I MPACT IONIZATION RATE
CALCULATIONS
FOR DEVICE SIMULATION
Diplomarbeit
Advisors:
Prof. Dr. D. A. Wharam (University of Tübingen)
Priv.-Doz. Dr. F. M. Bufler (ETH Zürich & Synopsys Switzerland LLC)
Dr. A. Erlebach (Synopsys Switzerland LLC)
submitted by Christian May
Integrated Systems Laboratory,
ETH Zürich
&
Fakultät für Mathematik und Physik,
Eberhard–Karls–Universität Tübingen
Zürich, October 27, 2005
2
About this Thesis
This work has been carried out by Christian May (* August 25th, 1978) from November 1st,
2004 to October 31st, 2005 at the Integrated Systems Laboratory (IIS) of ETH Zurich in cooperation with Synopsys Switzerland LLC.
The thesis is being submitted to the University of Tübingen in partial fulfillment of the
requirements for the degree of Dipl.–Phys.
Thesis advisors were Professor Dr. David A. Wharam at the University of Tübingen, Priv.Doz. Dr. Fabian M. Bufler at ETH Zürich and Dr. Axel Erlebach at Synopsys Switzerland.
Acknowledgements
I would like to express my thanks to: Fabian Bufler for his competent supervision, the people
at Synopsys for their immediate response to any help request, as well as to my family and Olga
for their support.
Declaration of Authorship
I certify that the work presented here is, to the best of my knowledge and belief, original and
the result of my own investigations, except as acknowledged, and has not been submitted, either
in part or whole, for a degree elsewhere.
Place and date
Signature
Contents
Abstract
7
1
9
Introduction
1.1
2
Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Underlying Basics
11
2.1
Important Crystallographic Terms . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1.1
Bravais Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1.2
Coordination Number . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.1.3
Primitive Unit Cell / Primitive Cell . . . . . . . . . . . . . . . . . . .
12
2.1.4
Conventional Unit Cell / Unit Cell . . . . . . . . . . . . . . . . . . . .
13
2.1.5
Wigner–Seitz Primitive Cell . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.6
Crystal Structure: Lattice with a Basis . . . . . . . . . . . . . . . . . .
13
2.1.7
Diamond Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.8
Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.9
Brillouin Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2
Silicon Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.3
Monte–Carlo Transport Simulation . . . . . . . . . . . . . . . . . . . . . . . .
16
3
4
3
4
CONTENTS
Impact Ionization
19
3.1
Phenomenological Description . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2
Impact Ionization Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.2.1
Quantum Mechanical Derivation . . . . . . . . . . . . . . . . . . . . .
20
3.2.2
Random-k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.2.3
Momentum Conserving Impact Ionization Rate . . . . . . . . . . . . .
24
3.3
Determining the Threshold of Impact Ionization . . . . . . . . . . . . . . . . .
25
3.4
Impact Ionization Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Numerical Considerations
29
4.1
Choosing the Delta Distribution Approximation . . . . . . . . . . . . . . . . .
29
4.2
Choosing the Integration Method . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.2.1
Monte–Carlo Integration . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.2.1.1
Crude Monte–Carlo . . . . . . . . . . . . . . . . . . . . . .
33
4.2.1.2
Variance Reduction Techniques . . . . . . . . . . . . . . . .
34
4.2.1.2.1
Importance Sampling . . . . . . . . . . . . . . . .
34
4.2.1.2.2
Stratified Sampling . . . . . . . . . . . . . . . . .
34
Divonne Algorithm . . . . . . . . . . . . . . . . . . . . . .
35
Applying the Divonne Algorithm to the Problem . . . . . . . . . . . .
36
4.2.2.1
Interpolation Method . . . . . . . . . . . . . . . . . . . . .
36
4.2.2.2
The Peak Finder Routine . . . . . . . . . . . . . . . . . . .
37
4.2.2.3
Complete Integration Approach . . . . . . . . . . . . . . . .
38
4.2.1.3
4.2.2
5
Results for Silicon
41
5.1
Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
5.2
Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
CONTENTS
5.2.1
Impact Ionization Rate Integration Results . . . . . . . . . . . . . . . .
46
5.2.2
Modifications to the Phonon System . . . . . . . . . . . . . . . . . . .
47
5.2.2.1
Temperature Dependence of the Band Gap . . . . . . . . . .
47
5.2.2.2
Current Phonon Model . . . . . . . . . . . . . . . . . . . .
48
5.2.2.3
Addition of a Second Acoustic Phonon Branch . . . . . . . .
50
5.2.2.4
Field Dependent Phonon Energies . . . . . . . . . . . . . .
55
Phonon–assisted Impact Ionization . . . . . . . . . . . . . . . . . . . .
55
5.2.3
6
Results for Strained Silicon
57
6.1
Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
6.1.1
Silicon–Germanium Mixed Crystals (Six Ge1−x ) . . . . . . . . . . . . .
57
6.1.2
Strained Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Impact Ionization Rates for Different Stress Levels . . . . . . . . . . . . . . .
59
6.2
7
5
Conclusion and Outlook
61
A Software Documentation
63
A.1 New Keywords Introduced in DESSIS Command Files . . . . . . . . . . . . .
B χ2 Test
63
69
B.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
B.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
C Thresholds of Impact Ionization Processes in Silicon
73
C.1 Hole Initiated Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
C.2 Electron Initiated Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
D Derivation of the Phonon–assisted Impact Ionization Rate (in Progress)
91
6
CONTENTS
List of Figures
94
List of Tables
95
References
97
Index
101
Abstract
We present a method to calculate the impact ionization rate
based on band structure information. Both accuracy and efficiency requirements are thoroughly addressed. The method is
then applied to Silicon as well as strained Silicon. Upon comparing the results to impact ionization coefficient data obtained
from measurements, our simulated electron–initiated impact
ionization is found to be in excellent agreement. However, we
suggest that it is required to take phonon–assisted impact ionization into account to match the hole–initiated rate as well.
7
Chapter 1
Introduction
Impact ionization has been drawing considerable attention since the beginning of investigations
of hot carrier transport. However, impact ionization rates reported in the literature often differ by
several orders of magnitude. A comparison done by [Cartier93] impressively proves this point
by comparing results obtained from empirical pseudo–potential methods as well as simplified
band structures. Therefore, the need for thorough numerical studies arises.
In an (electron–induced) impact ionization event, a high–energy conduction electron collides with a valence electron, thus lifting the latter from the valence to the conduction band.
Upon introducing the electron–hole picture, this process eventually generates two electrons and
one hole. Two properties of this process render it particularly significant: First, it is autocatalytic; second, it can be induced by both electrons and holes.
From a technological point of view, the process is important because it represents a limiting
mechanism for devices under high fields as well as a basis of device functionality (for example
in the design of avalanche photo–diodes). On the other hand, since an impact ionization event
cools down the primary charge carrier significantly, it plays a decisive role in the determination
of the distribution function, which makes it equally interesting from a physical point of view.
In order to achieve an accurate treatment of the ionization process, a full band approach is
required. This complexity along with the large phase space available for this process make an
analytical solution prohibitive. The aim of this thesis therefore is to find a numerical approach
to the problem.
9
10
1.1
1. Introduction
Overview of the Thesis
This thesis is organized as follows: in chapter 2, a short review is given of some important crystallographic terms along with some fundamental concepts of Monte–Carlo simulation. Chapter
3 provides information on impact ionization; different impact ionization rates (Random-k and
momentum conserving) are derived there. After the physical problem is stated, numerical issues are addressed in chapter 4, where we devise a method to integrate the impact ionization
rate. The results for Silicon are presented in chapter 5. Later, the method is applied to strained
Silicon in chapter 6.
Chapter 2
Underlying Basics
2.1
Important Crystallographic Terms
This section on crystallographic terms relies primarily on [Ashcroft76]. The most important
crystallographic definitions and facts necessary for this work are briefly summarized.
2.1.1
Bravais Lattice
The periodic array in which the repeated units of a crystal are arranged is called a Bravais lattice.
Regardless of the actual units (single atoms, groups of atoms or molecules), the Bravais lattice
only points out the underlying periodic structure.
Definition: A three–dimensional Bravais lattice consists of all points with position vectors
of the form
R = n1 a1 + n2 a2 + n3 a3 ,
(2.1)
where a1 , a2 , a3 are not all in the same plane and n1 , n2 , n3 are any integer values. The vectors
ai , i ∈ {1, 2, 3}, are called primitive vectors and span the lattice.
The face–centered cubic (fcc) Bravais lattice can be constructed by adding to the simple
cubic lattice an additional point in the center of each square face (see figure 2.1). A symmetric
11
12
2. Underlying Basics
set of primitive vectors for the face–centered cubic lattice is
a
(ŷ + ẑ)
2
a
(ẑ + x̂)
=
2
a
=
(x̂ + ŷ) .
2
a1 =
a2
a3
(2.2)
We will later see how the diamond crystal structure (which Silicon crystallizes in) can be constructed from the fcc lattice.
Figure 2.1: FCC Unit Cell
2.1.2
Coordination Number
Each point in a Bravais lattice has the same number of closest points (called nearest neighbors),
which is therefore a property of the lattice. A simple cubic lattice has coordination number 6,
while the coordination number of a face–centered cubic lattice is 12.
2.1.3
Primitive Unit Cell / Primitive Cell
A volume of space that fills the whole space exactly when translated through all the vectors
in a Bravais lattice is called a primitive cell or primitive unit cell. To satisfy this condition, a
primitive unit cell must contain precisely one lattice point. Note that the way of choosing a
primitive cell is not unique.
2.1. Important Crystallographic Terms
2.1.4
13
Conventional Unit Cell / Unit Cell
A unit cell is a volume of space that fills the whole space without any overlapping when translated through some subset of the vectors of a Bravais lattice. The conventional unit cell is
generally bigger than the primitive cell, e.g. the face–centered cubic lattice is often described
in terms of a cubic unit cell which is four times as large as a primitive fcc cell.
2.1.5
Wigner–Seitz Primitive Cell
It is always possible to choose a primitve cell with the full symmetry of the Bravais lattice, the
most common one being the Wigner–Seitz Cell. The latter can be constructed by drawing lines
from one given lattice point to all others in the lattice, bisecting each line with a plane, and
taking the smallest polyhedron containing the point bounded by these planes. Therefore, the
Wigner–Seitz cell about a lattice point is the region that is closer to this point than to any other
lattice point.
2.1.6
Crystal Structure: Lattice with a Basis
A crystal structure consists of identical units of atoms, ions, etc. at all points of a Bravais lattice.
Again, a non–primitive conventional unit cell can be chosen to describe a Bravais lattice as a
lattice with a basis.
2.1.7
Diamond Structure
The diamond lattice consists of two face–centered cubic Bravais lattices interpenetrating each
other with a displacement along the body diagonal of the cubic cell by 4a (x̂ + ŷ + ẑ), where a is
the cube side of the conventional cubic cell. In figure 2.2, the two fcc lattices have been printed
in different colors for clarity. Both silicon and germanium, whose lattice constants (cube sides)
are given in table 2.1, crystallize in the diamond structure.
14
2. Underlying Basics
Figure 2.2: Conventional cubic cell of the diamond lattice.
Element
Lattice Constant a [Å]
Silicon
5.43
Germanium
5.66
Table 2.1: Silicon and Germanium Lattice Constants
2.1.8
Reciprocal Lattice
A plane wave eiK·r has the periodicity of the Bravais lattice (constituted by a set of points R) if
the relation
eiK·(r+R) = eiK·r
(2.3)
holds. K then belongs to the so–called reciprocal lattice. From equation (2.3), we get that all
vectors K satisfying
eiK·R = 1
(2.4)
for all R in the Bravais lattice are part of the reciprocal lattice. The Bravais lattice is sometimes
referred to as the direct lattice to distinguish it from the reciprocal lattice. Furthermore, it can
be shown that the reciprocal lattice of the reciprocal lattice is the direct lattice itself.
Let a1 , a2 , a3 be a set of primitive vectors for the direct lattice, then the reciprocal lattice
can be generated by
a1+(i mod 3) × a1+((i+1) mod 3)
,
a1 · (a2 × a3 )
where i ∈ {1, 2, 3}. From definition (2.5) it can be directly observed that
bi = 2π
bi · aj = 2π δij .
(2.5)
(2.6)
2.2. Silicon Band Structure
15
For the diamond structure, the basis vectors spanning the reciprocal lattice can be chosen to
be
b1
=
b2
=
b3
=
2π
a0
2π
a0
2π
a0
( −1
1
1
)
(
1
−1
1
)
(
1
1
−1
)
(2.7)
where a0 refers to the lattice constant.
2.1.9
Brillouin Zone
The Wigner–Seitz primitive cell of the reciprocal lattice is called the first Brillouin zone. As the
reciprocal of the body–centered cubic lattice is the face–centered cubic lattice (and vice versa),
the first Brillouin zone of the fcc lattice is the bcc Wigner–Seitz cell.
The first Brillouin zone for the diamond structure can be analytically described by the following set of equations
2π
a0
2π
2π
2π
|kx | ≤ , |ky | ≤ , |kz | ≤ .
a0
a0
a0
|kx | + |ky | + |kz | ≤ 1.5
(2.8)
A graphical representation is given in figure 2.3, where significant axes and points of symmetry
are labeled.
