-n+ structures with parameters which very close to those of the Si ATs discussed above demonstrated high current-low voltage switching within a time of about 200 ps.

The switching was characterized by the authors as "superfast" for two reasons.

First, the switching time was approximately 15 times shorter than that in analogous Si BJTs.

Second, it was shorter than the transit time required for the carriers to cross the base regions of the thyristor, provided the carrier velocity was saturated.

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Temporal dependences of the collector voltage and current across GaAs (solid lines) and Si (dashed lines) transistors in the high-current avalanche mode. The time dependence of the collector voltage characterizes the speed of the switch, while the dI/dt rates of the transistors are limited by the circuit parameter Lp/RL. As it can see in the Figure, an extremely fast switching is observed in a GaAs avalanche transistor. The time required for a reduction in the collector voltage from the initial value of 300 V to ~ 110 V is about 200 ps, i.e. shorter than the time required for the carriers to traverse the n0 collector region structure at the maximum possible (saturation) velocity. It should be noted that the observed low residual voltage manifests relatively deep modulation of the conductivity of the collector region. Meanwhile, the essential modulation of the conductivity in this depleted layer by the carrier drift or diffusion from the emitter should take much longer than the minimum time for the passage of a carrier across the region.

Any attempts to explain such QUALITATIVE difference in the switching times of Si and GaAs ATs taking into account: a) the difference in electron and hole mobilities, which should be larger in GaAs than in Si, (b) the difference in the ionization rates of electrons and holes in GaAs comparing with Si.

29

c) effect of the small operating area

failed. The observed effects were understood and explained taking into account the negative electron mobility in GaAs.

In Si ATs, a stationary "anode" high field domain appeared at n-n+ boundary, and just the avalanche ionization within this anode domain provided further current growth and low-voltage high-current switching

Electric field profiles across the GaAs AT simulated in 1D model

30

In GaAs ATs, the avalanche ionization in multiple Gunn domains of extremely high amplitude provided quasi-volume generation of excess carriers, current growth and low-voltage high-current switching

The temporal evolution of the field domains clearly demonstrates an effective positive feedback which is characteristic of impact ionization in high-field domains, namely that growth in the domain amplitude causes an increase in ionisation rates, which in turn leads to an appreciable increment in the carrier density. An increase in the carrier concentration causes growth in the domain amplitude Fm, followed by a further increase in ionisation intensity, etc.

Measured (solid line) and simulated (dashed line) voltage waveforms during the switching process. Despite marked high-frequency oscillations in the simulated collector voltage, the agreement between the simulations with the 1D model and the experiment is very good, and even the switching delay is comparable.

31

The simulations show that after nucleation every domain travels towards the anode with a velocity of about 107 cm/s. A domain disappears after it travels a distance of L0, a value which ranges from several microns to 20-30 μm for different domains. Hence the travelling domains can be considered "quasi-stable", because their "lifetime" is much larger than the characteristic time constant of domain formation τf . It is worth to note that the situation looks very unusual from the point of view of the "classical" Gunn effect. First of all, just one stable travelling domain can usually exist in a sample. A "multidomain regime" can be achieved only when the bias increases with time very fast. In this case the field outside the domain can even rise despite domain formation. but in the case studied here the bias applied to the collector decreases in time.

Second, Gunn domains of a very high amplitude (Fm~ 4-6×105 V/cm) provide very large values for the ionization coefficients αi, βi for both electrons and holes (~1.6×104 – 105 cm-1). These values are quite comparable to the reciprocal domain width, thus providing a considerable probability of an ionization event occurring within a single domain. This situation differs significantly from the approach adopted in the “classical” theory of Gunn effect, where Fm is assumed to be not too high, so that the characteristic time required for electron-hole pair generation is larger than the domain transit time.

Third, the theory of the Gunn effect in the presence of free holes developed in Refs. [150,151] assumes "ohmic" hole behaviour, i.e. that Ohm's law is valid for holes in any electric fields: vp = μp F. It is obvious that this assumption is not valid in the very strong fields under consideration. Hence one can conclude that further efforts are needed in order to obtain a detailed description of the physical nature of the domains responsible for superfast switching in GaAs avalanche transistors.

There is no doubt, however, that the simulations performed in Refs. mentioned above give a qualitatively correct description of the effect.

