Bachelor of Science ThesisStockholm, Sweden 2012

TRITA-ICT-EX-2012:182

A L E X E L L G R E N

Simulation of Avalanche breakthroughin Si and SiC

K T H I n f o r m a t i o n a n d

C o m m u n i c a t i o n T e c h n o l o g y

Simulation of Avalanche breakthrough in Si and SiC

Alex Ellgren

27th August 2012

Abstract

Denna uppsats beskriver en Monte Carlo-simulering av avalanchegenombrott i kisel och kiselkarbid, med

tanked att denna ska användas som ett hjälpmedel i undervisningen i introduktionskurser i halvledarkomponenter.

Modellen som används approximerar laddningsbärarna som hårda klot som studsar på varandra och på så sätt

överför kinetisk energi sinsemellan. Jonisationer registreras vid kollisioner där EKin > Eg. De simulerade

resultaten stämmer väl överens med verkligheten för kisel, men är bedydligt högre än väntat för kiselkarbid.

Det föreslås att detta pekar på en för författaren okänd materialparameter som sänker elektronhastigheten i

kiselkarbid, då bandgapsenergin i sig är för låg för att förklara det höga kritiska fält som uppmätts i kiselkarbid.

This paper describes a Monte Carlo simulation of avalanche breakthrough in Si and SiC, which is intended

as a teaching aid in introductory courses on semiconductor devices. The model used approximates the charge

carriers as hard balls bouncing o� each other and transferring kinetic energy between themselves, with ionizations

being recorded when EKin > Eg. The simulated results are close to their expected values for Si, and signi�cantly

higher than expected for SiC. It is suggested that this indicates the existence of an hitherto unknown to the

author material parameter in SiC that would suppress the carrier velocities, since the bandgap energy itself is

too low to counteract the high critical �eld that has been measured in SiC.

1

Introduction

In just about all introductory courses on semiconductor and device physics, students are introduced to the PN-junction, the area in a semiconductor device where two oppositely doped regions come into contact. Understandinghow carriers move through this junction is fundamental, but many important concepts are often glossed over intextbooks. So was the case with the book used in KTH course IH1611 in 2010, Principles of Semiconductor Devicesby Sima Dimitrijev, where the physics presented caused a confusing paradox when trying to understand high velocitycarriers. Motivated by this, it has been the goal of this project to understand the necessary processes of how carriersmove and use this knowledge to properly explain avalanching to potential future students.

The explicit purpose of this study is twofold: To a): Using MATLAB, write a simpli�ed Monte Carlo-simulationprogram that allows the user to follow individual carriers through an avalanche breakthrough, and b): Enablethe author to write up a short presentation, using graphics from the simulation, that can be given to students inintroductory device physics courses so that they may better understand avalanche breakthroughs.

Contents

0.1 Glossary of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

0.1.1 Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.1.2 Solid state physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.1.3 Industry terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I Theory & Background 4

1 Si & SiC 4

1.1 Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 SiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Crystal quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.3 Electrical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 The PN-junction 7

2.1 The unbiased PN-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The PN-junction under bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Forward bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Reverse bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Avalanching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Ionization coe�cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Multiplication factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Carrier movement 10

3.1 Semiclassical carrier transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Scattering mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Phonon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.2 Impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.3 Crystal defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

II Simulating the Avalanche breakthrough 12

2

4 The model 12

4.1 Bandgap and critical �eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 Carriers as an ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.2 Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.3 E�ective mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.4 Dielectric constant for 4H-SiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2.5 Intrinsic carrier concentration of SiC at 600 K . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 MATLAB Program 14

5.1 Time scale and step size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.2 Flight mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3 The MATLAB functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3.1 avasim.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.3.2 widthcalc.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3.3 �eldcalc.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3.4 trajcalc.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Calibrating the model 19

6.1 Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

III Results 20

7 Graphics 20

7.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.1.1 Far below breakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.1.2 Below breakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.1.3 Breakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7.1.4 Beyond breakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7.2 Silicon Carbide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.2.1 Far below breakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.2.2 Below breakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.2.3 Breakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.2.4 Beyond breakthrough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Conclusion 28

IV Appendix: MATLAB code 30

0.1 Glossary of terms

This glossary is intended to quickly refresh the memory of a reader already familiar with basic solid state physics,or to make the report readable to a reader who isn't. It is not intended as a full explanation of these concepts.

3

0.1.1 Crystallography

fcc - Face-center cubic, a basic crystal structure. The fcc lattice is made up out of cubic unit cells that in additionto sharing eigth 1

8 atoms on the corners of the cube also shares one 12 atom on every six faces of the cube.

hcp - Hexagonal close packed, a basic crystal structure.

Diamond crystal structure - One of the structures of pure group 4 semiconductors, most notably silicon, germaniumand diamond-form carbon.

Isotropy - An isotropic crystal has the same properties in all directions. In contrast, an anisotropic crystal may forexample have di�erent atomic spacing in di�erent directions.

Polymorphism - A crystal with the same chemical formula may crystalise in several di�erent polymorphs withwidely di�erent physical properties. Diamond and graphite are a popular example.

0.1.2 Solid state physics

Hole - Conceptual counterpart to the electron. Despite technically being an absence of a particle, the hole isconsidered to have the exact same properties as the electron but with opposite charge. When a free electron and ahole meet they recombine, binding the electron to the empty orbital slot occupied by the hole.

Bandgap - Energy range between valence band and conduction band where neither holes nor electrons can exist.

Conduction band - Available energy levels above the bandgap. Free electrons travel through the conduction band.

Valence band - Available energy levels below the bandgap. Holes travel through the valence band.

Reciprocal lattice - A mathematical construct that describes the crystal in terms of momentum instead of whereatoms are located.

E�ective mass - A simpli�cation that allows us to pretend the conduction and valence bands are �at with someaccuracy. The greater the band bending, the greater the distortion of carrier momentum, which can be consideredas the e�ective electron/hole mass being di�erent from the free electron mass.

0.1.3 Industry terms

Boule - A rough tapered cylinder of manufactured semiconducting material from which wafers are cut.

Part I

Theory & Background

1 Si & SiC

1.1 Si

Silicon is the second most abundant chemical element on earth 1 and by far the most used material in semiconductorelectronics. It's usefulness is mostly due to its relative ease of processing as well as its native oxide SiO2, whichcombines being a good insulator with �tting together with the bulk silicon excellently, so that a minimum ofdiscontinuities is formed at the semiconductor-oxide interface. This makes it comparatively easy to constructintegrated circuits with millions of separate devices on one chip.

While it is impossible to produce large absolutely pure silicon crystals, modern processing techniques manage afair approximation thereof. Typically, intrinsic silicon has an impurity ratio on the order of 1015cm−3, which is afew defects for every ten million silicon atoms. Impurity in this context can mean a multitude of things - it canbe a contaminant atom of a di�erent element at a lattice site, or a silicon atom that hasn't taken its proper placein the lattice and as such cannot participate in the electron exchange, or a sudden change in the ordered patternof the lattice. Many of these defects, such as atoms out of place, can be helped by annealing, which is basicallyheating the crystal up so that the atoms vibrate into place, much as one might shake a bucket full of sand to even

1http://hyperphysics.phy-astr.gsu.edu/hbase/tables/elabund.html 2012-06-26

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the level. However, heating obviously can't turn one element into an other. 'Pure' silicon is rarely used in practice,but commonly doped by purposely introducing non-Si atoms into the crystal, either by letting it grow out of amix of liquid or gaseous silicon with the desired doping concentration or by bombarding the silicon surface withhigh-velocity particles. Doping concentrations are typically between 1015cm−3, enough to overpower the unknowncontaminants, and 1020cm−3, which is about as much impurity silicon can take and still be recognizable as silicon.

1.1.1 Crystal structure

Silicon is tetravalent like its upward neighbour carbon, and crystallizes in the diamond structure with a latticespacing of about 5.5 Å. Every atom connects covalently to four other atoms, with four equally strong bondsradiating out from the center atom 120◦ apart to form a local tetrahedron. Alternatively, the diamond unit cellcan be visualized as two fcc unit calls hooked through each other, one shifted 1/4th of the lattice spacing aside inall three dimensions. The packing fraction of this structure is only 0.34, quite a bit lower than the close-packedstructures of fcc and hcp. The low packing fraction is due to the covalent nature of the bond: Forming regulartetrahedrons takes energetic precedence over squeezing many atoms in close together.

Figure 1: Ball and stick-model of silicon unit cell, courtesy of Ben Mills/Wikimedia Commons

1.2 SiC

Silicon carbide is a compound non-metal solid composed of equal parts silicon and carbon in a variety of crystalstructures. All structures stable enough to grow macroscopic crystals give very hard, very heat-resistant materials,wherefore low-grade SiC-powder is and has been a common industrial abrasive since the late eighteen hundreds.In its pure, macroscopic form SiC is colorless and transparent, very similar to diamond, but crystals this pure aredi�cult and expensive to grow. Most industrially produced SiC is instead a grainy powder that can be sinteredtogether to an extremely hard ceramic, but advances in process technology has made it possible to grow large-scaleregular crystals suitable for use as semiconductors.