2.2
Silicon Band Structure
Figure 2.4 shows the band structure of Silicon along some important axes of symmetry (Λ and
∆). The lower group forms the subbands of the valence bands, the upper those of the conduction
bands. Their separation amounts to Eg = 1.12 eV at room temperature.
The upper edge of the valence band lies at the Γ point (k = 0), where two different subbands
(heavy hole and light hole) share a common extremum. Holes can therefore appear near the
extremum in two subbands with differing effective masses. If the extrema of a band lie outside
k = 0, a number of equivalent extrema must exist from symmetry considerations. For example,
the lower edge of the lowest conduction band in Silicon is situated at the X valley (at about 85%
of the ∆ axis; see figure 2.3), where it is six–fold degenerate.
16
2. Underlying Basics
2.3
Monte–Carlo Transport Simulation
Eventually, the impact ionization rate calculated in this work is to be integrated in SPARTA
(Single–PARTicle Approach), a Monte–Carlo simulator.
The aim of this section is to provide a short overview of the Monte–Carlo method in transport simulation. Most of its content is based on [Jacoboni83] and [Bufler02].
In a typical Monte–Carlo program, the simulation starts with a charge carrier in given initial conditions with wave vector k0 . Then the duration of its first free flight is chosen with a
probability distribution determined by the scattering probabilities. During the free flight phase,
external forces are acting according to
h̄k̇ = q E.
(2.9)
k and q here represent the carrier wave vector and charge respectively.
After that, a scattering mechanism is chosen to be responsible for terminating the free flight
according to the relative probabilities of all possible scattering mechanisms
Pi =
Si (k)
.
∑ j S j (k)
(2.10)
Depending on the scattering mechanism, a new k state is chosen as initial state for the following
free flight phase; this procedure is iteratively repeated until the quantities of interest are known
to within the desired precision. One way to determine the precision would be to divide the
entire history into a number of successive subhistories, determine the average value and take its
standard deviation as an uncertainty estimate.
Obviously, the simulation time must be long enough for the chosen initial conditions not
to influence the final results. A reasonable choice therefore has to be a compromise between
achieving ergodocity (t → ∞) and constraints imposed by the availability of computational resources.
2.3. Monte–Carlo Transport Simulation
Figure 2.3: The first Brillouin zone for Silicon
17
2. Underlying Basics
Energy [eV]
18
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-0.5
[111] <- [000] -> [100]
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
kx [2π/a0]
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 2.4: Silicon band structure (empirical pseudo–potential method) along Λ and ∆.
Chapter 3
Impact Ionization
This chapter provides an overview of impact ionization. After a phenomenological description
in section 3.1, different impact ionization rates (Random-k and momentum conserving) will be
presented in section 3.2. Furthermore, the impact ionization threshold will be determined in
chapter 3.3. Finally, the experimentally accessible impact ionization coefficient will be introduced in section 3.4.
3.1
Phenomenological Description
In high field transport situations in semiconductors, a charge carrier can gain enough energy
to generate another electron–hole pair. This scattering process is called impact ionization and
can be divided into electron and hole initiated impact ionization. In an electron initiated impact ionization event, an electron with an energy greater than the impact ionization threshold
can make the transition to a secondary electron and a secondary electron–hole pair. Analogously, the process can be initiated by holes. The corresponding inverse process is called Auger
recombination.
Since a charge carrier loses at least the gap energy Eg in an impact ionization event, this
process cools down the distribution and is therefore important for determining the tail of the latter. Moreover, there are practical reasons for the wide interest in this subject: On the one hand,
impact ionization limits device operation under high fields, on the other hand, the generation of
another electron–hole pair can be exploited in the design of avalanche photodiodes.
19
20
3. Impact Ionization
3.2
Impact Ionization Rate
We will only consider electron initiated impact ionization here; the ideas are valid for hole initiated impact ionization in an analogous manner with the obvious substitutions. From a quantum
mechanical point of view, the two post–collision electrons are indistinguishable. Therefore, we
only have a one–particle pre–scattering (primary) state and a three–particle post–scattering (secondary) state without the possibility to identify one of the secondary electrons with the primary
one. As a consequence, the transition rate per unit time
S({c0 , kc0 }, {c00 , kc00 }, {v0 , kv0 }|{c, kc })
(3.1)
(c and v denote conduction band and valence band indices respectively, while kc and kv represent the corresponding crystal momenta) has to be summed over different post–collision states
only (c00 ≥ c0 ) in order to obtain the scattering rate
∑00 ∑0 ∑ ∑ ∑ S({c0, kc0 }, {c00, kc00 }, {v0, kv0 }|{c, kc}).
S({v, kv }) = ∑
c0
3.2.1
c
c00 ≥c0
(3.2)
v kc0 kc00 kv0
Quantum Mechanical Derivation
In order to derive an expression for the scattering rate occurring in equation (3.2), we follow
[Bufler97]. A two–band Hamiltonian with contributions necessary to include impact ionization
can be found there. The term leading to the electron initiated impact ionization rate reads
WII,c = III
†
cσi + III∗ ∑ ∑ c†σi d−σi c−σi cσi .
∑ ∑ c†σi c†−σi d−σi
i
σ
i
σ
(3.3)
c†σi and cσi refer to creation and annihilation operators of electrons at site Ri with spin label σ
†
respectively, while dσi
and dσi refer to hole creation and annihilation operators in an analogous
manner.
Note that in order to compute III , overlap integrals have to be calculated in a numerically
complex way, e.g. using pseudo wave functions obtained from the empirical pseudo–potential
method. Instead, it is taken to be constant here; this assumption is supported by the successful
description of impact ionization using a constant matrix element (e.g. [Bude95]). It can later be
used as an adjustable parameter to fit simulation results to experimental findings.
3.2. Impact Ionization Rate
21
The Heisenberg equation of motion for an operator AH (t) in the Heisenberg picture reads
ih̄
d
AH (t) = [AH (t), HH (t)],
dt
(3.4)
where HH (t) is the Hamiltonian.
Applying the Heisenberg equation of motion (3.4) to the diagonal elements of the density
matrix and contribution (3.3) of the Hamiltonian
i h̄
d †
c ck = [c†k ck ,WII,c ]
dt k
(3.5)
and taking the expectation value yields four–point operators. We apply the Heisenberg equation
of motion again to these terms and arrive at a differential equation of the form
d
y(t) = −i Ω y(t) + Γ(t).
dt
(3.6)
Γ(t) now consists of four–point and six–point operators again. If subsequent Heisenberg
equations were actually set up for each of these terms, it would result in an infinite hierarchy
of equations. In order to obtain a solution, the procedure can be stopped at this level and
some approximation has to be introduced. A time–dependent Hartree decoupling scheme can
be applied to approximate the four–point and six–point operators by products of two–point
operators.
Equation (3.6) now can be solved by
Zt
y(t) =
Γ(t − t 0 + t0 ) e−i Ω (t −t0 ) dt 0 .
0
(3.7)
t0
Assuming that Γ(t) in equation (3.7) is slowly varying in time compared to the occurring
exponential factor, a suitable approximation (the so–called Markov approximation) is
y(t) ≈ Γ(t)
Zt
e−i Ω (t −t0 ) dt 0
0
(3.8)
t0
= π δ− (Ω) Γ(t),
(3.9)
which makes the expression local in time. In the limit, we obtain for δ− (Ω)
lim δ− (Ω) = δ(Ω) −
t0 →−∞
with the principal value P.
i P
πΩ
(3.10)
22
3. Impact Ionization
Using the approximation (3.8) for equation (3.7) and collecting all terms, we observe that
the odd part of δ− (Ω) in the limit (3.10) cancels.
Finally, dropping the spin labels (as distribution functions with opposite spin signs are equal
in the system considered) and neglecting degeneracy, we arrive at the following outscattering
term
∂ fc
∂t
II,c
=∑∑∑
k0c k00c k0v
2π
|III |2 δ(Ec (k00c ) + Ec (k0c ) + Ev (k0v ) + Eg − Ec (kc )) δk00c +k0c +k0v ,kc .
h̄
(3.11)
Detailed calculation also yields an inscattering term as well as a generation term, which are
not given here for brevity (please consult [Bufler97]). Note that we are still within a two–band
model. However, the generalization to multiple bands is straightforward and will be carried out
in the following sections.
3.2.2
Random-k
The idea of the Random-k method is to choose the post–collision momenta randomly neglecting momentum conservation. This way we arrive at a rate which is only energy–dependent.
The method was introduced in [Kane67] and found to be in reasonable agreement with the
momentum–conserving impact ionization rate, especially far away from the threshold energy.
Close to the threshold energy, imposing momentum conservation turns out to be a more severe
restriction, which will be discussed later in section 3.3.
rk will serve as an upper bound for the momentum–
It is clear that the Random-k rate SII
conserving impact ionization rate since we omit one restriction. As discussed in section 3.2.1,
we will again assume a constant matrix element M̃ here.
The Kronecker symbol is being replaced according to
δ (kv , kv0 + kv00 + kc0 ) →
1
,
N
(3.12)
where N is the number of discrete states (which equals the number of unit cells considered).
The impact ionization rate then amounts to
rk
SII
(v, kv ) =
2π 2
M̃
h̄
1
∑ ∑ ∑ ∑ N δ(Ev(kv), Ev0 (kv0 ) + Ev00 (kv00 ) + Ec0 (kc0 ) + Eg)
v0
v00 ≥v0
c0
kv0 kv00 kc0
(3.13)
3.2. Impact Ionization Rate
23
Introducing two integrals over delta functions evaluating to unity, we get
rk
SII
(v, kv ) =
2π 2
M̃
h̄
∑
v0
v00 ≥v0
c0
1
∑∑∑ N
k 0 k 00 k 0
v
v
c
Z∞
dEv0 0
Z∞
dEc0 0 δ(Ev (kv ), Ev00 (kv00 ) + Ev0 0 + Ec0 0 + Eg ) · · ·
0
0
· · · δ(Ev0 0 − Ev0 (kv0 )) δ(Ec0 0 − Ec0 (kc0 )) , (3.14)
where we can now identify the density of states twice so that
2π 2
··· =
M̃
h̄
∑
Ω2
∑
k 00 N
v0
v
v00 ≥v0
0
c
Z∞
dEv0 0
Z∞
dEc0 0 Zc0 (Ec0 0 )Zv0 (Ev0 0 )δ(Ev (kv ), Ev00 (kv00 ) + Ev0 0 + Ec0 0 + Eg ).
0
0
(3.15)
In equation (3.15), there is now another expression containing a density of states leading to
2π
· · · = M̃ 2
h̄
∑
v0
v00 ≥v0
c0
Ω3
N
Z∞
dEv0 0
0
Z∞
dEc0 0 Zc0 (Ec0 0 )Zv0 (Ev0 0 )Zv00 (Ev (kv ) − Ev0 0 − Ec0 0 − Eg ).
(3.16)
0
Considering that Z(E) = 0 ∀ E < 0, we are now in a position to restrict the upper limits of
integration. From the argument of the last density of states function we get
Ev (kv ) − Ev0 0 − Ec0 0 − Eg > 0 ⇒ Ec0 0 < Ev (kv ) − Ev0 0 − Eg
(3.17)
and from the argument of the first density of states function combined with condition (3.17) it
follows that
Ev0 0 < Ev (kv ) − Eg .
(3.18)
The final expression for the Random-k rate is now only energy–dependent and amounts to
rk
SII
(Ev ) = srk
0
∑
v0
v00 ≥v0
c0
Ev (k
Zv )−Eg
Ev (kv )−Ev0 0 −Eg
dEv0 0
0
Z
dEc0 0 Zc0 (Ec0 0 )Zv0 (Ev0 0 )Zv00 (Ev (kv ) − Ev0 0 − Ec0 0 − Eg )
0
(3.19)
where we have defined a constant
2
srk
0 = M̄
e4 a70
.
4 (2π)3 h̄ε20
(3.20)
24
3. Impact Ionization
In order to have a dimensionless constant M̄, we have used the following substitution1
e2 a0 2
M̃ = M̄
.
Ω ε0 2π
(3.21)
Furthermore, the relationship
a30
Ω
= Ω0 =
N
4
for the unit cell volume Ω0 has been used to arrive at (3.19).
3.2.3
(3.22)
Momentum Conserving Impact Ionization Rate
Introducing the Kronecker symbol again, the momentum conserving impact ionization rate SII
gets
SII (v, kv ) =
2π 2
M̃
h̄
∑ ∑ ∑ ∑ δkv,kv0 +kv00 +kc0 +G · · ·
v0
v00 ≥v0
c0
(3.23)
kv0 kv00 kc0
· · · δ(Ev (kv ) − Ev0 (kv0 ) − Ev00 (kv00 ) − Ec0 (kc0 ) − Eg )
=
2π 2 e4 a40
M̄ 2 2
h̄
Ω ε0 (2π)4
∑ ∑ ∑···
v0 kv00
v00 ≥v0
c0
kc0
· · · δ(Ev (kv ) − Ev0 (kv − kv00 − kc0 + G) − Ev00 (kv00 ) − Ec0 (kc0 ) − Eg )
= s0
∑
Z Z
d 3 kv00 d 3 k0c · · ·
v0
v00 ≥v0
c0
· · · δ(Ev (kv ) − Ev0 (kv − kv00 − kc0 + G) − Ev00 (kv00 ) − Ec0 (kc0 ) − Eg ),
where we have introduced the abbrevation
s0 :=
a40 M̄ 2 e4
.