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5. Dynamic Breakdown 5.1 Introduction

5.2 Impact ionization front (TRAPATT -zone) 5.3 Silicon Avalanche Sharpers (SAS) 5.3.1 Computer simulations and comparison with experimental results 5.3.2 Stability of the plane ionization front 5.3.2.1 Short-wavelength instability of plane ionization front 5.3.2.2 Long-wavelength instability of plane ionization front 5.3.3 The problem of the initial carriers

5.4 GaAs diodes with delayed breakdown 5.5 Superfast switching of GaAs thyristors 5.6 Main features of streamer breakdown 5.6.1 Introduction 5.6.2 Analytical theory of a streamer discharge 5.6.3 Computer simulation

As mentioned Chapter 2, the breakdown scenario depends critically on the ramp dV0/dt. Over a very wide range of magnitudes of dV0/dt the conventional (static or quasi-static) breakdown scenario discussed in Chapter 3 is followed, but if dV0/dt ramp becomes extremely large the picture changes dramatically.

Qualitative field distribution across the base of a reverse-biased p+-n junction

when a large reverse current I0 is flowing through the diode.

00

00

00

εεεε

εε

/jS/Idt/dF

;dtdFj

==

=

33

At every point in the space charge region (SCR) the field increases in time according to the law: 0000 εεεε /jS/Idt/dF == . Point x0

moves to the right with a velocity v0 = vs Njj0 , where jN = e.Nd

.vs.

The velocity v0 of the point x0 where the field F = Fi is equal to

deNjv

tx 0

0 ==

If v0 is less than the saturation carrier velocity vs, the electrons created in the avalanche region (in which F > Fi ) will move faster than the point x0. The carriers generated in the avalanche region will modify the field distribution along the base, and we will be dealing with a conventional static (or “quasi-static") avalanche breakdown as considered in Chapter 3.

A qualitatively different situation arises if v0 > vs. In this case the "avalanche point" moves faster than the maximum possible carrier velocity vs, which means that the region to the right of the point x0 has no chance "to know" anything about the processes occurring to the left of this point. Until a constant current I0 flows through the diode, the current density in the SCR to the right of the point x0 is merely equal to the displacement current, and the field at every point of this region increases with the same ramp 00 εε/jdt/dF = . Unlike the situation in static breakdown, when the main breakdown processes are concentrated at the boundaries of the diode in the case under consideration (dynamic breakdown, svv >0 ), the whole base is included in the breakdown process.

It has been shown for many years ago A. S. Clorfine, R. J. Ikola, L. S. Napoli, "A theory for the high-efficiency mode of oscillation in avalanche diodes" RCA Review, 30 (3), 397-421 (1969); B. C.DeLoach, D. L. Scharfetter "Device physics of TRAPATT oscillators" IEEE Trans. Electron Devices, ED-17 (1), 9-21 (1970)

that so called plane Impact ionization front (TRAPATT zone) appears in a diode in such condition

34

Formation and motion of the impact ionization front (TRAPATT zone) at

dsNo Nevjj =>

The processes of the formation and moving of the plane TRAPPAT zone have been discussed in detail. It has been noted that according to direct adequate simulations the velocity of the TRAPATT zone can not exceeds (2-3)vs at experimentally used dV/dt values of ~ 1012 V/s.

However, experiments made in the Ioffe in 1979 I. V. Grekhov, A. F. Kardo-Sysoev "Subnanosecond current drop in delayed breakdown of silicon p-n junctions" Sov. Tech. Phys. Lett. 5 (8), 395-396 (1979) see also A. F. Kardo-Sysoev "New Power Semiconductor Devices for Generation of Nano- and Subnanosecond Pulses" in Ultra-Wideband Radar Technology ed. by J. D. Taylor, CRC Press, (2000) pp., 205-290

show that in fact, the velocity of the experimentally observed superfast switching corresponds to the values of "zone" velocity of (20-30) vs.

According to rather general physical reasons, the moving of the plane wave with such a velocity is impossible at given dV/dt magnitudes.

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The difference between the theory and the experiment makes it possible to suppose that the plane ionization front is subsequently transformed into some other kind of breakdown phenomenon.

A transition from a plane ionization front to a streamer discharge is apparently possible

It is well-known that the spread velocity of streamers can be as high as ~ 109 cm/s and the electron-hole density in the channel behind the streamer head can reach ≥ 1018 cm-3.

Transition from the plane ionization front to streamer discharge could thus explain both the short switching time (~ 100 ps) in long diodes (L ~ 300 μm) and the relatively small values for the residual voltage drop after switching.

A streamer is a thin, highly conductive filament that elongates at a very high velocity (~ 108- 109 cm/s) due to impact ionization in the vicinity of its head. Streamer discharge in liquids, solids and gases (including lightning) has been investigated in innumerable papers

36

Schematic diagram of the forefront and longitudinal field distribution of a streamer

37

The much higher velocity of streamer discharge than of a plane ionization front is caused by the substantially higher field Fm at the streamer head relative to the characteristic maximum field Fm in case of a plane front.