1.2.1 Crystal structure

Silicon carbide does not have one single crystal structure, and with that no single set of either electrical, thermalor even visual properties. It exhibits extensive polymorphism, and several hundred di�erent structures have beenfound in nature2. Though all chemically identical, they can be as far apart in physical properties as diamond isfrom graphite. Though some of the di�erent structures are radically di�erent (some have lattice spacings hundredsof times larger than the common types3) most are structurally similar. They are so called polytypes, which means

2Bechstedt et al, Polytypism and Properties of Silicon Carbide, p. 353Kelly, Correlation between layer thickness and periodicity of long polytypes in silicon carbide, p. 250

5

that they are the same in two dimensions and vary in the third. If we visualize the atoms as billiard balls and stackthem, the �rst two layers will be the same (actually there are two possibilities for the second layer that will givecrystals that are mirror images of each other, but for our purposes those crystals are identical) but the third can beplaced in two di�erent ways that are energetically equivalent. So can the next layer, and the one after that. Thesedi�erent stacking possibilities create a pattern, and only a few of these patterns are stable enough to allow for largecrystals to grow.

There are four commonly used, stable con�gurations of the SiC crystal: 2H-SiC, 4H-SiC, 6H-SiC and 3C-SiC. Ofthese, 2H and 3C represent familiar crystal structures while 4H and 6H can be seen as combinations of these.

2H-SiC has the double hexagonal structure known from wurtzite, where two hcp-type lattices, one with carbonatoms and one silicon, are interwoven. 3C has the double-fcc structure of zinc-blende, which is like the diamond-structure of silicon with every second atom exchanged for carbon. The di�erence can be understood if we studythe following images, where hard spheres approximate atoms.

Figure 2: Two layers of spheres, tightly packed

When spheres are packed close together, the �rst layer will have a triangular symmetry between the highest pointof the spheres. If we refer to the sphere center-points for the �rst layer as position A, and then add a second layer,the spheres in this layer can be cradled in two di�erent sets of energetically equivalent valleys. Let us call thesevalleys B and C and assume the second layer settles in the B valleys. For layer three, there are now two di�erent,energetically equivalent choices: The A position, directly above the spheres in the �rst layer giving the hexagonalsymmetry of the hcp-structure, or with a 60◦twist respective to layer one to position C, plugging the white-spaceholes we can see on the image above. This ABC-stacking sequence makes up the isotropic zinc-blende structure of3C.

Figure 3: Three layers of tightly packed spheres in two di�erent patterns. Left: ABA, Right: ABC

The possible crystal structures are combinations of these elements. 3C-SiC is alternating silicon and carbon layersin a repeating ABC pattern, and 2H is just alternating A and B, one layer of silicon and one of carbon. The 4Hstructure has a longer unit cell that can be likened to a 2H-cell with a twist in it, denoted in this notation asABAC, and 6H-SiC is longer yet with two twists, repeating the sequence ABCACB. Of these crystals all but 3Care anisotropic, which means they have di�erent properties in di�erent crystal directions. Mobility, for an example,is much higher within a hexagonal plane than moving 'up' one level to the next extended hexagon. The structurebeing simulated in this project is the anisotropic 4H-SiC, which means we must also de�ne a crystal direction toadequately describe the crystal.

Hexagonal crystals like the H-polytypes of SiC are frequently described by four axes of symmetry: a, b and c in thehexagonal plane 60◦ apart and h as the perpendicular axis to the next hexagonal plane. This four-axial symmetryis not mathematically necessary, as the two-dimensional hexagonal plane can be described fully by two axes, but israther an elegant way to display the symmetry.

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1.2.2 Crystal quality

Even with modern process technology it is di�cult to grow large, pure SiC crystals. In semiconductor technology,we are mostly interested in the large bandgap varieties since they are well suited for power electronics. Of thecommon polytypes, 6H and 4H are best suited for wafer production4 and are available commercially since the earlynineties5.There are several problems inherent in SiC manufacturing. Since it does not melt but instead sublimes, one cannotgrow the SiC crystal by pulling a crystal seed out o� a melted mix as is usually done with silicon6. Instead, asimilar process is used where polycristalline SiC is heated to a vapour (at about 2400◦C) and made to condense ona crystal seed of the desired polytype, where it tends to grow in the h-direction into a nominally cylindrical boule.However, these crystals have so far not been successfully grown in the same sizes as silicon crystals routinely are,and wafers typically have a diameter of 100 mm instead of the customary 200 to 300 mm.SiC crystals also have a variety of troublesome crystal defects. Among the most common are small volumes of puresilicon or carbon crystals, sudden changes in polytype, random voids in the lattice and long hexagonal pipes thatextend along the h-direction of the crystal. These pipes in particular impose a limit on SiC technology as they willnot only locally interrupt current �ow but might endanger the structural integrity of the entire wafer. In additionto this, the hardness of the material makes cutting it into wafers and polishing them smooth very di�cult.

1.2.3 Electrical properties

SiC's usefulness as a semiconductor largely relies on it's high bandgap. For this reason, the polytype most used is theone with the highest Eg: 4H-SiC at 3.23 eV7. This high bandgap (compare 1.12 eV for Si) means that it takes veryhigh temperatures (about 1000◦C8) before the number of thermally generated free carriers surpasses the numberof ionized doping atoms and the device current is drowned in thermal noise, as compared to a silicon device thatbecomes inoperable at about 300◦C and has usually broken before then. The large bandgap also means that SiChas a very high critical �eld before breakthrough voltage occurs, which makes it useful for high �eld applications.The high critical �eld also means that many SiC devices can be made up to ten times smaller than their siliconcounterparts, which in turn gives them faster response time.These same qualities also cause some problems in simulating SiC. Due to the high energy needed to thermally excitean electron to the conduction band, SiC has extremely low intrinsic carrier concentration at room temperature (onthe order of 10−9cm−3)9. Even dopants, which are selected to be easily ionized, frequently need more of a push thanSiC can deliver at 300 K so the amount of avaliable carriers can be noticeably lower than the doping concentration.To see SiC's more semiconductor-like properties (as opposed to its possible use as an insulator), it is usually bothsimulated and measured at 600 K10.

2 The PN-junction

An intrinsic semiconductor has no particular directionality. Current �ows in one direction as easily as in any other,and on its own a piece of semiconducting material is really no di�erent than a piece of poorly conducting metal.However, put together semiconductors can be controlled to conduct or block current without any mechanical changebeing wrought upon them. This makes it possible for us to build devices where current or potential control current�ow, which enables all form of digital logic. The most basic of these semicondictor interfaces is the junction betweenone electron-starved p-region and an electron over-saturated n-region, the pn-junction, which will be discussed inthis section.

2.1 The unbiased PN-junction

When a semiconductor is doped, the Fermi level - which can be considered the average electron energy, or alternatelythe probability of free electrons - shifts upwards with n-doping, since it gives more free electrons, which increases

4http://web.it.kth.se/~bellman/docs/ZetterlingDocentLecture.pdf p.65http://www.iue.tuwien.ac.at/phd/ayalew/node2.html6http://www.iue.tuwien.ac.at/phd/ayalew/node13.html7http://www.io�e.rssi.ru/SVA/NSM/Semicond/SiC/bandstr.html 2012-03-288Zetterling, Process Technology for Silicon Carbide Devices9http://www.iue.tuwien.ac.at/phd/ayalew/node21.html 2012-04-02

10Zetterling, Process Technology for Silicon Carbide Devices

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the probability of free electrons in that region and forces the electrons to higher energy states since they cannot alloccupy the same one, or downwards with p-doping. When a p-region and an n-region are brought together11, thefree carriers quickly di�use to minimize potential di�erences but the doping atoms, normally ionized at the device'soperating temperature, are stuck in place in the crystal lattice and their electric charge creates a �eld in the device.This �eld further sweeps away any free electrons and holes that enter the charged area, so that any free carrier thatenters this �eld's zone of in�uence will be sent on toward the region with opposite net charge, thus depleting thejunction-region of free charges and giving it the name depletion zone.

How deep into the doped regions the depletion zone extends depends on how strongly doped the region is. Strongerdoping gives smaller �eld extension, since there are more mobile carriers to counteract the potential di�erence. Inthe case where the p and n-sides of the junction are di�erently doped, the depletion zone will be asymmetrical,extending further into the weaker doped side in proportion to the relative doping levels. However, since both holesand electrons are equally a�ected (although sent in opposite directions), there is no net current �ow through anunbiased junction. Whenever a carrier drifts into the depletion zone or is thermally excited away from its depletionzone-bound atom it will start to drift, but this drift also concentrates the same kind on carrier in one place and theytend to di�use toward areas with less carrier concentration. In an unbiased pn-junction, these di�usion currentsbalance the drift caused by the junction �eld.