(2π)9 h̄ε20
(3.24)
It is understood that G is a reciprocal lattice vector over which has to be summed. The sums
over k vectors have been converted to integrals using
Ω
lim ∑ f (k) =
(2π)3
Ω0 /Ω→0 k
1 Since M̄
Z
d 3 k f (k)
(3.25)
is not known a priori (see also section 3.2.1), it will be set to 1 in the algorithm. The impact ionization
rate returned by our code is therefore already given in s−1 , but has to be multiplied by M̄ 2 . Its value can be
obtained by means of a least square fit of the impact ionization coefficient α (cf. section 5.1). Moreover, we
note that M̃ 2 is real–valued as it is an abbreviation for a combination of the direct and exchange matrix elements
M̃ 2 := |M̃d − M̃e |2 + |M̃d |2 + |M̃e |2 .
3.3. Determining the Threshold of Impact Ionization
25
(see for example [Reigrotzki98]). Ω and Ω0 denote the volume of the crystal and unit cell
respectively.
The energy averaged rate then amounts to2
∑ d 3 kv δ(E − Ev (kv ))SII (v, kv )
R
R(E) = v
,
∑v d 3 kv δ(E − Ev (kv ))
R
(3.26)
which finally is the integral we will have to compute. The results are presented in chapters 5
and 6.
3.3
Determining the Threshold of Impact Ionization
A simple rule for determining the threshold energy of impact ionization was derived by
[Capasso85] under the assumption of parabolic conduction and valence bands:
Eth =
3
Eg ,
2
(3.27)
where Eg is the band gap. However, especially for wide–band–gap materials, this rule is not
applicable. A more precise calculation procedure will be presented in this section.
A way of obtaining the threshold of impact ionization using only energy conservation, k
conservation and full band structure information has been proposed in [Bude92]. We carry out
the straightforward calculation, which has (to our knowledge) not been done yet for Silicon
without any further restrictions. The threshold energy Eth has to satisfy
Eth =
2 Note
min
kv ,kv0 ,kv00 ,kc0
[Ev (kv )]
(3.28)
that formula (3.26) is equivalent to performing an energy averaging for each band separately and then
weighting the resulting function Rv (E) with the relative density of states:
Zv (E)
Zv (E) d 3 k δ(E − Ev (k)) SII (v, k)
∑ Zv (E)Rv (E)
∑ Z(E) Rv (E) = v∑v0 Zv0 (E) = ∑
(2π)3 ∑v0 Zv0 (E) Zv (E)
v
v
R
R(E) =
∑v d 3 k δ(E − Ev (k)) SII (v, k)
(2π)3 ∑v0 Zv0 (E)
R
=
26
3. Impact Ionization
subject to
Ev (kv ) = Ev0 (kv0 ) + Ev00 (kv00 ) + Ec0 (kc0 ) + Eg
(3.29)
kv = kv0 + kv00 + kc0 + G,
(3.30)
Eth =
(3.31)
which is equivalent to
min [Ev (kv0 + kv00 + kc0 + G)]
kv0 ,kv00 ,kc0
subject to
Ev (kv0 + kv00 + kc0 + G) = Ev0 (kv0 ) + Ev00 (kv00 ) + Ec0 (kc0 ) + Eg .
(3.32)
In order to implement condition (3.32) easily, we minimize the following function
f (kv0 , kv00 , kc0 ) = · · ·


M
kv0 6∈ BZ ∨ kv00 6∈ BZ ∨ kc0 6∈ BZ
(3.33)
···

Ev + |Ev − Ev0 − Ev00 − Ec0 − Eg | elsewhere
instead of (3.31) so that both the energy of the primary charge carrier and the energy conservation error are minimized. (Note that in (3.33) all energy functions have the same arguments as in
(3.32), which have been omitted for better readability.) To ensure that the problem is bounded,
the function returns an artificially high number M whenever any of the three free k variables
lies outside the Brillouin zone defined by (2.8).
After the minimization routine has converged, the energy conservation error obviously has
to be subtracted from the result to get the threshold energy. It turns out that this method works
well for Silicon and energy is conserved to within 10−4 eV.
In order to carry out the minimization, the Downhill Simplex Method by Nelder and Mead
[Nelder65] does not need any derivatives (which would even be discontinuous in the case of
equation (3.33)) and is therefore suitable for the problem. Given an initial simplex (N+1 points
in N dimensions), the algorithm executes a series of expansion, contraction and reflection steps,
mostly reflecting the point where the function value is largest to the opposite face, thus eventually reaching the lowest point. An overview of this minimization approach can also be found in
[Press92].
Special attention has to be paid not to end up with a local minimum. To avoid this and
to cover the nine–dimensional space fairly well, 500000 independent runs are performed for
3.3. Determining the Threshold of Impact Ionization
27
each band combination and the smallest value where the density of states is nonzero is being
returned. Trying to increase the number of runs did not change the results any more. A list of
processes and their respective threshold energies for all allowed band combinations (v, v’, v”,
c’) can be found in appendix C.
The threshold values found in this work can be obtained from table 3.1. Note that they agree
well with the values obtained by an ab initio method used in [Kuligk05] (where the lowest nonvanishing contribution was taken).
Electron threshold [eV]
Hole threshold [eV]
This work
1.140
1.367
Values from [Kuligk05]
1.196
1.305
Table 3.1: Threshold energies for impact ionization in Si
The physical reason why the hole threshold energy lies farther away from the band gap
energy than the electron threshold can be realized by looking at a band structure plot (cf. figure
2.4). In order for the threshold energy to be approximately equal to the band gap energy, the
following would have to hold:
In the case of hole initiated impact ionization, the two secondary holes have to be located at
the center of the Γ valley and the secondary electron at an X valley in order to have the lowest
energy of secondary particles possible. Due to momentum conservation, the primary hole then
must have had the same k vector as the secondary electron. However, energy conservation
would require it to have an energy equal to the band gap energy. Those two conditions can
never be fulfilled at the same time using a realistic band structure. Therefore, no hole initiated
impact ionization processes are possible close to the band gap energy.
On the other hand, the conditions are easier to satisfy for electron initiated impact ionization.
Again, the secondary hole must be situated at the center of the Γ valley. However, the two
secondary electrons can now be situated at two different X valleys, e.g. at X+X and X+Z . In that
case, there is now the possibility to add an umklapp vector in such a way that both momentum
and energy are conserved. These processes can be found in appendix C, where all involved
bands, valleys and umklapp vectors are listed with their respective threshold energies.
28
3.4
3. Impact Ionization
Impact Ionization Coefficient
The impact ionization coefficient is an experimentally accessible quantity which will later be
used to compare our impact ionization rate calculations to measurement data. First, we state the
definition of the generation rate
φ
= < SII (v, kv ) >
n
Z
1
Ω0
=
SII (v, kv ) fv (kv ) d 3 kv ,
∑
3
n v (2π)
(3.34)
(3.35)
Ω0
which represents the mean number of generated pairs per unit time; n here denotes the primary
charge carrier density. Then, the impact ionization coefficient can be defined as
α=
φ
.
n vd
(3.36)
By dividing generation rate by drift velocity, we obtain the mean number of electron–hole
pairs generated per unit length. One way to measure the impact ionization coefficient uses the
substrate current of a field effect transistor. If the electrons gain enough energy due to the drain–
source voltage, the pairs generated through impact ionization can be counted by measuring the
substrate hole current.
Chapter 4
Numerical Considerations
The aim of this chapter is to develop a method to integrate an integral of type (3.26) both
efficiently and accurately. First, possible approximations of the delta distribution occurring in
this integral are examined in section 4.1. Second, different integration methods are tried in order
h
i
to find a suitable one for this kind of problem. In the given simulator, the |k| interval 0, 2π
a0 is
divided into 96 equidistant points. Taking all of these points into consideration would require a
number of evaluations on the order of 1015 , which already strongly suggests looking for a more
efficient way, which will be done in section 4.2.
4.1
Choosing the Delta Distribution Approximation
Special care has to be taken how to approximate the delta distribution. Throughout the existing literature, many different approaches can be found, e.g. a simple box as in equation (4.3)
of width 0.1 eV is chosen in [Reigrotzki98], whereas [Sano92] and [Kane67] both use an energy interval of width 0.2 eV. Since these choices appear somewhat arbitrary ([Sano92] even
explicitly states that in annotation 29), we are looking for well–founded criteria which support
our choice. In order to estimate the accuracy of possible delta distribution approximations and
to choose the best approximation for our purpose, we consider some special cases. First, the
density of states is given by
Z(ε) =
1
(2π)3
Z
d 3 k δ(ε − E(k)),
29
(4.1)
30
4. Numerical Considerations
which can be written as
m3/2
Z(ε) = √
2 π2 h̄3
if E(k) =
Z
√
du δ(u) ε − u
(4.2)
h̄2 k2
2m .
When implementing the substituted integral (4.2) using a box approximation of the form
δ(x) =


1
|x| ≤ δx /2

0
elsewhere,
δx
(4.3)
high accuracy can easily be achieved, e.g. 10−5 with δx = 0.1 and 100 supporting points; also
note that the accuracy dependence on δx is only moderate.
The approximation is getting significantly worse when implementing formula (4.1), again
with a parabolic band structure, but without any integral substitution. It can be improved by introducing a modified Lorentz profile (which was also used in [Reigrotzki98] for the calculation
of the density of states) instead of a simple box:
δ(x) =



η
1 2 +η2
δ
|x|
2 arctan 2η

0
|x| ≤ δx /2
(4.4)
elsewhere,
There is a strong dependence on the width η and cutoff parameter δ. For example, the optimum accuracy for 100 supporting points (per dimension) is 3% when choosing δ = 0.29, η =
0.07, whereas the optimum for 20 supporting points is at δ = 1.44, η = 0.31 and yields an
accurate result to within 4%.
As a result, one should try a similar substitution (4.1 → 4.2) for the impact ionization rate
(3.23). However, this is impossible as the mapping from k values to a linear combination of
band structure energies (see expression (3.23)) is not a function which can easily be split into
bijective parts.
As a next step, (4.1) will be integrated with E(k) obtained from the empirical pseudopotential method. When trying box approximations and modified Lorentz profiles of variable width,
the sum of squared errors then reaches its minimum with a modified Lorentz profile at
4.2. Choosing the Integration Method
31
δ = 0.25 eV
(4.5)
η = 0.15 eV
using a Monte–Carlo integration method (see next paragraph).
4.2
Choosing the Integration Method
5e+27
4.5e+27
density of states [eV-1 m-3]
4e+27
3.5e+27
3e+27
2.5e+27
2e+27
1.5e+27
1e+27
look-up table
48 points
24 points
monte carlo, 1E4 evaluations
5e+26
0
0
0.05
0.1
0.15
0.2
0.25
energy [eV]
0.3
0.35
0.4
0.45
0.5
Figure 4.1: Density of states for the first conduction band in bulk Si calculated with different
integration routines
The last three–dimensional example can already serve to demonstrate the inappropriateness
of classical integration methods with equally spaced abscissae: Figure 4.1 shows a comparison
between the look–up table (solid line), an equally spaced abscissae approach with 24 points per
dimension (dashed line) and 48 points per dimension (dotted line) and a Monte–Carlo approach
with 10000 evaluations (dashed–dotted line). We note that the Monte–Carlo method already
yields almost equally accurate results with only 10000 evaluations as the 48 point method does
with 59725 (one octant of the Brillouin zone).
32
4. Numerical Considerations
Trying to refine the mesh inside the X and Γ valleys respectively (e.g. halve the step size
in these regions) to cure the inaccuracy of the equally spaced abscissae approach only fixes the
problem for a small range of low energies, where the valleys are populated. The advantage
of the Monte–Carlo method even increases with higher dimensionality and makes it a feasible
method for the nine–dimensional impact ionization rate integral (3.26). A similar accuracy can
never be achieved within reasonable computing times when using equidistant supporting points.
Next, the parameter set (4.5) has been used to reproduce the Random-k approximation
(3.19), this time performing the nine–dimensional integral (3.13). Figure 4.2 shows the accuracy of this integration method: our results match the Random-k curve well. It suggests that
both our choice of the delta distribution approximation (modified Lorentz profile with parameter set (4.5)) and integration method (Monte–Carlo) are optimized. Some comparisons are
plotted in figure 4.1, two for checking our integration method and one for checking our delta
distribution approximation.
First, we compare our results to an equidistant integration point method using 16 points (per
dimension) and such a method with 10 points outside the X valleys and a refinement by a factor
of 8 inside the valleys. These two approaches turn out to overestimate the results; the significant
dependence on step sizes already suggests their inappropriateness.
Second, we implement a Monte–Carlo integration with a box of width 0.1 eV as in
[Reigrotzki98] serving as a delta approximation and compare the results to our optimized parameter choice (4.5). Here we get an underestimation (e.g. of up to 80% for holes at 1.2 eV)
compared to our results.
Note that we have carried out all comparisons with respect to the density of states or the
Random-k approximation as these are cases where the result is known (at least to within high
accuracy) — which is not true for the momentum conserving impact ionization rate (3.23).
As a result, we will choose the parameter set (4.5) and a Monte–Carlo integration algorithm
for further calculations.