The main properties of streamer discharge had already been described in the 1950s, but a quantitative theory has been lacking until now. A qualitative analytical theory of streamers, taking into account the specific properties of the semiconductors, is presented in the papers made in the Ioffe: M. I. Dyakonov, V. Yu. Kachorovskii "Theory of streamer discharge in semiconductors" Sov. Phys. JETP 67(5) 1049-1054 (1988) M. I. Dyakonov, V. Yu. Kachorovskii "Streamer discharge in a homogeneous electric field" Sov. Phys. JETP 68 (5), 1070-1074 (1989) M. I. Dyakonov, V. Yu. Kachorovskii "Velocity of streamer propagating from a point during linear voltage increase" Sov. Techn. Phys. Lett. 16 (1), 32-33 (1990)

Analytical theory of a streamer discharge developed in the papers by Dyakonov and Kachorovskii as well as corresponding computer simulations are posed and analyzed. It has been noted that transformation of the plane ionization wave into streamer discharge is partly supported by results obtained with GaAs devices:

GaAs diodes with delayed breakdown

Superfast switching of GaAs thyristors

Experimental results obtained are described and analyzed in detail. Unsolved problems of the dynamic breakdown are considered: The problem of the initial carriers The problem of the formation and spread of the streamers in semiconductor devices. The problem of multi-streamer charge in semiconductor devices.

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In spite of many unsolved problems, Silicon Avalanche Sharpers based on the dynamic breakdown phenomena has found practical applications in sub-nanosecond high voltage generators: http://www.fidtechnology.com/Products/fastpowergens.htm A series of fast power generators with pulse amplitudes from 1 kV to 200 kV, pulse rise times from 100 ps to 1 ns and a pulse width from 0.1 to dozens of nanoseconds has been developed and successfully applied.

Conclusion

The problems discussed in this book could perhaps be described more exactly by a title such as “Main approaches to electrical breakdown phenomena in "good" single-crystal semiconductors and devices based on them", The term "good semiconductors" is understood as referring to "semiconductors with relatively high carrier mobility". Roughly speaking, the phenomena considered in this book apply to semiconductors with a low-field mobility μ of a magnitude exceeding ~ (1- 10) cm2/Vs. The term "main approaches" means that the breakdown phenomena discussed in this book are considered, as a rule, in the context of fairly simple spatial configurations.

In semiconductors with relatively small values for the low-field mobility μ, i.e. in polycrystalline, amorphous, and most polymer semiconductors, etc., thermal breakdown plays a very important role. It is very difficult as a rule to separate the contributions of electrical and thermal breakdown in the case of semiconductors of low mobility. The history of breakdown switching in chalcogenide glassy semiconductors research appears to provide one of the most instructive examples of this problem. The prevention of surface breakdown (the edge termination problem) in reverse-biased devices is one of the most valuable examples of the importance of breakdown studies under conditions involving complicated spatial configurations. Many techniques have been developed for protecting the devices from surface breakdown (guard rings, junction termination extensions etc.), but

39

calculations of the field distribution at the surface and in its vicinity nevertheless require the solution of two-dimensional and sometimes even three-dimensional problems [22-24, 215]. A necessity for 2D and 3D simulations frequently arises when

• the breakdown in a high-field domain between the drain and the gate is analysed in different types of field effect transistors in a deep saturation regime,

• when breakdown in bipolar transistors is considered under conditions of a pronounced crowding effect, or

• when devices of fairly complicated geometry are being investigated, etc. It should be noted, however, that all the basic principles and approaches considered in this book are applicable to such calculations in a practically unchanged form. Although investigations of breakdown phenomena in solid states started about a hundred years ago, this is still a lively and powerful branch of the mighty tree of semiconductor physics. This situation will continue to prevail all the time that new materials are becoming involved in semiconductor electronics, all the time that still new devices are being proposed, and all the time that modifications and improvements are being developed for known devices within semiconductor electronics. To anybody who doubts in the usefulness of the efforts spent to get know such an interesting and important subject as "Breakdown Phenomena in Semiconductors and Semiconductor Devices" we would like to answer by the words of 76-th William Shakespeare sonnet:

Why is my verse so barren of new pride,

So far from variation or quick change?

Why with the time do I not glance aside

To new-found methods and compounds strange?....

For as the sun is daily new and old,

So is my love still telling what is told.