Figure 4: Diagram of the unbiased PN-junction with the depletion zone marked in red. Note that the Fermi level(green line) is �at through the image, indicating the junction is in equilibrium.

2.2 The PN-junction under bias

The PN-junction makes up basis of the standard diode, and when we bias it, we can observe it doing a range ofinteresting things, most famously acting as a recti�er. As this report is about avalanche breakthrough, it will onlygive a very shallow overview of forward and modest reverse bias, before delving deep into the intricacies of thepronounced reverse biasing where avalanching occurs.

2.2.1 Forward bias

When forward biasing a diode, the positive terminal is connected to the p-side and the negative to the n-side.This forces the electrons on the n-side and the holes on the p-side towards the junction, so that the depletion zonenarrows and the potential di�erence lessens. This in turn diminishes the electric �eld centered over the junction sothat carriers that enter the depletion zone are not accelerated as forcefully. The energy curves shown in �gure 4 aresmoothened, and with increasing bias �attened out completely so that current can �ow unhindered from terminalto terminal.

11Figuratively speaking. Doping is usually done on site, so that the bulk semiconductor structure is already in place when dopantsare injected into the material. It is possible to grow doped crystals from scratch and switch between acceptor and donor dopants forextremely sharp transitions, but it is expensive and di�cult for all but the simplest structures.

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2.2.2 Reverse bias

For a reverse biased diode, on the other hand, the positive terminal attaches to the n-doped region and vice versa,and as bias is increased so is the potential di�erence over the depletion zone. Under moderate voltages, this potentialbarrier is enough to stop any but a minute leakage current from coming through. The ideal diode is supposed toonly let current through one way and stop it from going the other, with resistance increasing with every addedreverse-bias Volt. In reality, a diode only acts this way up to the breakthrough voltage.

Once voltage increases enough to bring �eld strength up to the material's critical �eld, the thermally generatedminority carriers in the depletion zone will reach velocities high enough that their kinetic energy is enough to ionizelattice atoms. These newly freed electrons, along with their companion holes, are in their turn accelerated and may,if the �eld is great enough, knock other electron-hole pairs loose. Once carrier energies are large enough for thisprocess to not only get started but for the chain-reaction to be sustained through the depletion zone, the numberof free carriers hurtling about in the depletion zone grows exponentially, much like the number of snow crystals inthe avalanche that has given it its name. These avalanching carriers can carry current the 'wrong way' through thediode.

2.3 Avalanching

Consider a diode reverse-biased at far below the breakthrough voltage. Despite our best e�orts there is a smallleakage-current �owing backwards through it, caused by minority carriers either randomly falling into the depletionzone or being thermally generated in it. These carriers accelerate in the �eld over the depletion zone, headingfor the opposite side of the diode. This passage is not smooth and uneventful, but rather a sequence of randomcollisions with lattice atoms and impurities. In all of these collisions, the carrier loses some of its energy to thelattice, which becomes marginally warmer. As long as the electric �eld is low, the carrier eventually makes its waythrough the depletion zone and the heat left by its collisions disperses as lattice waves.

Now, imagine the reverse bias voltage increasing: With every added bit of voltage, carriers gain a little more energybetween each collision. Every collision leaves more heat behind. Eventually, a carrier comes along, crashes into alattice atom, and leaves behind enough energy to ionize a bound electron, a process referred to as impact ionization.

Suddenly, one carrier has turned into three - the original electron12 as well as a brand new electron-hole pair. Ifthese carriers reach the same energies and cause another impact each, we get nine free carriers, then twenty-seven,eighty-one, and increasing toward in�nity until the current causes resistive heat to build up and the device catches�re.

Avalanche breakthrough is not in itself harmful to the device - as long as the current is limited by something outsidethe depletion zone, the process is fully reversible. Lattice atoms that have lost one of their bonds will quickly �nda replacement electron and re-order themselves to a uniform crystal potential. No side of the device will ever 'runout' of carriers. However, if the current is left unchecked, heat will build up quicker than it can be dissipated andthe device will burn itself out.

It is important to note that while avalanche breakthrough is frequently described as a phenomena that happens ata very distinct material-dependent �eld strength, it is actually more of a gradual process. The carrier velocities,while a product of the applied �eld, are not one and the same. The velocity given by the �eld is altered by therandom thermal velocity the carrier had before it entered the depletion zone, which is randomly distributed over awide set of possibilities.

This means that impact ionization can and will occasionally occur even in an unbiased junction. Instead, avalanchebreakthrough is more properly described as when impact ionization becomes so probable that current can be ex-pected to spiral out of control when the ratio of impact-generated carriers which themselves impact-ionize approachesone.

2.3.1 Ionization coe�cients

Baliga13 de�nes the impact ionization coe�cient for holes as 'the number of electron-hole pairs created by a hole

transversing 1 cm through the depletion layer along the direction of the electric �eld '.

12It is almost always an electron. While impact ionization works just the same for holes, their lower mobility means electrons areusually the ones to start the avalanche.

13Baliga, Power semiconductor devices, p. 67

9

The de�nition is similar for electrons. Baliga also gives experimentally determined values for these coe�cients insilicon14. For electrons,

αn = an × exp(−bnE

) (1)

with an= 7×105per cm, bn=1.23×106V/cm, and for holes

αp = ap × exp(−bpE

) (2)

with ap = 1.6× 106 per cm and bp=2×106V/cm.

2.3.2 Multiplication factor

The ionization coe�cient is in the dimension ionizations per unit of length. If it is multiplied with an arbitrarylength dx, αdx, we have the number of ionizations caused while traversing this length. If a carrier pair is generatedat a distance x from a pn-junction with width w, the amount of new pairs it generates can be calculated by

M(x) = 1 +

ˆ x

0

αnM(x)dx+

ˆ w

x

αpM(x)dx (3)

A solution to this equation that is slightly easier to visualize,

M(x) =M(0)× exp[ˆ x

0

(αn − αp)dx] (4)

utilizes M(0), the number of electron-hole pairs at the edge of the depletion zone as a result of the original pair. Toobtain a value of M(0), then, one substitutes expression 4 into equation 3 and extracts M(0):

M(0) = {1−ˆ w

0

αpexp[

ˆ x

0

(αn − αp)dx]dx}−1 (5)

Finally, substituting this expression into equation 4 gives the unwieldy but workable expression

M(x) =exp[´ x0(αn − αp)dx]

1−´ w0αpexp[

´ x0(αn − αp)dx]dx

(6)

The only variable for this equation is the electric �eld, inherent in αn/p15 .

So, using Baliga's experimentally determined values and knowing how to calculate the �eld strength as a functionof position, these equations should give sound results. Of course, with a and b only known for Si, this equationcannot be directly applied to SiC. However, if the predictions of these equations can be made to match up with thesimulation results it would indicate that the same simulations could be applied to the known material parametersof SiC and yield plausible results.

3 Carrier movement

A semiconductor has two types of carriers, electrons and holes. Both play equal parts in conduction: Electrons dom-inate in n-doped materials, holes dominate in p-doped, and in both cases the non-dominating kind have importante�ects most noticeable when a device is in its 'o�' state. For the purpose of understanding avalanche breakthrough,however, we will focus on electrons. This is because the electron current will always reach the avalanche conditionbefore the hole current does due to the higher mobility of the free electrons. This might seem counterintuitive:How is it possible for the electrons to move faster than the absence of electrons? Shouldn't they reasonably beexpected to move as each other's mirror images? The paradox is due to another concept poorly communicated inmany elementary textbooks: The mobilities as given in reference books are not given as the mobilities of individualcarriers, they are given as the mobilities in the valence and conduction bands. The valence band is lower in energy

14Baliga, Power semiconductor devices, p. 6715Baliga, Power semiconductor devices, p. 68-70

10

and more �rmly under control of the atomic nucleus, therefore it takes more energy to move a slot (whether occupiedby electron or hole) in the valence band than it does in the less restricted conduction band. Compare running freeand running tied to a bungee line - the same amount of e�ort will take you very di�erent distances. The 'e�ort' inthis analogy is then the forces accelerating the electrons - electric �eld and thermal motion both.

3.1 Semiclassical carrier transport

In a typical semiconductor, atoms are placed about 5 Å apart. Atomic and ion sizes range between tens to hundredsof picometers. While it is di�cult to prescribe what is classically meant as size to an electron - de�ning it as adelocalized wavefunction we could in essense consider the conduction band as a sort of massive pseudoparticle - wecan for the moment consider it much smaller than an atom. If we temporarily step out of quantum mechanics andchoose to consider the electron wave packet as a localized tiny particle - justi�able if we assume both momentumand position uncertainty to be small compared to the active regions of a device16 - then to this tiny electron thereis no perceivable di�erence between an external �eld applied to a device and the local �eld around an atom in thecrystal lattice.