4.2. Choosing the Integration Method
33
1E+17
1E+16
1E+15
impact ionization rate [s-1]
1E+14
1E+13
1E+12
1E+11
1E+10
holes, random k
electrons, random k
holes, random k, MC
electrons, random k, MC
holes, random k, MC, box approx.
holes, 16 equidistant points
holes, valley refined
1E+09
1E+08
1E+07
1
1.5
2
2.5
3
3.5
energy [eV]
4
4.5
5
5.5
Figure 4.2: Random-k approximation
4.2.1
Monte–Carlo Integration
4.2.1.1
Crude Monte–Carlo
This section basically follows [Hammersley75]. Consider the following integral
Θ=
Z 1
f (x)dx
(4.6)
0
as an easy one–dimensional example. Letting ξ1 , ..., ξn be independent random numbers, then
fi = f (ξi )
(4.7)
can be used to calculate an estimator f¯ of the expectation Θ using
1 n
f¯ = ∑ fi .
n i=1
(4.8)
Its variance amounts to
σ =
2
Z 1
0
( f (x) − Θ)2 dx.
(4.9)
34
4. Numerical Considerations
f¯ is then called the Crude Monte–Carlo estimator with standard error
σ
σ f¯ = √ .
n
4.2.1.2
4.2.1.2.1
(4.10)
Variance Reduction Techniques
Importance Sampling For any function g(x) which has no zeros other than those
of f (x), we define
Z x
G(x) =
g(y)dy.
(4.11)
0
It follows that
θ=
Z 1
f (x)dx =
Z 1
f (x)
0
0
g(x)
g(x)dx =
Z 1
f (x)
0
g(x)
dG(x).
(4.12)
Restricting our choice to a positive function g(x) satisfying
Z 1
G(1) =
g(y)dy = 1,
(4.13)
0
G(x) represents a distribution function. Equation (4.12) now tells us that
f (η)
g(η)
also has expecta-
tion Θ provided that η is sampled from the distribution G(x). Its variance now amounts to
σf =
2
g
Z 1
f (x)
0
g(x)
−Θ
2
dG(x) .
(4.14)
The idea behind this is to be able to choose g(x) in such a way that it reduces the error.
When choosing g(x) =
f (x)
Θ
, the error would vanish, as can easily be seen from formula (4.14).
Since Θ is not known, one can only try to make g(x) as similar to f (x) as possible. This way,
the ratio
f (x)
g(x)
will vary little and lead to a decreased standard error. Because we have to keep in
mind to satisfy equation (4.13), g(x) still has to be easy to integrate. As a conclusion, g(x) has to
be a trade–off between a function that significantly reduces the standard error by approximating
f (x) and a function which can easily be integrated.
4.2.1.2.2
Stratified Sampling
Another error reducing strategy is to divide the integration
domain into several pieces [α j , α j+1 [ with 0 = α0 < α1 < ... < αk = 1. We choose to sample n j
points from the j−th piece, so the estimator for Θ is of the form
k
t=
nj
1
∑ ∑ (α j − α j−1) n j f (α j−1 + (α j − α j−1)ξi, j ).
j=1 i=1
(4.15)
4.2. Choosing the Integration Method
35
Its variance amounts to (cf. [Hammersley75], [Veach97])
σt2
α j − α j−1
=∑
nj
j=1
k
Z αj
α j−1
k
1
f (x) dx − ∑
j=1 n j
2
Z
αj
α j−1
2
f (x) dx
.
(4.16)
This expression can be less than the crude Monte–Carlo variance if the stratification is carried out in such a way that the differences between the mean values of f (x) in the various pieces
are greater than the variations of f (x) within the pieces.
4.2.1.3
Divonne Algorithm
The Divonne algorithm was devised by T. Hahn mainly for multidimensional integrals occurring
in elementary particle physics and is described in [Hahn05]. It mainly uses stratified sampling
for its variance reduction by dividing the integration domain into subregions r in which the
spread s(r), defined as
s(r) =
1
max f (x) − min f (x) ,
x∈r
2 x∈r
(4.17)
will be approximately equal.
Divonne works in three phases: in phase 1 (partitioning phase), minimum and maximum
values are obtained by means of numerical optimization. Thus, strata can be chosen in such a
way that s(r) stays as constant as possible for different r. In phase 2 (final integration phase),
subregions are sampled and a χ2 test determines whether they are within their error bounds.
Failing subregions are passed to phase 3 (refinement phase) and are subdivided again.
36
4. Numerical Considerations
4.2.2
Applying the Divonne Algorithm to the Problem
The Divonne algorithm proved to be reliable and fast enough to approximate one integral of
type (3.26) within an accuracy of 5% in about 8 hours on a 3 GHz CPU. Note that this is still a
nine–dimensional integral as three dimensions are necessary for the averaging and SII already
is obtained through a six–dimensional integral. This section provides information on how the
algorithm was embedded to fit our needs.
First, we have to think how the function we integrate exactly looks. Since the band structure
is available only at discrete points, there has to be some interpolation method which will be
described in section 4.2.2.1.
Second, we want to provide the integration procedure with as much information as possible
beforehand to speed up integration. This will be described in detail in section 4.2.2.2.
Combining all this, we will describe the overall procedure to integrate the impact ionization
rate in section 4.2.2.3.
4.2.2.1
Interpolation Method
k(8)
k(5)
k(7)
k(6)
k(4)
k(3)
z
y
6
k(1)
k(2)
-x
Figure 4.3: Unit cube for interpolation
The |k| range
h
0, 2π
a0
i
is split into 96 equally spaced discrete points for which the band
structure is tabulated in the given simulator. When using Monte Carlo integration, energy values
4.2. Choosing the Integration Method
37
in between have to be obtained by some interpolation. Letting E(k) be the values stored in the
look–up table, we obtain the interpolated values Ẽ(k) using
8
Ẽ(k) =
∑ V ( j)(k)E(k( j))
(4.18)
j=1
with k( j) being the nearest discrete points (see figure 4.3). For the three components kx , ky , kz of
k, the coefficients V ( j) (k) are given by:
(7)
(7)
(7)
(8)
(8)
(8)
(5)
(5)
(5)
(6)
(6)
(6)
(3)
(3)
(3)
(4)
(4)
(4)
(1)
(1)
(1)
(2)
(2)
(2)
V (1) (k) = (kx − kx )(ky − ky )(kz − kz )
(4.19)
V (2) (k) = (kx − kx )(ky − ky )(kz − kz )
V (3) (k) = (kx − kx )(ky − ky )(kz − kz )
V (4) (k) = (kx − kx )(ky − ky )(kz − kz )
V (5) (k) = (kx − kx )(ky − ky )(kz − kz )
V (6) (k) = (kx − kx )(ky − ky )(kz − kz )
V (7) (k) = (kx − kx )(ky − ky )(kz − kz )
V (8) (k) = (kx − kx )(ky − ky )(kz − kz )
4.2.2.2
The Peak Finder Routine
In order to provide the integration routine with some information on contributions to the integral, a peak finder routine has been implemented which is being run before each integration.
The assumption is that it is easier to find the minimum of the modulus of the argument of a
delta distribution than to search for the delta peak itself because more information is available
in regions away from the peak. We therefore spend a significant amount of computational effort
to find these minima first.
In a similar way to the threshold determination in section 3.3, we have to minimize
f (kv , kv00 , kc0 ) = · · ·


M
kv 6∈ BZ ∨ kv00 6∈ BZ ∨ kc0 6∈ BZ
···
(4.20)

|E − Ev | + |Ev − Ev0 − Ev00 − Ec0 − Eg | elsewhere
with respect to kv , kv00 , kc0 . Again, kv0 can be obtained by means of momentum conservation and
the arguments of Ev , Ev0 , Ev00 , Ec0 are the same as in equation (3.32), while E is now the primary
38
4. Numerical Considerations
charge carrier energy for which we are trying to find contributions to the impact ionization rate.
M is a high number telling the algorithm that it is far from the minimum and BZ again denotes
the Brillouin zone as defined in equation (2.8).
For each allowed band combination (i.e. E ≥ min Ev0 + min Ev00 + min Ec0 + Eg ), 500000
independent minimization runs are performed and the lowest minimum is being returned. Since
the minimization procedure returns a value which consists of the sum of two arguments of delta
distributions, we compare this returned value to the implemented width of our delta distribution
approximation. If it is greater than twice the implemented width, we reject the point as it cannot
give any contribution to the integral. Otherwise, it may contribute something and we therefore
add it to the list of starting points for the integration.
4.2.2.3
Complete Integration Approach
After determining the threshold energy using the procedure described in section 3.3 (step 1),
our algorithm starts at this energy. There it searches for peaks according to section 4.2.2.2 (step
2) and then starts a Monte–Carlo Integration of formula (3.26) using the Divonne algorithm as
presented in section 4.2.1 (step 3). It may seem redundant to generate a list of peaks obtained
from numerical optimization and pass them to the Divonne algorithm because the latter one
starts with an optimization phase anyway (see there). However, the peak finder routine uses
more information to locate the peaks than the integrand itself can provide (see 4.2.2.2) so Divonne can exploit it in its partitioning phase. Steps 2 and 3 are repeated at each desired energy
depending on the resolution one wants to obtain. A flowchart of the algorithm is given in figure
4.4.
4.2. Choosing the Integration Method
39
Find Threshold Eth
E := Eth
Find Peaks at E
Integrate S(E)
E := E + DE
E>Emax?
no
yes
Termination
Figure 4.4: Algorithm Flowchart
40
4. Numerical Considerations
Chapter 5
Results for Silicon
In appendix C, a list of possible processes is given which has been compiled during the integrations. Whereas the list of hole initiated processes is complete, the list for electrons had
to be cut for brevity. We chose to print the list sorted by bands, valleys and umklapp vectors
up to the processes with lowest overall threshold energies (bold face). Note that the threshold
processes for holes are normal processes whereas the ones for electrons are umklapp processes.
The physical reason for that was already given in section 3.3.
5.1
Electrons
Figure 5.1 shows the electron initiated impact ionization rate in Silicon obtained from the previously described procedure. In order to provide the simulator with a continuous rate at each
energy, we could either tabulate the discrete values and apply some interpolation or fit them to
an analytical expression, the latter turning out to work well. Our choice of a fitting formula is
based upon the original Keldysh formula introduced by [Keldysh60]
S(E) = Θ(E − Eth ) P
E − Eth
Eth
where Θ(x) denotes the Heaviside step function, defined as


1 x ≥ 0
Θ(x) =

0 x < 0.
41
2
,
(5.1)
(5.2)
42
5. Results for Silicon
In [Cartier93], a so–called ”multi component Keldysh–type formula”
3
S(E) =
∑ Θ(E − Eth, j ) Pj
j=1
E − Eth, j
Eth, j
2
has been introduced which we now generalize even further:
n
E − Eth, j a j
.
S(E) = ∑ Θ(E − Eth, j ) Pj
Eth, j
j=1
(5.3)
(5.4)
In accordance with [Fischetti88], a three component formula (n = 3) turns out to be suitable
for our results plotted in figure 5.1. The results given in table 5.1 were obtained by means of
a least square fit. Comparing the first value to the one determined by direct minimization (see
table 3.1), we note an agreement that suggests confidence in the reliability of the fitting.
j
Eth, j [eV]
Pj s−1
aj
1
1.13
2.0 · 1012
2.981
2
1.6
2.3 · 1014
2.978
3
2.6
1.8 · 1016
2.490
Table 5.1: Fitting parameters for the electron impact ionization rate in Si
In order to find a physical reason for the placement of the three threshold energies Eth, j , we
have compiled a list of possible impact ionization processes. A process is characterized by the
bands and valleys of the primary and secondary charge carriers as well as the umklapp vector (if
applicable), so we have defined two scattering processes to be the same when all involved bands,
valleys and umklapp vectors are the same. When looking at figure 5.2, a justification of those
steps can now be given: For each 100 meV interval in the considered range, the density of states
times the number of processes whose threshold energy lie in that interval is plotted. Therefore,
it gives an impression of the number of processes which start in a given energy range. Both the
steps at 1.6 eV and 2.6 eV are situated shortly before peaks in the number of possible processes
times the density of states. A detailed list of possible processes is also given in appendix C.
Experimental values for the impact ionization coefficient have been taken from [Robbins85]
and [Takayanagi92]. The dimensionless fitting parameter M̄ 2 has been used to match simulation
results to the measured values. When plotting the logarithm of the impact ionization coefficient
5.1. Electrons
43
1e+16
-1
impact ionization rate [s ]
1e+14
1e+12
1e+10
1e+08
1e+06
electrons, random k
electrons, MC
electrons, fit
10000
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
energy [eV]
Figure 5.1: Electron Impact Ionization Rate in Silicon
as a function of the inverse field, a straight line with negative slope is obtained. Figures 5.3
and 5.4 show a comparison between measured values and simulation results. Table 5.2 lists the
fitting result for M̄ 2 and compares it to a value obtained in [Bartels97]. We notice a considerable difference that can be explained by the approximation made in [Bartels97] by assuming
parabolic valleys for post–collision states, which have been treated in a full–band approach in
this work.
M̄ 2
This Work
0.14
Value from [Bartels97] 0.02
Table 5.2: Matrix element, electron initiated impact ionization
44
5. Results for Silicon
3.5e+30
number of processes times density of states
3e+30
2.5e+30
2e+30
1.5e+30
1e+30
5e+29
0
.
[2
.
[2
.
[2
.
[2
.
[2
.
[2
.
[2
.
[2
.
[2
.
[2
.
[1
.
[1
.
[1
.
[1
.
[1
.
[1
.
[1
[
[
.0
.3
9.