3.2 Scattering mechanics

3.2.1 Phonon scattering

Any crystal warmer than absolute zero temperature will have lattice vibrations. These vibrations cause atoms toshift minutely among themselves, like interconnected beach balls on a wavy sea, which in turn means that thecrystal potential in any one place is always changing. Doing calculations under these conditions would be extremelytedious. Instead, these waves of crystal vibrations are modeled as separate particles - phonons - that move througha lattice that can then be considered stationary.

Phonons are a major cause of scattering in semiconductors, increasing in importance with temperature and decreas-ing with material bandgap. They are divided into two types: Acoustic and optical, whereof optical phonons onlyoccur in crystals with more than one type of atom. If we picture the lattice as a one-dimensional chain of atoms,acoustic phonons can be visualized as either rythmically stretching the chain like a rubber band so the atoms movetoward and away from each other, referred to as longitudinal, or by shaking one end of it like a rope so that itweaves in a sine shape. Shaking the end up-down or left-right gives two di�erent, independent vibrational wavescalled the transverse acoustic modes. Optical phonons occur when the di�erent types of atoms react di�erently toa change: For example, an electric �eld over doped silicon will not noticeably a�ect the silicon atoms but it willtug at the dopants, causing them to move out of sync with the rest of the lattice. Optical and acoustic phononshave di�erent temperature dependence. Simply put, the mobility in a semiconductor dominated by acoustic phononscattering is proportional to T−

32while in an optically dominated semiconductor it is proportional to T−

12 17.

Since silicon is a one-atom substance and SiC is dominated by other scattering mechanisms, this project will onlyutilize acoustic scattering to model how temperature in�uences carrier scattering. Too many mechanisms wouldoverly complicate the code, and considering the other approximations this one seems justi�ed.

3.2.2 Impurity scattering

Though the mental image of electrons bouncing o� atoms is appealingly simple, it is quite far from the truth.Electrons do not scatter from some hard, well-de�ned atomic surface, they scatter from interaction with the electric�eld around an atom. We can then assume with fair accuracy that any silicon atom with four fully formed covalentbonds is then electrically satis�ed and shielded by its associated electrons, so we only get signi�cant electric �eldsaround dopant ions and crystal defects. Dopant ions are fairly self-explanatory: They are atoms of a di�erentelement (typically group III and V in Si and SiC) intentionally introduced to the crystal lattice to change it'selectrical properties. Only dopants that have found their intended place in the lattice is electrically available, soimplanted dopants that are out of place instead count as crystal defects. At the temperatures we are simulating(300 K for Si and 600 K for SiC) we assume all dopants to be ionized.

16Lundstrom, Fundamentals of carrier transport, p.4117http://ecee.colorado.edu/~bart/book/transpor.htm 2012-04-06 section 2.8.4

11

3.2.3 Crystal defects

Defects in crystals can be said to have di�erent dimensionality. One misplaced atom is like a zero-dimensionaldot, a line that breaks up the crystal symmetry has one dimension, a sudden change of stacking sequence can beconsidered a two dimensional defect, and in some cases - quite commonly in SiC - an entire chunk of matter can becompletely absent, giving a three-dimensional defect. All of these break up the crystal lattice and decrease carriermobility, and they also function as centers where scattering is far more likely to occur than in a normal lattice atom.While defects are present in all semiconductors - in fact, 'intrinsic' silicon is a wholly theoretical concept - theyplay a larger role in some. Silicon, having been produced in large scale with high demands on purity for decades,can usually be considered fairly pure so that dopants and interfaces cause more of a stir than unintentional crystalimpurities, but SiC still has many process technological problems to solve.

Part II

Simulating the Avalanche breakthrough

4 The model

For the simulation part of this project, I have used a very simpli�ed model of how carrier transport works. The coreof the code is a scattering model, where a modeled carrier has a certain probability to scatter at each iteration. Ifit does scatter, its kinetic energy is measured to see if it would have created a secondary electron-hole pair. Sinceit would be highly ine�cient to attempt to simulate a number of carriers that approaches in�nity, these secondarypairs are instead added to a counter as the initial carrier continues on. This means that the simulation will nevershow actual avalanching - rather by watching it we get an idea at approximately what voltage it can be expected.This model does not model how the current increases as the avalanche progresses, but if we assume that ionizationson average happen in the middle of the statistical range (an assumption that is certainly valid at large numbersof ionizations) then we should get a fair idea of the current if we multiply the number of carriers by 2 for everymarked ionization.

4.1 Bandgap and critical �eld

The model employed in this simulator calculates the kinetic energy of a carrier as Ekin = mv2

2 , and checks forcollisions where the kinetic energy is greater than what is required to ionize a new electron-hole pair. This seemsto work well for silicon, but for SiC breakdown occurs earlier and more suddenly than expected, and in the latestages some peaks reach very high energies. This seems to be due to a �aw in the model: SiC has roughly threetimes the bandgap of Si, but almost ten times the critical �eld. In reality, there must be some mechanism asidefrom ionization energy that holds the avalanching back in SiC, but the author does not know what it is.Some (admittedly hurried) research into the matter has not brought much to light. In a paper from 2006, Li-Mo Wang suggests a general equation relating critical �eld to bandgap for all semiconductors, EBI = 1.36 ×107(Eg/4.0)

318, where EBI is the critical �eld in a perfect crystal. This works rather well for silicon - 1.36 ×107(1.12/4.0)3 = 2.99×105 V/cm - but is less on the mark for SiC: 1.36×107(3.23/4.0)3 = 7.16×106 V/c is almostthrice the measured critical �eld. Granted, the SiC available has a ways to go before it can be labeled a perfectcrystal, but even if we let ourselves be satis�ed by that explanation this equation does not help the project at hand.If applied, it would only add an even higher �eld, causing even higher ionization.

4.2 Approximations

The biggest di�culty when simulating semiconductor processes is the staggering number of interdependent events,so that for an accurate simulation the minute change of position and/or energy of a single electron means that everyother parameter must be recalculated. For all but very small systems, this simply isn't possible. Instead, one mustmake approximations, and to make those approximations one must be able to evaluate the magnitude of the errorintroduced.

18Li-Mo Wang, Relationship between Intrinsic Breakdown Field and Bandgap of Materials, 25th International Conference on Micro-electronics, 2006

12

4.2.1 Carriers as an ideal gas

The most glaring simpli�cation in this model is that the carrier particles neither interact with the Coulomb potentialover the depletion zone nor with each other. This can partly be justi�ed by calling the system large in comparisonto the carriers. However, as impact ionization is a very localized process, these collisions have to be studied ina much 'smaller' context than carrier transport in a diode normally would be, and in that case the charge of thecarrier should play a part in the probability that a fast-moving carrier will collide with a stationary (in comparison)atom. The error margin introduced by this is drowned out by the random probability, but if this project is revisitedwith a better thought out probability function, then the question of how close to an ideal gas the carrier particlescan be considered should be evaluated.

4.2.2 Isotropy

As has been explained, Si is an isotropic material but 4H-SiC is not. In the real world this should result in somedi�ering transport properties between the two, but these di�erences are not taken into account in this project.

4.2.3 E�ective mass

While we often describe the bandgap as the distance between two levels of energy, we know that in reality theselevels are not so level at all. Instead both conduction and valence bands are made up of several bands that havedi�erent energies at di�erent points in the reciprocal lattice, as can be seen in �gure 519. The material bandgap isthe minimum distance between these groups of energy bands, which is where it takes the least energy for a valenceelectron to enter the conduction band.

Figure 5: Band structure of silicon at 300 K

The e�ective mass is a measurement for how steep these curves are, as a sharper turn causes a larger shift inmomentum which can be percieved as added weight. The steeper the curve, the higher the e�ective mass. However,the e�ective mass is not generally the same in all crystal directions, particularly not for an anisotropic materiallike SiC. In the simulation the e�ective mass used is a weighted average value of the transverse and longitudinale�ective masses calculated using Lundstrom's equation20 for bulk silicon. Evaluating how well it works for SiC isbeyond the scope of this project, but should be done if the simulator is ever to be expanded upon.

1

m∗c≡ 1

3m∗l+

2

3m∗t(7)

19http://www.io�e.ru/SVA/NSM/Semicond/Si/bandstr.html 2012-04-2820Lundstrom, Fundamentals of Carrier Transport, p. 140

13

4.2.4 Dielectric constant for 4H-SiC

There doesn't seem to be a published value for the dielectric constant of 4H-SiC. The closest available seems to bea measurement for the 6H polytype with ε⊥ = 9.66 and ε‖ = 10.03 published in 1970 by Patrick and Choyke21, thataccording to Io�e Physico-Technical Institute22 is used for 4H-SiC as well. In his dissertation SiC Semiconductor

Devices Technology, Modeling, and Simulation Tesfaye Ayalew amends these values to ε⊥ = 9.76 and ε‖ = 9.9823,and suggests that since 4H has a somewhat smaller bandgap than 6H-SiC, we may expect these values to be slightlylower in 4H-SiC. For the purpose of this paper, I judge that the di�erence in directionality will be inconsequentialwhen submerged in the other approximations and that in this case 4H-SiC can be treated as isotropic with a relativepermittivity of 9.9.