[
.9
.2
8.
[
.8
.2
7.
[
.7
.2
6.
[
.6
.2
5.
[
.5
.2
4.
[
.4
.2
3.
[
.3
.2
2.
[
.2
.2
1.
[
.1
.2
0.
[
.0
.2
9.
[
.9
.1
8.
[
.8
.1
7.
[
.7
.1
6.
[
.6
.1
5.
[
.5
.1
4.
[
.4
.3
.1
3.
.1
2.
2
1.
.
[1
<
Figure 5.2: Number of processes times density of states in 100 meV intervals
100000
FB MC, electrons, <100>
FB MC, electrons, <111>
Robbins, electrons, <100>
Robbins, electrons, <111>
Takayanagi, electrons, <100>
impact ionization coefficient [cm-1]
10000
1000
100
10
1
0.1
0.01
2
4
6
8
10
12
14
inverse field [10-6 cmV-1]
Figure 5.3: Impact ionization coefficient for electrons in Silicon at 300K
5.1. Electrons
45
100000
10000
-1
impact ionization coefficient [cm ]
FB MC, electrons, <100>
FB MC, electrons, <111>
Robbins, electrons, <100>
Robbins, electrons, <111>
Takayanagi, electrons, <100>
1000
100
10
2
3
4
5
6
7
inverse field [10-6 cmV-1]
Figure 5.4: Impact ionization coefficient for electrons in Silicon at 300K, zoomed
46
5. Results for Silicon
5.2
5.2.1
Holes
Impact Ionization Rate Integration Results
Figure 5.5 shows our results for the hole initiated impact ionization rate. Here, a one component
fitting formula of type (5.4) turns out to be sufficient. The values obtained from our least square
fit are given in table 5.3.
1.00E+17
1.00E+16
1.00E+15
impact ionization rate [s-1]
1.00E+14
1.00E+13
1.00E+12
1.00E+11
1.00E+10
1.00E+09
1.00E+08
holes, random k
holes, MC
holes, fit
1.00E+07
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
energy [eV]
Figure 5.5: Hole Impact Ionization Rate in Silicon
Comparing our results to impact ionization coefficient measurements, the simulation of
holes turns out to be more problematic than electrons. In order to match our findings to established measurements like [Robbins85], the matrix element has to be raised to M̄ 2 = 660,
which is far away from physically meaningful values.
j
Eth, j [eV]
Pj s−1
aj
1
1.33
6.58 · 1013
4.172
Table 5.3: Fitting parameters for the hole impact ionization rate in Si
5.2. Holes
47
In order to try to get closer to a more realistic matrix element, we change our model and
explore its limits. Since the distribution function plays a critical role in the number of holes
initiating an impact ionization event, we will focus on different approaches to shift its tail to
higher energies. One way will be to add a further acoustic phonon branch in section 5.2.2.3,
which is intended to enable us to lower the acoustic coupling constant, therefore pushing the tail
of the distribution to higher energies. Furthermore, we will make our averaged phonon energies
hole energy dependent (i.e. field dependent) and observe the outcome in section 5.2.2.4. Every
changed phonon model then has to be checked for consistency by comparing drift velocities as
a function of the applied field at different temperatures to experimental findings. We therefore
also need the temperature dependence of the band gap, which will be given in section 5.2.2.1.
5.2.2
Modifications to the Phonon System
Before describing our changes in the upcoming sections 5.2.2.3 and 5.2.2.4, we first outline the
temperature dependence of the band gap and the currently implemented phonon model.
5.2.2.1
Temperature Dependence of the Band Gap
Since a band gap change also affects the impact ionization rate by shifting the whole curve
along the energy axis, it is important to reproduce the temperature dependence of the band gap
in our model. The band gap exhibits only small temperature dependence at low temperatures
while approaching a linear asymptote above Debye temperature. As derived in [Paessler00], it
can be written in terms of an integral of the form
α
Eg (T ) = E(0) −
kB
Z
dε
w(ε) ε
,
exp ( kBεT ) − 1
(5.5)
where α denotes the limit of the slope of Eg (T ) and w(ε) is a normalized weighting function:
α =
lim
T →∞
dE(T )
−
dT
(5.6)
Z
dε w(ε) = 1.
(5.7)
48
5. Results for Silicon
We follow [Paessler01] in choosing the weighting function as a finite sequence of discrete peaks
at certain phonon energies εi = h̄ωi = kb Θi . Inserting this ansatz
N
w(ε) =
∑ Wi δ(ε − εi),
(5.8)
i=1
N
∑ Wi
= 1
(5.9)
i=1
into equation (5.5), we immediately get
N
Eg (T ) = Eg (0) − α ∑
i=1
Wi Θi
.
exp (Θi /T ) − 1
(5.10)
For Silicon, numerical fittings to a two parameter model yield the parameter set given in table
5.4.
Eg (0)
α
Θ1
Θ2
[eV]
−4
10 eV K−1
[K]
[K]
1.17
3.21
160 596
W1
W2
0.36
0.64
Table 5.4: Parameters describing the temperature dependence of the Si band gap
5.2.2.2
Current Phonon Model
In order to simplify calculations, the optical phonon branch is approximated by a constant
abs/em
phonon energy h̄ω0 . Using a constant matrix element, the transition probability Sopt
(k0 |k),
where k and k0 denote the momenta before and after the scattering event respectively, amounts
to (cf. [Bufler02])
abs/em
Sopt (k0 |k) =
π
1 1
2
Dt K nopt + ∓
δ(E(k0 ) − E(k) ∓ h̄ ω0 ).
ρV ω0
2 2
(5.11)
ρ denotes the mass density and V stands for the crystal volume. After carrying out the summation over all post–collision momenta, we arrive at the scattering rate
S(k) =
∑
0
k ∈VBZ
S(k0 |k).
(5.12)
5.2. Holes
49
Equation (5.11) then yields
abs/em
Sopt (k)
=
=
1 1 1
π
2
Dt K nopt + ∓
δ(E(k0 ) − E(k) ∓ h̄ ω0 )
∑
ρ ω0
2 2 V k0 ∈V
BZ
π
1 1
Dt K 2 nopt + ∓
Z(E(k) ± h̄ω0 ),
ρ ω0
2 2
(5.13)
(5.14)
i.e. the scattering rate is primarily dependent on the density of states.
In a simple model, the acoustic phonon dispersion is linearized according to ω(q) = uq,
with u representing the velocity of sound. Moreover, the Bose–Einstein distribution is approximated as
h̄ ω
kB T
−1
1
kB T
− ,
(5.15)
h̄ ω
2
which is justified as long as h̄ ω kB T . Finally, the energy associated with the phonon can
n(q) =
e
−1
≈
be neglected in the delta function. This elastic approximation enables us to factor out the delta
function and add the absorption and emission terms, arriving at
Sac (k0 |k) =
π
2 kB T
Ξ2
δ(E(k0 ) − E(k)).
ρV u
h̄ u
(5.16)
This leads us to the acoustic phonon scattering rate
Sac (k) =
2 π kB T 2
Ξ Z(E(k)).
ρ h̄ u2
(5.17)
Using the elastic model, the experimental drift velocities can be reproduced to a high degree
of accuracy in both Silicon and Germanium (see e.g. [Bufler98]). Furthermore, it yields a
scattering rate that is only energy–dependent but not wave vector–dependent, which facilitates
many calculations, e.g. by turning the mobility calculation into a one–dimensional integral over
energy (cf. equation 5.23).
On the other hand, the elastic approximation underestimates the number of hot holes. The
following model has been proposed in [Bufler01] to compensate for this shortcoming and will
be used as a starting point for the modifications in sections 5.2.2.3 and 5.2.2.4.
Instead of applying the elastic approximation, both q2 and h̄ ω(q) are replaced by constants.
The modulus of the phonon wave vector, q = ||k0 − k|| is averaged over a sphere resulting in
q = 43 ||k||. Then, the phonon energy is weighted with the equilibrium distribution function
according to
R
h̄ωac =
dE h̄ω( 43 k(E)) Z(E) e−βE
R
,
dE Z(E) e−βE
(5.18)
50
5. Results for Silicon
where Z(ε) is the density of states given by (4.1) and β ≡ kB1T . The inverse of the band
p
structure is taken to be k(E) = h̄1 2m(E) E, where m(E) represents the density–of–states mass.
Analogously, the mean value of the modulus of the phonon wave vector can be obtained.
Both < q2 > and < h̄ ω(q) > then turn into temperature dependent values, which are later
plotted in figure 5.7. The acoustic coupling constant then also becomes temperature dependent
via
∆(q) = ε q.
(5.19)
Simply lowering the optical coupling constant and increasing the acoustic one leads to an
underestimation of the saturation velocity at room temperature and is therefore not a legitimate
way to achieve a hole distribution enhancement.
5.2.2.3
Addition of a Second Acoustic Phonon Branch
Figure 5.6 shows the phonon dispersion in Silicon along (100) obtained from neutron scattering
data. So far, only the LA branch has been implemented in the simulator. Now the idea is to add
the TA branch which will enable us to lower both the optical and acoustic coupling constants,
which will eventually lead to a distribution enhancement for holes. The averaging procedure
described in section 5.2.2.2 has to be performed for both acoustic branches yielding the values
given in table 5.5. Figure 5.7 also shows the phonon energy and mean modulus of the wave
vector as a function of the lattice temperature.
Again, the hole–phonon scattering rate amounts to (compare to [Fischer00])
1 1 1
π
2
∆ (q) Nq + ∓
δ E(k0 ) − E(k) ∓ h̄ω(q) ,
S(k) = ∑
ρω(q)
2 2 Ω
k0
(5.20)
where ρ and Ω denote the mass density and crystal volume respectively. Nq is the Bose–Einstein
distribution Nq = (exp [β(ε − µ)] − 1)−1 and the overlap integral has been approximated as 1.
Converting the sum to an integral using (3.25) and assuming a constant energy h̄ω(q) = h̄ω0 ,
comparison to the definition of the density of states Z(E) given in (4.1) yields
π
1 1
2
S(k) =
Z (E(k) ± h̄ω0 ) .
∆ (q) Nq + ∓
ρω(q)
2 2
(5.21)
After carrying out the addition of the new phonon branch and all averaging procedures, this
expression can explain why we obtain the same scattering rate even though we have lowered
5.2. Holes
51
70
60
Phonon Energy [meV]
50
LA branch
TA branch
LO branch
TO branch
40
30
20
10
0
0
0.2
0.4
0.6
0.8
1
q [2π/a]
Figure 5.6: Phonon dispersion in Silicon along (100). Fitted by [Pop04] to neutron scattering
data from [Dolling63].
both the optical and acoustic coupling constant in the new model (see table 5.5; ∆ is obtained
from equation (5.19)). First, the sum ∑µ
∆2µ
ω0,µ
over all implemented phonon branches µ remains
almost the same in the new model. Second, the shift by h̄ω0 in the argument of the density
of states can make up for an additional (small) difference. Figure 5.8 shows the sum over all
electron–phonon scattering processes where a phonon is emitted by an electron and where a
phonon is absorbed by an electron. As the scattering rate for absorption processes is increased,
the one for emission is lowered, leading to a constant overall electron–phonon scattering rate.
Furthermore, we observe that the characteristic kink in the density of states at about 4.5 eV is
reproduced in the curves. This kink is shifted by h̄ω0 to the left for absorption processes, so
the curve for the new model repeats the kink at 3 meV to the right from the old model as the 3
meV lower averaged TA energy results in an accordingly smaller shift to the left (discernible at
increased zooming into figure 5.8).
As a result of the procedure, the average hole energy, defined as
Z
Ē =
E f (E)dE,
(5.22)
where f(E) is the distribution function, is only slightly increased. For example, the increase
5. Results for Silicon
Phonon Energy (K)
52
100
LA branch
TA branch
50
(a)
Si−holes
−1
Phonon Vector (A )
0
0.2
0.1
(b)
0
50
150
250
350
Lattice Temperature (K)
450
Figure 5.7: Phonon energy (a) in terms of temperature and mean modulus of the phonon wave
vector (b)
amounts to about 3% at 300 K and an electric field of 250 kV/cm along the direction <100>.
The results of a simulation with the latter parameters are also plotted in figure 5.9 where the
slight distribution enhancement can be read off.
In figure 5.10, we check the consistency of our model by comparing the drift velocities to
data obtained by [Canali71], [Canali75] and [Smith81]. We also calculate the mobility according to equation (5.23) given in [Bufler02], yielding a value of 487.7 cm2 /(Vs) in reasonable
agreement with [Madelung04] listing 505 cm2 /(Vs).
µii = −
|q| ∑ν
R
eq
fν
dE Sν 1(E) v2νi (E) ddE
(E) Zν (E)
∑ν
R
eq
dE fν (E) Zν (E)
(5.23)
5.2. Holes
53
-1
Scattering Rate [s ]
1.0E-05
absorption, new model
absorption, old model
emission, new model
emission, old model
absorption+emission, new model
absorption+emission, old model
1.0E-06
1
2
3
Energy [eV]
4
5
Figure 5.8: Phonon Scattering Rate
Dt K
h̄ωopt
8
10 eV cm−1 [meV]
ε (LA) h̄ωac (LA)
ε (TA) h̄ωac (TA)
[eV]
[meV]
[eV]
[meV]
Previous Model
8.7
63
5.03
7.23
-
-
This Work
8.5
63
2.505
7.23
2.505
4.06
Table 5.5: Coupling constants and averaged phonon energies at 300K
54
5. Results for Silicon
1.0E+01
1 ac. branch
2 ac. branches
1.0E+00
Distribution
1.0E-01
1.0E-02
1.0E-03
Si holes
T = 300 K
E = 250 kV/cm
E || <100>
1.0E-04
0
0.5
1
1.5
Energy [eV]
Figure 5.9: Distribution function comparison between the 1- and 2 acoustic branches model
1.0
77 K
E || <111> Monte Carlo
E || <100> Monte Carlo
E || <100> ToF Exp.
E || <111> ToF Exp.
E || <111> MToF Exp.
0.1
300 K
7
Drift Velocity (10 cm/s)
(a)
1.0
Si−holes
(b)
245 K
0.1
370 K
1
10
Electric Field (kV/cm)
100
Figure 5.10: Hole drift velocities in Silicon using the 2 acoustic branches model
5.2. Holes
5.2.2.4
55
Field Dependent Phonon Energies
So far we have introduced a new phonon branch and changed the coupling constants. Apart
from that, the averaged phonon energies are additional parameters in this model which can also
be changed. Since the phonon energies are obviously field dependent, we try to determine the
average phonon energy iteratively: after running a bulk simulation, the hole energy is determined. This energy corresponds to a temperature which we use for β in equation (5.18) in order
ph
to calculate the new phonon energy. Starting with E0 set to the average phonon energy at the
desired temperature without any external field, we therefore have to apply
h
ph
hE ii = f hE ii
hE ph ii+1 = g hE h ii
(5.24)
(5.25)
iteratively until both relative errors are as small as desired,
|hE ph ii+1 − hE ph ii |
< ε
hE ph ii
|hE h ii+1 − hE h ii |
< ε.
hE h ii
(5.26)
(5.27)
( f and g just indicate that the left hand side is a function of the argument of the right hand side.)
It turns out that this procedure converges after a few iterations. However, it does not have a
significant effect due to the increased flatness of the dispersion curve at higher energies.
Another cure might be to consider phonon–assisted impact ionization, which will be done
in the upcoming section.
5.2.3
Phonon–assisted Impact Ionization
As both the addition of a new acoustic phonon branch and the consideration of the field dependence of phonon energies did not improve the distribution significantly, we now turn to
phonon–assisted processes.
Phonon–assisted impact ionization may seem negligible in comparison to direct impact ionization since it is a second order process. However, it relaxes momentum conservation and may
therefore play a critical role by shifting the threshold energy (cf. [Sano92]). In our calculations
we have so far neglected phonon–assisted processes. Still, the direct impact ionization rate
calculation yielded results in accordance with measurements for electrons, but not for holes.
56
5. Results for Silicon
We suggest that this difference is due to the following: Neglecting momentum conservation,
we arrive at the Random-k rate which starts at the gap energy. Observing energy and momentum
conservation, the threshold energy for electrons is still close to the gap energy, whereas it is
considerably higher for holes (see also section 3.3 and table 3.1). Therefore, due to Silicon
band structure characteristics, imposing momentum conservation has a larger threshold shifting
effect on the hole induced than on the electron induced impact ionization rate. Considering
phonon assisted processes can therefore have a more significant effect on the hole induced than
on the electron induced rate.
Phonon–assisted impact ionization has already been treated in [Eagles60], but only in a
parabolic band approximation. In the framework of second order perturbation theory, the
phonon–assisted impact ionization rate is compared to the direct rate in [Isler01], yielding a 10
to 100 times higher rate near threshold. Using second order perturbation theory, the phonon–
assisted impact ionization rate can be written as
ph
Rii
2π
=
h̄
< f |Hee |ν >< ν|Hep |i > 2
δ(E f − Ei ),
∑ ∑
Ei − Eν
f
ν
(5.28)
where i and f denote the initial and final state respectively. The inner summation has to be
carried out over all possible intermediate states and Hee and Hep denote the electron–electron
and electron–phonon interaction Hamiltonian respectively. However, it is not clear how to deal
with divergent terms occurring in expression (5.28). The case Ei = E f = Eν cannot generally be
ruled out and would lead to diverging behaviour (which is avoided in [Isler01] by just treating
special cases). Therefore, we do not pursue the second order perturbation theory approach here.
Due to the limited time frame of this thesis, our work on a quantum mechanical derivation
(analogous to [Bufler97], [Bufler02]) of the phonon–assisted impact ionization rate is still on–
going. Preliminary results are documented in appendix D.
Chapter 6
Results for Strained Silicon
Before applying our method to strained Silicon and presenting the results thereof in section 6.2,
the material considered will be introduced in section 6.1.
6.1
6.1.1
Material
Silicon–Germanium Mixed Crystals (Six Ge1−x )
The lattice constant of Six Ge1−x exceeds the Silicon lattice constant. However, it does not
change with Germanium content in a linear way (Vegard’s rule) but takes the form of a convex function staying slightly below the straight line. Numerical values for both Silicon and
Germanium were given in table 2.1.
Calculated band gap values taken from [Madelung04] along the Γ − X and Γ − L lines
amount to
Eg (Γ − X) = (0.8941 + 0.0421 x + 0.1691 x2 ) eV
(6.1)
Eg (Γ − L) = (0.7596 + 1.0860 x + 0.3306 x2 ) eV.
(6.2)
These curves are also plotted in figure 6.1. Band gap measurements show a change from silicon–
like behavior (band gap between Γ and X) to germanium–like behavior (band gap between Γ
and L) exhibiting a kink at about 15% silicon content where the change takes place.
57
58
6. Results for Strained Silicon
1.1
Eg [eV]
1
0.9
0.8
Gamma-X
Gamma-L
0.7
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 6.1: Gap Energy in Six Ge1−x along Γ − X and Γ − L lines as a function of x
6.1.2
Strained Silicon
Strained silicon is a layer of silicon atop a substrate of silicon–germanium. As of consequence
of the larger interatomic distance in SiGe (cf. section 6.1.1), atoms in the silicon layer get
stretched farther apart as they align with atoms in the silicon–germanium layer.
A relaxed Six Ge1−x layer has the same lattice constant as a volume Six Ge1−x crystal. The
silicon layer on top of it is then called under biaxial tensile stress, i.e. the lattice constant in the
direction perpendicular to the layer is smaller than the parallel ones. Under these circumstances,
the band structure is being changed significantly: the six–fold degeneracy of the X valleys is
lifted as the conduction band splits into a two–fold degenerate ∆2 band and a four–fold degenerate ∆4 band. According to [People85], the conduction band offset is approximately given as 60
meV per 10% Germanium. The energy level of the two valleys on the axis perpendicular to the
growth plane (∆2 ) is lowered with respect to the energy levels of the four in–plane valleys (∆4 ).
Electrons therefore occupy the ∆2 valleys where they only experience the transverse effective
mass for in–plane transport (note that mt = 0.19 m0 , whereas ml = 0.98 m0 ), which eventually
enables faster charge transport.
6.2. Impact Ionization Rates for Different Stress Levels
6.2
59
Impact Ionization Rates for Different Stress Levels
In this section, we observe the influence of stress on the impact ionization rate. Table 6.1 lists the
germanium content in the silicon–germanium layer, the resulting band gap in the strained silicon
(e− )
layer and the threshold energies Eth
(h+ )
and Eth
for electron and hole initiated impact ionization
respectively. We observe that the threshold is shifted towards lower energies with increasing
stress levels, but by a smaller amount than the band gap is lowered, which can be explained by
the availability of fewer possibilities to satisfy both energy and momentum conservation at the
same time.
Ge content [%]
Band gap [eV]
0
(e− )
(h+ )
Eth
Eth
1.12
1.140
1.367
10
1.063
1.091
1.337
20
1.003
1.036
1.314
Table 6.1: Band gap energies and impact ionization thresholds in strained Si
Figures 6.2 and 6.3 show the impact ionization rate in strained silicon for holes and electrons
respectively. For holes, we have fitted an analytical expression and list the results in table 6.2.
Pj s−1
aj
1
1.23 3.2 · 1013
5.144
1
1.15 1.9 · 1013
6.117
Ge content [%]
j
10
20
Eth, j [eV]
Table 6.2: Fitting parameters for the hole impact ionization rate in strained Si
60
6. Results for Strained Silicon
1E+14
Impact Ionization Rate [s-1]
1E+13
1E+12
1E+11
1E+10
1E+09
1E+08
Holes, 10% stress
Holes, 20% stress
Holes, 10% stress, fit
Holes, 20% stress, fit
1E+07
1
1.5
2
Energy [eV]
2.5
3
Figure 6.2: Hole initiated impact ionization rate in strained Si
1E+14
Impact Ionization Rate [s-1]
1E+13
1E+12
1E+11
1E+10
1E+09
1E+08
Electrons, 10% stress
Electrons, 20% stress
1E+07
1
1.5
2
Energy [eV]
2.5
Figure 6.3: Electron initiated impact ionization rate in strained Si
3
Chapter 7
Conclusion and Outlook
In this work, a comprehensive method to obtain the impact ionization rate has been developed and presented. The nine–dimensional integrals occurring in these calculations represent
a numerical challenge. Previously, numerical methods and approximations have been applied
throughout the literature without giving supporting reasons. In the present study, however, special care was taken to find criteria which support the appropriateness of our choice of numerical
methods, especially the integration method and the delta distribution approximation. We developed an algorithm taking into account both efficiency and accuracy requirements.
Our results for electron–initiated impact ionization were in agreement with measurements.
We were also able to state physical reasons for the result of the fitting to an analytical rate.
However, the results obtained for holes yielded a matrix element suggesting that the model has
to be refined. We therefore systematically explored the capabilities of our phonon model and
concluded to have reached its inherent limitations.
We pointed out the importance of phonon–assisted impact ionization and explained why
it could resolve the discrepancies between measurements and simulation. A possible way to
derive the phonon–assisted impact ionization rate in a quantum mechanical approach has been
outlined and preliminary results were stated. However, more work has to be done on this subject
for further clarification.
Moreover, we have investigated the impact of introducing stress on the impact ionization
rate. We observed that the impact ionization threshold is shifted to lower energies, but by a
smaller amount than the band gap is lowered. This can be explained by the fact that there are
61
62
7. Conclusion and Outlook
fewer possibilities to satisfy both energy and momentum conservation conditions at the same
time.
Appendix A
Software Documentation
This section describes the usage of software developed while writing this thesis. To make the
software easily available, the new module has been integrated in the device simulator DESSIS.
The keywords necessary to activate the impact ionization rate calculations are described below.
A.1
New Keywords Introduced in DESSIS Command Files
In order to invoke the impact ionization rate calculation, the keyword WithIICalculation
has to be specified. The keyword InducedBy allows the user to choose between hole induced
and electron induced impact ionization, where 0 and 1 denote the impact ionization rate induced
by holes and electrons respectively.
There are four parameters controlling the placement of energy values for which the impact ionization rate has to be calculated: NumberOfIIPoints, RefinedIntervals,
Refinement and MaxEnergy. The energy interval between the threshold energy and
MaxEnergy (given in eV) is being divided by NumberOfIIPoints equidistant points. If
RefinedIntervals is greater than zero, it specifies the number of these intervals to be refined starting at the lowest one. The ones which have to be refined are then subdivided into
as many smaller equidistant intervals as specified by Refinement. Therefore, there will be
NumberOfIIPoints + RefinedIntervals (Refinement − 1) points to be calculated.
Furthermore, ProcessesFile determines whether all processes will be logged. They
will be saved to a file if set to 1 and suppressed if set to 0. For more information on file names
and their content, please consult table A.3.
63
64
A. Software Documentation
The next five parameters are passed to the integration algorithm itself: EpsRel is the
desired accuracy, i.e. the algorithm tries to find an estimate Iˆ for the integral I so that
|Iˆ − I| ≤ I · EpsRel holds. MinMegaEval times one million is the minimum number of integrand evaluations. If set too low, the error estimates could yield unreliable results. Analogously,
MaxMegaEval times one million determines the number of integrand evaluations after which
the integration will be terminated (even if the desired accuracy has not been achieved) in order
to ensure a finite running time. SubregionConfidence specifies the desired confidence for
the χ2 test in each nine–dimensional subregion of the integration volume. For details on how
the χ2 test is performed, please refer to appendix B. If a subregion fails the χ2 test, it will be
treated further only if two sampling averages differ by more than MinDeviation · EpsRel. This
way, spending too much computational time in dividing subregions is avoided.
The keywords ModifLorentzDelta and ModifLorentzEta can be specified to alter the shape of the modified Lorentz profile used for approximating the delta distribution.
ModifLorentzDelta and ModifLorentzEta correspond to δ and η in equation (4.4),
respectively. Both of them are given in eV. Note that a simple box approximation as in equation
(4.3) can also be represented this way.
1
When IICalcBandgap is not present in the command file, Silicon is assumed and the
band gap is taken to be 1.12 eV. For any other material, please adjust this parameter accordingly.
Finally, the keyword WithoutUmklappProc can be specified if umklapp processes shall
not be considered.
Tables A.1 and A.2 list all keywords including their data types, default values and restrictions.