4.2.5 Intrinsic carrier concentration of SiC at 600 K

This value was found by applying a ruler to a �g. 3.7 in Ayalew's paper24, which yielded a value of 107cm−3for theintrinsic carrier concentration in SiC at 600 K. A more exact measurement would be preferable, but as the value isonly used to calculate the built-in potential di�erence in the pn-junction, the importance of which decreases withincreasing bias, it is not a major concern.

5 MATLAB Program

5.1 Time scale and step size

One of the �rst concerns inherent in this kind of simulation is what kind of time step one should use. In this model,no set time step is speci�ed; instead, we work with a set step size in the horizontal direction and check our otherparameters after each such step. The particle is expected to travel at an angle to this straight horizontal path,which means it will travel further and take longer the wider the angle is.

Of course, the size of the step must be carefully considered. If it is too big, the carriers will rush forward and out ofthe monitored zone in only a few iterations, which tells us nothing. If it is too small, the carriers will never build upspeed and will consequently loiter near the injection end of the depletion zone until the code runs out of iterationsand the program ends.

To have an idea about what kind of step can be reasonable, we can start with comparing it to the mean free pathin a real semiconductor, which is given by multiplying the average carrier velocity by the scattering time τsc.

lm = v × τsc (8)

As this is the average velocity under avalanching potential, v is the saturated drift velocity which is on the order of107 cm/s for most semiconductors25. To calculate the scattering time τsc, we use Singh's equation26 for mobility.

µ =qτscm∗

(9)

As the carriers responsible for avalanching are always electrons, µ would be the electron mobilities, and m* is thee�ective mass. The e�ective mass varies depending on crystal direction, but is for Si and SiC always of about thesame magnitude as the free electron mass. As the simulation should preferably be as similar as possible, with anexception for speci�c material di�erences, for both materials, m will replace m* in this equation. Furthermore, theelectron mobility is not constant, but depends on doping and temperature. Let's say, for argument, that we areusing n-type Si with a doping concentration Nd=1018cm−3 at 600 K and n-type SiC with Nd=1018cm−3 at 300 K.We then have µSi=140027 and µSiC= 20028 cm2V −1s−1. Knowing the mobilities and e�ective masses for siliconand SiC, we can then calculate the scattering time and �nd lm, which is the average distance a carrier travels before

21Patrick & Choyke, Static Dielectric Constant of SiC,197022http://www.io�e.rssi.ru/SVA/NSM/Semicond/SiC/basic.html 2012-04-0323http://www.iue.tuwien.ac.at/phd/ayalew/node69.html 2012-04-0324http://www.iue.tuwien.ac.at/phd/ayalew/node62.html 2012-04-0225Singh, Semiconductor Devices, p. 9826Singh, Semiconductor Devices, p. 9427http://www.io�e.ru/SVA/NSM/Semicond/Si/electric.html#Hall28The diagram '4H-SiC. Electron Hall mobility vs. temperature', http://www.io�e.rssi.ru/SVA/NSM/Semicond/SiC/hall.html

14

scattering. Knowing that the carriers that cause ionization will be the unusually speedy outliers. The calculationsgive that the average scattering times are τscSi

= 8× 10−13 and τscSiC= 1× 10−13. This would put the mean free

path in the micrometer range.

With the way the acceleration is implemented in the code a carrier that is slow during one step will acceleratewildly in the next, so to limit velocities that otherwise strove towards and beyond the speed of light I have limitedthe time taken per step to a maximum of 1×10−13 seconds. This restriction mainly applies in the �rst few hundrediterations, as once carriers are up to speed they will generally be moving in the 10−14 to 10−15 seconds per stepregion.

As for step size, due to how scattering probability is implemented there must be the same number of steps forSi and Sic, or Si will have ten times as many ionizations. It then seems sensible to de�ne step = wd

loopSize, with

with typical valueswdSi= 6.7 × 10−5, wdSiC

= 8 × 10−6 and loopSize=1000 gives stepSi = 6.7 × 10−8 m andstepSiC = 8× 10−9.

5.2 Flight mechanics

The model relies on ball-like carriers bouncing o� each other to simulate current transport. The kinetic energy ofthese carriers is calculated using a very familiar equation from Newtonian mechanics:

EKin =1

2mv2 (10)

The velocity is always changing; when a carrier enters the simulated depletion zone, it will have a randomizedvelocity in the range expected for a thermally excited electron at a temperature of 300 K (600 K for SiC). As ittravels further into the zone, it is exposed to an electric �eld of increasing (up to the abrupt junction) strength thatwill sweep it forward and accelerate it over a evenly spaced grid in space stretching between x = 0 and x = wd ,over time t which is the time it takes to cross a step of length s at last step's velocity vx0 .

vx = vx0+qFEt

m0(11)

This velocity acts on x , and in concordance with the y-velocity brings the carrier one step forward. At this newposition, the electric �eld strength FE is updated, and acts again to update vx , which changes x , and so on andso forth. Now we introduce scattering to the model: What happens when a fast-traveling electron charges into a,comparatively, massive and stationary atom?

For every iteration, the function probcalc calculates the probability that a carrier will hit something. This proba-bility is compared to a random number to see if collision occurs. If it does, the kinetic energy of the carrier will bechecked and compared to the bandgap energy. Should EKin be larger, that means that the particle carried enoughpunch to knock another electron free from the ion it hit. The coordinates of this event are then recorded, and ared star is drawn in the plot to mark the occasion. Of course, in the real case we would now have a secondaryelectron being accelerated and possibly ionizing yet another electron, that in turn may ionize another, giving us theproverbial avalanche. In this model, this e�ect has been ignored to save on computing time, so that we only recordthe path of the initial electrons. Thus the number ioncount is not truly the number of ionizations, but more thefactor by which the original number of carriers has been ampli�ed.

To compare ioncount to Baliga's M(x) as the actual numbers of ionizations reached throughout the depletion zone,our best bet seems to be to hope that the carrier we follow is typical (which is impossible to tell for any one run, butshould approximate a true average if the run is long enough and we follow one carrier from start to �nish enoughtimes). As M(x) is the number of new carriers generated and ioncount ticks up by one every time the carrierdoubles itself, the relationship between ioncount and M(x) should go toward M(x) = 2× ioncount− 1.

5.3 The MATLAB functions

The program �ow can perhaps best be illustrated by a �owchart, as seen below.

15

Figure 6: avasim program �ow

5.3.1 avasim.m

avasim is the main function in this program. avasim receives the necessary data, sends it on to the appropriatefunctions and redirects it to the plotting tools, before returning to the user with the number of ionizations achieved.For ease of use, avasim only takes two input parameters: material (Si or SiC, given as a string) and voltage overthe junction.

avasim is con�gured to simulate a diode with expected avalanche breakdown at V=1 kV, from which doping levelsare calculated. Should one wish to change the target breakdown voltage, the parameter is called V_target. Donordoping levels Nd are set to be a hundred times higher than acceptor levels Na.

All other parameters one might wish to set, such as doping levels, are set in the code itself. avasim starts by settingall values, after which it calculates the e�ective mass. Of course, Si and SiC have di�erent crystal properties andas such also di�erent e�ective masses.

16

Unfortunately, the equation used to calculate the e�ective mass is only valid for isotropic materials29, and is alsorecommended for weak electric �elds, something the several MV �elds required for avalanching are unlikely toqualify as. However, it still ought to be better than going with the free electron mass.

Most of avasim's real work is executed in two loops, one which loops through a speci�ed number of carriers and onethat takes each carrier on its journey across the depletion zone. These loops are doubled so that there are in facttwo sets of nearly identical nestled loops, one to track �ve separate carriers to give an example of what a carrierpath can look like and one to track 1000 carriers so a set of reliable mean values can be calculated.

The �rst of these loops starts as soon as all constants and starting values are assigned, and initializes an electron-likecarrier in the middle of the left hand edge of the simulation region.

Since an ideal pn-junction has all of its electric �eld concentrated over the depletion zone, the carrier motion outsidethis region will be thermal in origin. The velocity of thermal carriers can be found from 1

2mv2= 3

2kT , so that theaverage velocity of carriers dropping into the depletion zone will be

vth =

√3kT

m0(12)

Of course, we cannot assume that every instantiated carrier will have this velocity. Therefore, vth is multiplied witha normally distributed random number with a mean value of 1, an example of which can be seen in �gure 6.

Figure 7: Example of the random velocity factor distribution for 10000 samples.

The velocity is then randomly divided into x and y-direction velocity, with a �nal random operation that randomizesthe y-direction between up and down.

At this point, the function widthcalc is called, returning the width wd of the depletion zone. This wd de�nes theactive area of our device, and any carrier that escapes the region is terminated.

29See section 4.1.3

17

After widthcalc, the initialized carrier enters loop number two, where it is exposed to an electric �eld FE courtesyof the function fieldcalc. This function returns a value for the electric �eld strength in a given x-coordinate, andis called at every iteration. For every new value of FE , the velocity vx is updated by equation 11.

The other velocity number, vy , is not subject to an accelerating �eld and does not change unless there is a scatteringevent. This, along with a few updates of the matrices tracking position, time and velocity, is followed by a condition:If the next set of x values (y values that get too far away from the middle are instead set to bounce back) areoutside the depletion zone boundaries, assign all remaining x :s and y :s with the current x and y values and quitthe loop. If not, simply carry on like previously until

i) The values are outside the boundaries, or

ii) The loop counter has reached the set maximum value

If the loop continues, it progresses to the next function: probcalc. probcalc started out as one of the moreambitious parts of this project, in that in it the author attempted to divine what factors govern the scatteringprobabilities and weighing them against the values predicted by the ionization integral. However, it was not to be:The probability kept coming back as one, which �ts poorly with an observable reality where current can actually becontrolled fairly well. In the end the ambitious probcalc was replaced by the somewhat less impressive 'a randomnumber between 0 and 1', which sadly performed better. These momentary setbacks aside, the author will continueto mull it over on sleepless nights.

The call to probcalc is followed by a call to the function trajcalc, which takes position, velocity, probabilityand material properties into account to move the carrier one step ahead. The carrier path is returned to avasim,whereupon the loop conditional determines whether to run the loop one more time or step out of it. Once the loopis �nished, the results are plotted for easy perusal by the reader.

5.3.2 widthcalc.m

widthcalc calculates the built-in potential Vbi as well as the width wd of the depletion zone, taking doping andbias voltage into consideration. Vbi is temperature dependent, and the temperature is set to 300 K for silicon and600 K for SiC. Should we wish to run it at a di�erent temperature, it can be arranged with a simple adjustmentin the main function avasim. While Vbi is not directly interesting for the purpose of understanding avalanchebreakthrough, it is a necessary part of the equations for wpand wn , the two halves30 of the depletion zone. Theseequations follow below:

wn =

√2Es(Vbi + V )

q

Na

Nd(Na +Nd)(13)

wp =

√2Es(Vbi + V )

q

Nd

Na(Na +Nd)(14)

Aside from the widths, widthcalc also returns the unbiased depletion zone with wd0 to be used by fieldcalc.

In the case where one side is about a hundred times more heavily doped than the other, which is the standard casein power diodes, the di�erence between wd and the lower doped side's width can largely be disregarded. In thiscase one may approximate wd ≈ wp when Nd is large as in the case of this simulation.

5.3.3 �eldcalc.m

fieldcalc employs two equations to calculate the electric �eld, one for each side of the depletion zone. Theseequations had to be rewritten somewhat from their textbook form, as

• x here moves between0 and wp+wn rather than the more familiar −wp ≤ x ≤ wn

• There seemed to be no ready-made equation for the �eld when an external voltage was applied, so one wasderived

For0 ≤ x ≤ wp , we have

FE =qNa(wp − (wp − x))

Es(15)

30Note that these 'halves' are generally not of the same size. They do, however, have equal but opposite charge.

18

For wp < x ≤ wp + wn , the equation is

FE = −qNd(x− wn− wp)Es

(16)

Note that in this simulation, the carriers move one after the other. If they were all moving at once, they wouldin�uence each other and try to diverge as a result of their negative charge, and results would look quite di�erent.

As wn is in this case very small, fieldcalc exclusively uses equation 15.

5.3.4 trajcalc.m

The function trajcalc has several purposes:

• tracking carrier movement

• simulating the scattering events

• monitoring energy levels at scattering to see whether ionization occurs

When a carrier is treated by trajcalc, the �rst question asked is whether it scatters. If it doesn't, it is simplyaccelerated by equation 11 and sent on its way. If it does scatter, a secondary question is asked of it: Does thecollision carry with it enough kinetic energy enough to create a new carrier? If yes, the new velocities are the old ones

minus the velocity corresponding to the ionization energy vion =√

2qEgm0

and multiplied by the �eld acceleration,

vx(i) = vx(i−1) − vion +qFEt

m0(17)

vy(i) = vy(i−1) − vion (18)

whereupon the x and y-velocities are scrambled directionally to mimic hard particles bouncing o� each other andthe coordinates for the scattering are recorded.

If no, some energy is still lost in the scattering and direction is randomly scrambled as in the ionizing case.

vx(i) = vx(i−1) +qFEt

m0× loss (19)

vy(i) = vy(i−1) × loss (20)

limit is also the main calibration tool for this simulation: The total number of ionizations measured dependslogarithmically on limit, which has no real world equivalent since the modeled carriers do not correspond to singleelectrons. The correct value for limit can be found by comparing ioncount to the ionization integral as describedin section 5.2.

6 Calibrating the model

Once all the code is written, the parameters that do not directly correspond to physical entities can be tweakedto �t the simulation to reality. These are parameters such as loss, which models the elasticity of the scatteringevents, and limit, which controls the scattering probability. The reasoning in section 5.2 suggest that avalanchingcan be said to occur when a freshly generated carrier has a high likelihood of generating a new carrier of its ownbefore leaving the depletion zone, but without better data as to what can be a good average number of ionizationsper carriers further tweaking seems pointless.

6.1 Loss

The factor loss is applied to all scattering, whether it is ionizing or not. Changing it does a�ect the number ofionizations, but it does so acting together with limit. It should still be included for completeness, currently set at70% energy loss per collision, but more data is needed to ascertain whether that is a reasonable value or not.

19

6.2 Limit

The randomly generated probability prob is weighted against limit, so that the lower (1−limit) is the quicker thecarrier is likely to scatter. In essence, limit is closely related to the failed probability function probcalc, thoughit has been derived by trial and error rather than deep contemplation of the mysteries of carrier transport.

The trial-and-error method, with the goal of giving an approximate average of two ionizations per carrier when atthe measured breakthrough �eld (3 × 107V/m for Si and 2.5 × 108 V/m for SiC31), suggest that silicon and SiCrequire di�erent limits to give results that agree with reality. However, it seems like poor scienti�c practice tomultiply your results with di�erent numbers until you �nd one you like, so it is now set at 0.5 in lieu of a betteroption.

Part III

Results

7 Graphics

Shown here are MATLAB plots of the simulation output. The two black lines over the energy plots 'Carrier/Meanenergy by position' marks the bandgap energies for silicon (1.12 eV) and silicon carbide (3.23 eV), which are theminimum energies needed to ionize a new particle. The diode simulated is designed for breakthrough at 1000 V.

31Singh, Semiconductor Devices: Basic Principles, p. 523

20

7.1 Silicon

7.1.1 Far below breakthrough

Figure 8: Applied Voltage = 100 V

At voltages much lower than the measured breakthrough voltage, the �eld acceleration is too low to give rise toperceptible kinetic energy. Ionizations are vanishingly unlikely. Standard velocities are in the magnitude of 105m/s.As can be seen in the mean carrier velocity graph at bottom left, the shape of the velocity curve closely echoes the�eld intensity curve at top right.

21

7.1.2 Below breakthrough

Figure 9: Applied voltage = 500 V

At half the expected breakthrough voltage, we begin to see scattered energy peaks building up. The mean energystays far below what is required for ionization, but several individual carriers break above the average and causeionization. Standard velocities are about twice what they were for V=100 V.

22

7.1.3 Breakthrough

Figure 10: Applied voltage = 1000 V

With voltage at the designed-for breakthrough voltage 1 kV we start to get a lot of ionizations. The electric �eldpeaks at the critical �eld strength, and the mean velocity curve can be seen to curve slightly in a way that is notseen in the �eld strength curve as ionization turns kinetic energy into a greater number of free carriers.

23

7.1.4 Beyond breakthrough

Figure 11: Applied Voltage = 1500 V

At 1.5 times the expected breakthrough voltage, we are in full avalanche mode. Still, we can see that the �eld isnot very much higher than it was for V = 1kv, still less than 4 × 107V/m as compared to 3 × 107V/m, and theother results follow in that spirit. Velocities are higher, but on average not that much higher, and while there arecertainly more ionizations the mean carrier energy remains �rmly below the ionization energy.

24

7.2 Silicon Carbide

7.2.1 Far below breakthrough

Figure 12: Applied voltage = 100 V

Much as in the low-voltage silicon model, the low-voltage SiC simulation produces no energy peaks high enough tocause ionization. However, there is a visible curl to the mean velocity curve that was not apparent for silicon. Thisis due to the maximum time step value of 1× 10−13s, which has more in�uence in the higher �elds of SiC.