1A
simple box is the limiting case of a modified Lorentz profile in the following sense: Setting δ equal to the
desired width, we introduce ε := ηδ . Then


lim 
ε→0
1
2 arctan
δ
2η
η
1
|x|2 2
1
4
 = lim 1 +
−
ε
+
O(ε
)
= ,
ε→0 δ
|x|2 + η2
12 δ
δ3
δ
(A.1)
i.e. a simple box of width δ and height 1δ . For example, δ = 0.25 and η = 10000 already yields a box of height 4
within the tested machine precision.
A.1. New Keywords Introduced in DESSIS Command Files
65
Keyword
Data Type
Default Value
Unit
WithIICalculation
Boolean
InducedBy
Integer
0
NumberOfIIPoints
Integer
8
RefinedIntervals
Integer
2
Refinement
Integer
2
MaxEnergy
Double Precision
5 eV
ProcessesFile
Integer
0
EpsRel
Double Precision
10−2
MinMegaEval
Double Precision
200
MaxMegaEval
Double Precision
3000 106
SubregionConfidence
Double Precision
0.95
MinDeviation
Double Precision
0.25
ModifLorentzDelta
Double Precision
0.25 eV
ModifLorentzEta
Double Precision
0.15 eV
IICalcBandgap
Double Precision
1.12 eV
WithoutUmklappProc
Boolean
FALSE
FALSE
Table A.1: Keywords: Data Types and Default Values
106
66
A. Software Documentation
Keyword
Restrictions
WithIICalculation
InducedBy ∈ {0, 1}
InducedBy
0: hole induced, 1: electron induced
NumberOfIIPoints
NumberOfIIPoints > 0
RefinedIntervals
0 ≤ RefinedIntervals ≤ NumberOfIIPoints−1
Refinement
Refinement ≥ 1
MaxEnergy
MaxEnergy < 6
ProcessesFile
ProcessesFile ∈ {0, 1}
0: do not save processes, 1: save processes to file
(please consult table A.3 for file names)
EpsRel
0 < EpsRel < 1
MinMegaEval
0 < MinMegaEval < 1019
MaxMegaEval
MinMegaEval < MaxMegaEval < 1019
SubregionConfidence
0 < SubregionConfidence < 1
MinDeviation
0 < MinDeviation < 1
ModifLorentzDelta
ModifLorentzDelta > 0
ModifLorentzEta
ModifLorentzEta > 0
IICalcBandgap
IICalcBandgap > 0
WithoutUmklappProc
Table A.2: Keywords: Restrictions
A.1. New Keywords Introduced in DESSIS Command Files
File Name
Meaning
IICalc-Rate-Elec.txt
Electron induced impact ionization rate.
IICalc-Rate-Hole.txt
Hole induced impact ionization rate.
67
Columns: energy [eV], impact ionization rate s−1 ,
relative accuracy.
IICalc-Proc-Elec.txt
Electron induced impact ionization processes.
IICalc-Proc-Hole.txt
Hole induced impact ionization processes.
Columns: 4 band indices, 4 valley indices, 3 umklapp
vector components, threshold energy [eV],
energy error [eV].
Please see table C.1 for an explanation of band and valley
indices, as well as tables C.2 and C.3 for sample listings.
Table A.3: File Names
68
A. Software Documentation
Appendix B
χ2 Test
B.1
Fundamentals
A summary of the χ2 test can be found in [Bronstein00], on which this section is based. The
task is to determine whether a variable X is normally distributed. We divide X in k classes with
ξ j being the upper bound of the j-th class. The theoretical probability that X falls in the j-th
class is p j , i.e.
p j = F(ξ j ) − F(ξ j−1 ),
(B.1)
where F(X) is the distribution function of X. Since X is expected to be normally distributed,
ξj −µ
(B.2)
F(ξ j ) = Φ
σ
must hold.1
Given a sample (x1 , x2 , ..., xn ), calculate the frequency h j of the previously defined classes.
1 Φ(x)
denotes the distribution function of the normalized Gaussian distribution:
1
Φ(x) = √
2π
Z x
1
n
(B.3)
−∞
As µ and σ2 are generally not known, we use
x̄ =
t2
e− 2 dt
n
∑ xi
(B.4)
i=1
as an estimator for µ as well as
s2 =
1
n−1
n
∑ (xi − x̄)2
i=1
as an estimator for σ2 .
69
(B.5)
B. χ2 Test
70
χ2S =
k
∑
j=1
(h j − n p j )2
n pj
(B.6)
then approximately satisfies a χ2 distribution with m = n − 1 degrees of freedom. Note that
the classes have to fulfill n p j ≥ 5. For a desired accuracy 1 − α, take the quantile χ2α;k−1 for
which P(χ2 ≥ χ2α;k−1 ) = α. If
χ2S < χ2α;k−1 ,
(B.7)
then there is no contradiction to the assumption that the sample is taken from a normally distributed entity.
B.2
Implementation
The quantiles χ2α;k−1 appearing in section B.1 can be taken from tables available in the literature, e.g. [Bronstein00]. However, we want the user to be able to enter any desired accuracy
(cf. SubregionConfidence in section A.1) for the χ2 test in the individual subregions of
integration. We therefore need a continuous mapping and have approximated the χ2 distribution
function (which we actually have to numerically invert) by a quadratic polynomial expression
which mimics the correct behaviour accurately in the interesting region but can be inverted more
easily.
B.2. Implementation
71
Figure B.1: Parabolic approximation
72
B. χ2 Test
Appendix C
Thresholds of Impact Ionization Processes
in Silicon
Tables C.2 and C.3 list possible impact ionization processes in Silicon and their threshold energies sorted by bands, valleys and umklapp vectors for holes and electrons respectively. The
meaning of band and valley indices is explained in table C.1.
Band Index
Band
Valley Index
Valley
0
split-off valence band
0
X+X
1
light hole valence band
1
Γ
2
heavy hole valence band
2
L
3
1st conduction band
3
X+Z
4
2nd conduction band
4
X+Y
5
3rd conduction band
6
4th conduction band
Table C.1: Band and valley index nomenclature
73
74
C. Thresholds of Impact Ionization Processes in Silicon
C.1
v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Hole Initiated Processes
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v”
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
v, v’, v”, c’ valleys
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
3
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
G = (Gx , Gy , Gz )
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
-1
1
-1
1
1
1
1
-1
1
2
0
0
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
-2
0
0
0
2
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
-2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
0
0
0
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
-1
1
-1
-1
Eth [eV]
2.189
4.741
4.913
4.749
4.699
3.964
3.880
4.749
4.780
4.766
4.839
2.189
3.860
4.828
4.755
4.797
4.851
5.291
5.392
3.923
5.350
5.341
3.257
4.052
5.304
5.678
5.452
5.519
4.324
5.371
5.419
5.316
5.312
4.464
5.467
5.354
5.582
5.373
5.472
5.747
5.455
5.713
4.464
5.328
5.589
5.507
5.356
6.907
3.031
3.056
3.384
3.124
3.002
3.310
3.332
3.156
3.101
3.050
3.030
2.131
3.311
3.000
3.047
3.092
3.160
7.074
7.451
4.031
4.006
4.037
4.110
3.401
3.426
7.216
4.037
4.021
3.983
4.100
7.438
3.433
4.102
4.125
4.055
4.212
7.121
4.117
4.076
6.660
4.165
4.246
4.283
6.080
4.717
4.381
4.212
4.230
4.311
7.768
4.261
4.333
4.098
4.082
5.514
6.306
6.398
6.498
6.038
7.662
7.608
6.641
6.662
v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v”
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
v, v’, v”, c’ valleys
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
G = (Gx , Gy , Gz )
Eth [eV]
1
1
1
4.791
1
-1
1
4.839
2
0
0
3.953
-1
1
-1
4.763
-1
-1
-1
4.777
0
0
0
2.189
0
0
-2
4.029
1
1
-1
4.877
1
-1
-1
4.776
-1
1
-1
4.776
-1
-1
-1
4.728
0
2
0
3.885
1
1
1
4.770
1
-1
1
4.868
-1
1
1
4.763
-1
-1
1
4.706
0
0
0
3.258
1
1
-1
5.451
1
-1
-1
5.253
-1
1
1
5.326
-1
-1
1
5.446
-2
0
0
3.988
0
0
2
3.980
1
1
1
5.341
1
-1
1
5.329
-1
1
1
5.273
-1
-1
1
5.415
0
0
0
3.250
0
-2
0
3.990
1
1
-1
5.336
1
-1
-1
5.600
-1
1
-1
5.323
-1
-1
-1
5.503
1
1
1
5.761
1
-1
1
6.020
-1
1
1
5.481
-1
-1
1
5.364
0
0
0
4.464
1
1
-1
5.922
1
-1
-1
5.521
-1
1
-1
5.693
-1
-1
-1
5.313
0
2
0
7.314
1
1
-1
5.103
1
-1
-1
5.379
-1
1
-1
5.611
-1
-1
-1
5.716
0
0
0
2.131
1
1
-1
3.091
1
-1
-1
3.054
2
0
-2
7.233
-1
1
-1
2.999
-1
-1
-1
3.072
0
0
0
2.131
0
0
-2
3.394
1
1
-1
3.058
1
-1
-1
3.198
-1
1
-1
3.081
-1
-1
-1
3.052
0
2
0
3.326
1
1
1
3.240
1
-1
1
3.037
2
0
0
5.157
-1
1
-1
3.082
-1
-1
-1
2.990
0
0
0
3.092
1
1
1
3.935
1
-1
1
4.052
2
0
0
3.379
-1
1
-1
4.028
-1
-1
-1
4.450
0
0
0
3.092
0
0
-2
3.409
1
1
1
4.111
1
-1
1
3.927
-1
1
1
4.081
-1
-1
1
4.053
0
0
0
3.092
0
2
0
3.456
1
1
1
4.373
1
-1
1
4.007
-1
1
1
3.949
-1
-1
1
3.966
0
0
0
4.384
0
-2
0
7.111
1
1
-1
4.386
1
-1
-1
4.492
-1
1
1
4.135
-1
-1
1
4.539
-2
0
0
6.882
0
0
2
6.617
1
1
1
4.286
1
-1
1
4.280
2
0
0
7.075
-1
1
-1
4.370
-1
-1
-1
4.132
0
0
2
7.512
0
-2
0
6.464
1
1
-1
4.091
1
-1
-1
4.365
-1
1
-1
4.174
-1
-1
-1
4.262
1
1
1
6.672
1
-1
1
6.506
2
0
0
6.009
-1
1
-1
5.660
-1
-1
-1
6.706
0
0
0
5.546
1
1
1
6.653
1
-1
1
6.311
-1
1
1
6.632
continued on next page...
C.1. Hole Initiated Processes
v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
v”
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c’
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
v, v’, v”, c’ valleys
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
75
G = (Gx , Gy , Gz )
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
0
2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
2
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
Eth [eV]
6.729
6.566
7.546
6.621
6.586
6.443
6.625
2.037
6.821
2.877
3.001
2.933
2.856
3.144
3.171
5.734
2.853
2.928
2.823
2.879
3.952
3.145
2.889
2.858
2.844
2.876
5.492
6.657
3.853
3.827
3.555
3.750
3.237
3.300
3.762
3.651
3.421
3.832
2.909
3.223
3.651
3.553
6.246
3.715
3.583
4.044
4.063
4.050
6.197
3.974
3.943
4.162
6.489
6.436
3.885
4.070
4.001
3.967
7.244
6.234
3.892
3.882
3.933
3.961
7.380
6.474
6.121
7.312
6.476
6.476
5.451
7.242
6.406
6.490
6.334
6.329
7.223
6.485
6.477
6.493
6.285
2.072
3.053
3.048
3.099
3.059
3.086
3.092
3.083
3.005
2.855
2.939
2.072
3.100
2.888
2.881
2.974
2.977
2.972
3.557
3.518
3.109
3.523
3.531
2.974
3.097
3.922
3.580
3.775
3.516
3.084
3.510
3.420
3.509
3.333
4.070
4.588
v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
v”
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c’
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
v, v’, v”, c’ valleys
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
G = (Gx , Gy , Gz )
Eth [eV]
-1
-1
1
6.717
0
0
0
5.509
0
-2
0
7.433
1
1
-1
6.633
1
-1
-1
6.330
-1
1
-1
6.454
-1
-1
-1
6.494
0
0
2
4.600
1
1
1
2.880
1
-1
1
2.844
2
0
0
3.145
-1
1
-1
2.850
-1
-1
-1
2.892
0
0
0
2.037
0
0
-2
3.158
1
1
1
2.880
1
-1
1
2.842
-1
1
1
2.829
-1
-1
1
2.876
0
0
0
2.037
0
2
0
3.172
1
1
1
2.916
1
-1
1
2.892
2
0
0
4.828
-1
1
-1
2.854
-1
-1
-1
2.860
0
0
0
2.909
1
1
1
3.591
1
-1
1
3.671
2
0
0
3.213
-1
1
-1
3.603
-1
-1
-1
3.504
0
0
0
2.908
0
0
-2
3.266
1
1
-1
3.682
1
-1
-1
3.684
-1
1
-1
3.588
-1
-1
-1
3.558
0
0
2
6.293
0
-2
0
3.277
1
1
-1
3.638
1
-1
-1
3.315
-1
1
1
3.474
-1
-1
1
3.788
-2
0
0
5.836
0
0
-2
6.433
1
1
-1
4.036
1
-1
-1
4.055
-1
1
1
3.989
-1
-1
1
3.911
-2
0
0
5.963
0
0
2
6.754
0
2
0
6.473
1
1
1
3.959
1
-1
1
3.998
-1
1
1
3.919
-1
-1
1
3.882
0
0
0
4.058
0
2
0
6.309
1
1
1
3.886
1
-1
1
3.993
2
0
0
5.872
-1
1
-1
3.954
-1
-1
-1
4.038
0
0
0
5.786
1
1
-1
6.470
1
-1
-1
6.488
-1
1
1
6.541
-1
-1
1
6.432
-2
0
0
7.290
0
0
2
7.225
1
1
1
6.290
1
-1
1
6.470
-1
1
1
6.514
-1
-1
1
6.331
0
0
0
5.514
0
-2
0
7.010
1
1
-1
6.467
1
-1
-1
6.354
-1
1
-1
6.459
-1
-1
-1
6.510
1
1
1
3.024
1
-1
1
3.096
2
0
0
3.095
-1
1
-1
3.011
-1
-1
-1
2.961
0
0
0
2.072
0
0
-2
3.112
1
1
-1
2.966
1
-1
-1
2.987
-1
1
-1
2.923
-1
-1
-1
3.024
0
2
0
3.129
1
1
1
2.923
1
-1
1
2.950
2
0
0
4.274
-1
1
-1
3.005
-1
-1
-1
2.943
0
0
-2
6.905
1
1
-1
3.302
1
-1
-1
3.753
-1
1
1
3.686
-1
-1
1
3.551
-2
0
0
3.097
0
0
2
3.113
1
1
1
3.626
1
-1
1
3.515
-1
1
1
3.606
-1
-1
1
3.526
0
0
0
2.974
0
-2
0
3.121
1
1
-1
3.528
1
-1
-1
3.211
-1
1
-1
3.603
-1
-1
-1
3.378
1
1
1
4.211
1
-1
1
4.143
continued on next page...