25

7.2.2 Below breakthrough

Figure 13: Applied voltage = 500 V

At 500 V, we begin to see a lot of ionizations about halfway through the depletion zone where the �eld is about1× 108V/m, somewhat more than a third of the critical �eld for SiC. This is further discussed in section 4.1, andis thought to be a �aw in the model. It is however an interesting �aw, and a possible topic for further study.

26

7.2.3 Breakthrough

Figure 14: Applied voltage =1000 V

At breakthrough voltage, we have plenty of ionization and high average energies at upwards of 2.5 eV. As for the500 V run, this is probably higher than it should be but the mechanism that suppresses it in nature is unknown tome at this time.

27

7.2.4 Beyond breakthrough

Figure 15: Applied voltage =1500 V

At 1.5 KV we see about the same thing we saw at 1 kV, which is a lot of ionization and suspiciously high energiesfor carriers in a semiconductor - clearly, there is more to this to be discovered but it falls outside the projectspeci�cation of a bouncing ball-model with classically determined kinetic energy as the ionizing agent.

8 Conclusion

There were two separate goals of this project: One one hand, it was intended to let the author explore avalanchebreakthrough until able to write teaching materials, on the other it was intended to produce some form of simpli�edsimulator that could be used as a teaching aid in introductory device physics classes. Both these goals have been met- while the simulation holds too many approximations and simpli�cation to produce useful numerical predictions, itdoes work well for silicon and illustrates the impact ionization process and subsequent avalanche breakthrough well.As silicon is normally the �rst semiconductor students are introduced to in any detail, this should be an adequateteaching aid.

The simulation produces less plausible results for SiC, which the author thinks is in itself a valuable result. Itshows that the model used, which approximates the carriers as hard balls bouncing o� each other and transmittingenergy via inelastic scattering, is not as-is applicable for SiC but needs an unknown suppression mechanism to giverealistic results. This suppression mechanism seems to be hiding somewhere in the relationship between bandgapand critical �eld, which is non-trivial and is suggested as an avenue of further study.

28

References

1. Semiconductor Devices: Basic Principles, Jasprit Singh, 2001, ISBN 0-471-36245-X

2. SiC Semiconductor Devices Technology, Modeling, and Simulation, Tesfaye Ayalew, 2004, http://www.iue.tuwien.ac.at/phd/ayalew/mythesis.html2012-04-28

3. Fundamentals of Carrier Transport, Mark Lundstrom, 1990, ISBN 0-201-18436-2

4. Process Technology for Silicon Carbide Devices, Carl-Mikael Zetterling, Editor, 2002 IEE EMIS ProcessingSeries 2, ISBN 0-85296-998-8

5. Power Semiconductor Devices, B. Jayant Baliga, 1996, ISBN 0-534-94098-6

6. Relationship between Intrinsic Breakdown Field and Bandgap of Materials, Li-Mo Wang, 2006, presented at25th International Conference on Microelectronics

7. Process Technology for Silicon Carbide Devices: Lecture by Carl-Mikael Zetterling on 2000-03-21, notesavailable at http://web.it.kth.se/~bellman/docs/ZetterlingDocentLecture.pdf 2012-06-24

8. Io�e Physico-Technical Institute Semiconductor Materials Database, http://www.io�e.rssi.ru/SVA/NSM/Semicond/2012-06-24

9. Polytypism and Properties of Silicon Carbide, F. Bechstedt, P. Käckell, A. Zywietz, K. Karch, B.Adolph,K. Tenelsen, J. Furthmüller, 1997, DOI: 10.1002/1521-3951(199707)202:1<35::AID-PSSB35>3.0.CO;2-8

10. Correlation between layer thickness and periodicity of long polytypes in silicon carbide, J.F. Kelly, G.R.Fisher, P. Barnes, 2005, DOI: 10.1016/j.materresbull.2004.10.008

11. Principles of Semiconductor Devices,Bart Van Zeghbroek, http://ecee.colorado.edu/~bart/book/contents.htm2012-06-24

12. Static Dielectric Constant of SiC, Lyle Patrick and W.J. Choyke, 1970, DOI: 10.1103/PhysRevB.2.2255

13. Abundances of the Elements in the Earth's Crust, http://hyperphysics.phy-astr.gsu.edu/hbase/tables/elabund.html2012-06-24

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Part IV

Appendix: MATLAB code

30

function [x1,y1,v1,vx1,vy1,ionCount1,ionCount2,F_e1,t1] = avasim3(material,V)if nargin ~= 2 disp('Usage: avasim(material,V) where material is Si or Sic, V is the applied voltage');else clf m0=9.1e-31; % elektronmassa SI q=1.6e-19; % elektronladdning SI k=1.38e-23; %Boltzmann's constant loss=0.3; % part of energy kept in each scattering. 1 is all, 0 is none limit=0.5; if strcmpi(material,'Si') E_crit=3e7; % critical breakdown field V/m Eg=1.12; m_effl=0.98*m0; m_efft=0.19*m0; T=300; % Run Si at 300 K Es = 11.8*8.85*10^-12; % Silicon permittivity in F/m^2 ni=1.02e+16; % intrinsic concentration of silicon at 300K (m3) elseif strcmpi(material,'SiC') E_crit=2.5e8; % critical breakdown field V/m Eg=3.23; m_effl=0.29*m0; m_efft=0.42*m0; T=600; % Run SiC at 600 K to ensure carrier concentration ~ doping Es = 9.9*8.85*10^-12; % Approximate permittivity of 4H-SiC in F/m^2 ni=1.02e+16; % intrinsic concentration of silicon at 300K (m3) else disp('Enter either Si or SiC'); x=-1; y=-1; v=-1; ionCount=-1; F_e=-1; return end %ange Eg i eV, dopkoncentrationer i m^-3 tic loopSize = 1000; ionCoord1=[]; n=5; % Separately tracked carriers m=1000; %Only mean values tracked m_eff=3*m_efft*m_effl/(m_efft+2*m_effl); % giltig f�r Si, mindre s� f�r SiC (anisotropi). v1 = zeros(n,loopSize); %Preallocate v-vector vx1 = zeros(n,loopSize); %Preallocate vx-vector vy1 = zeros(n,loopSize); %Preallocate vy-vector

x1 = zeros(n,loopSize); %Preallocate x-vector y1 = zeros(n,loopSize); %Preallocate y-vector F_e1 = zeros(n,loopSize); %Preallocate F_e-vector t1= zeros(n,loopSize); v2 = zeros(m,loopSize); %Preallocate v-vector vx2 = zeros(m,loopSize); %Preallocate vx-vector vy2 = zeros(m,loopSize); %Preallocate vy-vector x2 = zeros(m,loopSize); %Preallocate x-vector y2 = zeros(m,loopSize); %Preallocate y-vector F_e2 = zeros(m,loopSize); %Preallocate F_e-vector t2= zeros(m,loopSize); % wd ~= wn (ojämn dopning) V_target=1000; wd_target=2*V_target/E_crit; % Hitta dopnivå från vald genombrottsspänning Na=Es*E_crit^2/(2*q*V_target); Nd=Na*100; wd=widthcalc2(Na,Nd,V,T,Es,ni); step=wd/(loopSize-10); % vill ha några steg utanför depletion zone for num=1:n; %% separate carrier loop ---------------------------------------------------------------------------------------------------- v_th=sqrt(3*k*T/(m0)); %Rimliga storlekar vstart=v_th*randn(1)*0.15+1; if(vstart<0) vstart=-1*vstart; end vx1(num,1)=vstart*rand(1);%Initial velocity x-direction - alltid positiv vy1(num,1)=sqrt(vstart^2-vx1(num,1)^2);%Initial velocity y-direction a=rand(1); if a<0.5 %vy can vara både upp eller ned % vy=vy; else vy1(num,1)=-vy1(num,1); end for i=1:loopSize F_e1(num,i)=fieldcalc2(wd,x1(num,i),Na,Es); v1(num,i)=sqrt(vx1(num,i)^2+vy1(num,i)^2); t1(num,i)=abs(step/vx1(num,i)); %vill inte ha negativ tid %begränsar t så inte får enorma värden if (t1(num,i)>1e-13) t1(num,i)=1e-13; end x1(num,i+1)=x1(num,i)+step; y1(num,i+1)=y1(num,i)+vy1(num,i)*t1(num,i); if (x1(num,i+1)>wd) || (x1(num,i+1)<0) % Carrier outside of simulation region disp('break') x1(num,i+1:loopSize)=x1(num,i+1); y1(num,i+1:loopSize)=y1(num,i+1); break; end

if (y1(num,i)>5e-4) || (y1(num,i)<-4e-4) % Bounce carrier off sides at incoming angle vy1(num,i)=-vy1(num,i); y1(num,i)=y1(num,i)+vy1(num,i)*t1(num,i); end % Calculate probability of collisions p=rand(1); % Calculate electron movement [vx1(num,i+1),vy1(num,i+1),ionCoord1]=trajcalc3(v1(num,i),x1(num,i),y1(num,i),vx1(num,i),vy1(num,i),Eg,p,loss,m_eff,ionCoord1,F_e1(num,i),t1(num,i),limit); end end %---End separate carrier loop------------------------------------------------------------------------------------------------------------- ionCount1=length(ionCoord1); ionCoord2=[]; for num=1:m; %mean values carrier loop-------------------------------------------------------------------------------------------------- v_th=sqrt(3*k*T/(m0)); %Rimliga storlekar vstart=v_th*randn(1)*0.15+1; if(vstart<0) vstart=-1*vstart; end vx2(num,1)=vstart*rand(1);%Initial velocity x-direction - alltid positiv vy2(num,1)=sqrt(vstart^2-vx2(num,1)^2);%Initial velocity y-direction a=rand(1); if a<0.5 %vy can vara både upp eller ned % vy=vy; else vy2(num,1)=-vy2(num,1); end for i=1:loopSize % separate carrier loop F_e2(num,i)=fieldcalc2(wd,x2(num,i),Na,Es); v2(num,i)=sqrt(vx2(num,i)^2+vy2(num,i)^2); t2(num,i)=abs(step/vx2(num,i)); %vill inte ha negativ tid %begränsar t så inte får enorma värden if (t2(num,i)>1e-13) t2(num,i)=1e-13; end x2(num,i+1)=x2(num,i)+step; y2(num,i+1)=y2(num,i)+vy2(num,i)*t2(num,i); if (x2(num,i+1)>wd) || (x2(num,i+1)<0) % Carrier outside of simulation region x2(num,i+1:loopSize)=x2(num,i+1); y2(num,i+1:loopSize)=y2(num,i+1); break; end if (y2(num,i)>5e-4) || (y2(num,i)<-4e-4) % Bounce carrier off sides at incoming angle vy2(num,i)=-vy2(num,i);

y2(num,i)=y2(num,i)+vy2(num,i)*t2(num,i); end % Calculate probability of collisions p=rand(1); % Calculate electron movement [vx2(num,i+1),vy2(num,i+1),ionCoord2]=trajcalc3(v2(num,i),x2(num,i),y2(num,i),vx2(num,i),vy2(num,i),Eg,p,loss,m_eff,ionCoord2,F_e2(num,i),t2(num,i),limit); end end %---End separate carrier loop------------------------------------------------------------------------------------------------------------- ionCount2=length(ionCoord2); v_mean=mean(v2); %-###########################--PLOTPLOTPLOTPLOT----##################################################################################-------- subplot(3,2,1) axis([0 wd min(min(y1)) max(max(y1))]) axis manual plot(x1(1,1:loopSize),y1(1,1:loopSize),x1(2,1:loopSize),y1(2,1:loopSize),x1(3,1:loopSize),y1(3,1:loopSize),x1(4,1:loopSize),y1(4,1:loopSize),x1(5,1:loopSize),y1(5,1:loopSize)); hold on if (isempty(ionCoord1)) disp('There were no ionizations') elseif (numel(ionCoord1)==2) plot(ionCoord1(1,1),ionCoord1(1,2),'r*'); disp('There was 1 ionization.') else plot(ionCoord1(1:length(ionCoord1),1),ionCoord1(1:length(ionCoord1),2),'r*'); ionStr=['There were ', int2str(length(ionCoord1)), ' ionizations.']; disp(ionStr); end title('Position of carrier in junction') ylabel('Position in device [m]') xlabel('Position in device [m]') %---------------------------------------------------------------------------------------------------------------------------------------------- subplot(3,2,2) %field plot(x1(1,1:loopSize),F_e1(1,1:loopSize)); % Alla fält kommer se likadana ut även om partikeln tagit olika många steg % plot(x(1,1:loopSize),F_e(1,1:loopSize)) title('Field at position in junction') ylabel('Field strength [V/m]') xlabel('Position in device [m]')

%------------------------------------------------------------------------------------------------------------------------------------------- subplot(3,2,3) %separate velocities plot(x1(1,1:loopSize),v1(1,1:loopSize),x1(1,1:loopSize),v1(2,1:loopSize),x1(1,1:loopSize),v1(3,1:loopSize),x1(1,1:loopSize),v1(4,1:loopSize),x1(1,1:loopSize),v1(5,1:loopSize)); title('Velocity by position') ylabel('Velocity [m/s]') xlabel('Position in device [m]') %------------------------------------------------------------------------------------------------------------------------------------------ subplot(3,2,4) %separate energies plot(x1(1,1:loopSize),1.12,'k--'); % silicon bandgap hold on plot(x1(1,1:loopSize),3.23,'k--'); % SiC bandgap plot(x1(1,1:loopSize),0.5*m_eff/q*v1(1,1:loopSize).^2,x1(1,1:loopSize),0.5*m_eff/q*v1(2,1:loopSize).^2,x1(1,1:loopSize),0.5*m_eff/q*v1(3,1:loopSize).^2,x1(1,1:loopSize),0.5*m_eff/q*v1(4,1:loopSize).^2,x1(1,1:loopSize),0.5*m_eff/q*v1(5,1:loopSize).^2); title('Carrier energy by position') ylabel('Energy [eV]') xlabel('Position in device [m]') %------------------------------------------------------------------------------------------------------------------------------------------- subplot(3,2,5) %mean velocity ionStr=['There were on average ', int2str(length(ionCoord2)/m), ' ionizations per carrier.']; disp(ionStr); plot(x2(1,1:loopSize),v_mean); title('Mean velocity by position') ylabel('Velocity [m/s]') xlabel('Position in device [m]') %--------------------------------------------------------------------------------------------------------------------------------------------- subplot(3,2,6) %mean energies plot(x2(1,1:loopSize),1.12,'k--'); % silicon bandgap hold on plot(x2(1,1:loopSize),3.23,'k--'); % SiC bandgap plot(x2(1,1:loopSize),0.5*m_eff/q*v_mean.^2); title('Mean energy by position') ylabel('Energy [eV]') xlabel('Position in device [m]') tocend

function wd=widthcalc2(Na,Nd,V,T,Es,ni)% All units SIq=1.6e-19; %Electron chargek=1.38e-23; %Boltzmann's constant in J/KVbi=k*T*log(Nd*Na/ni^2)/q;% Bias voltagewp=sqrt((2*Es*(Vbi+V)/q)*(Nd/(Na*(Na+Nd)))); %Biasedwd=wp;

function F_e=fieldcalc2(wd,x,Na,Es)q=1.6e-19; %Electron chargek=1.38e-23; %Boltzmann's constant in J/K %Ojämn dopning gör att så gott som hela utarmningsområdet på p-sidan,%wp~≃wdif (0<=x && x<=wd) F_e = q*Na/Es*(wd-(wd-x));else F_e=0; % Outside depletion zoneend

function [vx,vy,ionCoord]=trajcalc3(v,x,y,vx,vy,Eg,prob,loss,m,ionCoord,F_e,t,limit) q=1.6e-19;m0=9.1e-31;Ekin=0.5*m*v^2;EgJ=q*Eg; %bandgapsenergi i joulev_ion=sqrt(2*EgJ/m0); % energy lost upon ionizationdir=floor(8*rand(1)); %switch cases if ((1-limit)>= prob)%sprids inte vx=vx+q*F_e*t/m0;% velocity is increased elseif ((1-limit) < prob)%spridning if Ekin>EgJ %Energi nog att sl� loss ett elektron/h�l-par vx=(vx-v_ion+q*F_e*t/m0)*loss; vy=(vy-v_ion)*loss; ionCoord=[ionCoord;x,y]; switch dir case 0 % self scattering vx=vx; case 1 % momentum-conserving scattering below vx=-vx; case 2 vy=-vy; case 3 vx=-vx; vy=-vy; case 4

temp=vx; vx=vy; vy=temp; case 5 temp=vx; vx=vy; vy=-temp; case 6 temp=vx; vx=-vy; vy=temp; case 7 temp=vx; vx=-vy; vy=-temp; end return else % semi-elastic scattering -- non-ionizing bounce vx=(vx+q*F_e*t/m0)*loss; %The parameter loss models the energy loss during phonon scattering. vy=loss*vy; switch dir case 0 % self scattering vx=vx; case 1 % momentum-conserving scattering below vx=-vx; case 2 vy=-vy; case 3 vx=-vx; vy=-vy; case 4 temp=vx; vx=vy; vy=temp; case 5 temp=vx; vx=vy; vy=-temp; case 6 temp=vx; vx=-vy; vy=temp; case 7 temp=vx; vx=-vy; vy=-temp;

end endend

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