76
C. Thresholds of Impact Ionization Processes in Silicon
v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
v”
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c’
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
v, v’, v”, c’ valleys
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
G = (Gx , Gy , Gz )
1
-1
-1
-1
1
1
-1
-1
1
0
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
-1
1
-1
-1
-1
-1
1
-2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
1
1
-1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
Eth [eV]
4.307
4.164
4.265
4.107
4.525
4.483
4.335
4.646
4.072
5.267
4.721
4.316
4.491
5.948
6.639
6.742
7.767
6.309
6.044
6.615
6.685
5.511
6.718
6.384
6.258
1.941
2.797
2.801
2.715
2.813
2.759
2.767
2.762
2.725
2.728
2.709
1.941
2.812
2.698
2.753
2.758
2.680
3.174
3.420
2.838
3.236
3.284
2.713
2.842
3.328
3.328
3.047
3.403
2.712
2.842
3.302
3.285
3.245
3.353
4.123
3.936
4.169
4.102
3.849
3.975
4.303
3.886
4.009
6.955
4.161
4.231
3.795
3.836
6.411
6.091
7.261
6.583
5.944
5.186
6.660
6.426
6.412
6.300
5.341
6.422
5.849
6.212
1.663
2.529
2.538
2.517
2.583
2.321
2.298
2.575
2.536
2.509
2.594
1.663
2.334
2.583
2.551
2.513
2.483
2.844
2.815
2.287
2.867
2.850
2.248
2.294
2.816
2.852
2.824
2.883
2.369
2.835
v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
v”
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c’
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
v, v’, v”, c’ valleys
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
G = (Gx , Gy , Gz )
Eth [eV]
2
0
0
6.397
-1
1
-1
4.893
-1
-1
-1
4.276
0
0
2
6.322
1
1
-1
4.161
1
-1
-1
4.120
-1
1
-1
4.051
-1
-1
-1
4.089
1
1
1
4.450
1
-1
1
4.420
-1
1
1
4.260
-1
-1
1
5.013
0
0
0
5.646
1
-1
1
6.617
-1
1
1
6.615
-1
-1
-1
6.729
0
0
0
5.533
1
1
-1
6.728
1
-1
-1
6.948
-1
1
-1
6.094
-1
-1
-1
6.699
0
2
0
7.603
1
1
-1
6.662
1
-1
-1
6.826
-1
-1
-1
6.377
1
1
1
2.695
1
-1
1
2.816
2
0
0
2.759
-1
1
-1
2.709
-1
-1
-1
2.758
0
0
0
1.941
0
0
-2
2.828
1
1
-1
2.811
1
-1
-1
2.740
-1
1
-1
2.689
-1
-1
-1
2.706
0
2
0
2.797
1
1
1
2.742
1
-1
1
2.708
-1
1
1
2.761
-1
-1
1
2.683
0
0
0
2.714
1
1
-1
3.258
1
-1
-1
3.398
-1
1
1
3.344
-1
-1
1
3.331
-2
0
0
2.847
0
0
2
2.843
0
2
0
6.317
1
1
-1
3.362
1
-1
-1
3.359
-1
1
-1
3.332
-1
-1
-1
3.363
0
2
0
2.851
1
1
1
3.417
1
-1
1
3.364
-1
1
1
3.050
-1
-1
1
3.325
0
0
0
3.976
1
1
-1
3.973
1
-1
-1
3.810
-1
1
-1
4.267
-1
-1
-1
3.791
1
1
1
3.850
1
-1
1
3.868
-1
1
1
4.622
-1
-1
1
3.975
0
0
0
3.913
1
1
1
3.817
1
-1
1
4.059
-1
1
1
3.892
-1
-1
1
3.838
0
0
0
5.675
1
1
-1
6.347
1
-1
-1
6.476
-1
1
1
6.856
-1
-1
1
6.467
-2
0
0
7.269
0
0
2
7.096
1
1
-1
6.391
1
-1
-1
6.322
-1
1
-1
6.218
-1
-1
-1
6.563
1
1
1
6.410
1
-1
1
6.521
-1
1
-1
6.366
-1
-1
-1
5.865
1
1
1
2.499
1
-1
1
2.521
2
0
0
2.289
-1
1
-1
2.521
-1
-1
-1
2.494
0
0
0
1.663
0
0
-2
2.288
1
1
-1
2.557
1
-1
-1
2.513
-1
1
-1
2.624
-1
-1
-1
2.516
0
2
0
2.267
1
1
1
2.554
1
-1
1
2.541
-1
1
1
2.591
-1
-1
1
2.484
0
0
0
2.250
1
1
-1
2.849
1
-1
-1
2.889
-1
1
1
2.858
-1
-1
1
2.876
-2
0
0
2.305
0
0
2
2.316
1
1
1
2.912
1
-1
1
2.874
-1
1
1
2.825
-1
-1
1
2.832
0
0
0
2.248
0
-2
0
2.291
1
1
-1
2.823
continued on next page...
C.1. Hole Initiated Processes
v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
v’
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v”
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c’
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
v, v’, v”, c’ valleys
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
77
G = (Gx , Gy , Gz )
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
-1
0
0
2
1
-1
-1
-1
-1
-1
0
-2
0
1
1
-1
1
-1
-1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
-1
1
-1
-1
-1
1
-1
-2
0
0
0
0
2
1
1
-1
1
-1
-1
-1
1
-1
0
0
0
0
-2
0
1
1
-1
-1
-1
1
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
0
0
2
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
-2
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
Eth [eV]
2.838
2.848
2.816
3.723
4.209
3.832
3.622
4.643
3.656
3.931
4.014
4.109
3.722
3.640
3.689
4.847
3.629
6.265
6.282
6.234
7.089
6.371
6.404
7.143
6.239
5.960
1.819
2.735
2.784
3.005
2.662
2.849
2.628
2.940
2.798
3.061
2.550
1.819
2.614
2.811
3.339
2.533
2.730
3.562
3.551
3.224
2.619
2.759
3.219
4.069
3.599
2.423
2.647
3.601
3.532
1.783
2.352
2.306
2.500
2.430
2.355
1.783
2.463
2.400
2.319
2.525
2.449
1.783
2.434
2.492
2.443
3.894
2.314
2.295
3.021
2.716
2.490
3.252
3.941
2.371
2.482
3.036
2.844
3.361
2.921
2.504
3.242
3.173
2.890
2.804
1.709
2.059
2.078
2.302
2.078
2.074
1.709
2.294
2.128
2.128
2.035
2.124
1.709
2.303
2.050
2.061
2.027
2.044
3.936
2.497
2.451
2.353
2.750
2.347
2.284
2.360
2.470
v
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
v’
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v”
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c’
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
v, v’, v”, c’ valleys
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
G = (Gx , Gy , Gz )
Eth [eV]
1
-1
-1
2.908
-1
1
-1
2.819
-1
-1
-1
2.847
1
1
1
3.919
1
-1
1
3.895
-1
1
1
4.010
-1
-1
1
4.565
0
0
0
3.719
1
1
-1
4.111
1
-1
-1
3.702
-1
1
-1
3.691
-1
-1
-1
3.796
1
1
1
3.695
1
-1
1
4.358
-1
1
1
3.923
-1
-1
1
3.909
0
0
0
5.098
1
-1
1
6.472
-1
-1
1
6.241
0
0
0
5.015
1
1
1
5.966
-1
-1
1
6.329
0
0
0
4.809
1
1
1
6.236
1
-1
1
6.310
-1
-1
-1
6.264
1
1
1
3.038
1
-1
1
2.397
2
0
0
2.654
-1
1
-1
2.708
-1
-1
-1
2.635
0
0
0
1.819
0
0
-2
2.682
1
1
-1
2.772
1
-1
-1
2.666
-1
1
-1
2.918
-1
-1
-1
2.729
0
2
0
2.632
1
1
1
2.660
1
-1
1
2.681
-1
1
1
2.622
-1
-1
1
2.666
0
0
0
2.426
1
-1
1
3.436
2
0
0
2.698
-1
-1
1
3.445
0
0
0
2.428
0
0
-2
2.683
1
-1
1
3.956
-1
1
1
3.135
-1
-1
-1
4.157
0
2
0
2.661
1
1
1
3.720
-1
1
1
4.304
-1
-1
-1
3.115
0
2
0
3.783
1
1
-1
2.391
1
-1
-1
2.432
-1
1
1
2.459
-1
-1
1
2.285
-2
0
0
2.460
0
0
2
2.509
1
1
1
2.334
1
-1
1
2.434
-1
1
1
2.490
-1
-1
1
2.440
-2
0
0
3.896
0
0
2
4.183
0
-2
0
2.477
1
1
-1
2.439
1
-1
-1
2.536
-1
1
1
2.494
-1
-1
1
2.334
0
0
0
2.370
1
1
-1
2.592
1
-1
-1
2.786
-1
1
1
3.060
-1
-1
1
2.720
-2
0
0
2.504
0
0
2
2.496
1
1
1
2.621
1
-1
1
3.605
-1
1
1
2.850
-1
-1
1
3.233
0
0
0
2.371
0
-2
0
2.515
1
1
-1
2.641
1
-1
-1
2.979
-1
1
-1
4.036
-1
-1
-1
2.897
0
-2
0
3.834
1
1
-1
2.037
1
-1
-1
2.131
-1
1
1
2.045
-1
-1
1
2.124
-2
0
0
2.259
0
0
2
2.299
0
2
0
3.734
1
1
-1
2.064
1
-1
-1
2.152
-1
1
-1
2.054
-1
-1
-1
2.136
0
0
2
3.399
0
-2
0
2.298
1
1
-1
2.067
1
-1
-1
2.070
-1
1
-1
2.186
-1
-1
-1
2.041
0
0
0
2.283
1
1
-1
2.432
1
-1
-1
2.529
-1
1
1
2.495
-1
-1
1
2.378
-2
0
0
2.404
0
0
2
2.336
1
1
1
2.430
1
-1
1
2.531
continued on next page...
78
C. Thresholds of Impact Ionization Processes in Silicon
v
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
v’
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
v”
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c’
4
4
4
4
4
4
4
4
5
5
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
v, v’, v”, c’ valleys
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
G = (Gx , Gy , Gz )
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
1
1
1
1
1
1
1
-1
1
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
-1
-1
-1
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
1
-1
1
Eth [eV]
2.543
2.402
2.581
2.343
2.643
2.495
2.399
2.472
4.059
3.766
4.151
1.747
2.265
2.268
2.359
2.207
2.317
1.748
2.351
4.019
2.165
2.265
2.293
2.199
2.388
2.255
2.140
2.204
2.298
2.316
2.931
2.595
2.638
2.822
2.412
2.409
2.470
2.530
2.613
2.646
2.315
2.423
2.541
2.792
2.624
2.569
3.252
1.959
1.969
2.170
2.016
1.907
1.612
2.189
1.944
2.078
1.935
1.943
4.060
2.181
1.948
1.927
1.954
2.047
2.151
2.379
2.348
2.342
2.363
2.200
2.209
2.289
2.341
2.320
2.350
2.154
2.217
2.375
2.280
2.247
2.383
3.658
3.657
1.731
1.718
1.719
1.729
1.862
1.868
1.723
1.726
1.817
1.737
1.473
1.867
1.698
1.728
1.695
1.718
1.882
2.114
2.021
2.028
2.056
1.881
1.871
3.660
2.042
2.182
2.026
2.154
1.896
2.074
2.149
2.081
2.131
3.590
v
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
v’
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
v”
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c’
4
4
4
4
4
4
4
4
5
5
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
v, v’, v”, c’ valleys
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
G = (Gx , Gy , Gz )
Eth [eV]
-1
1
1
2.388
-1
-1
1
2.504
0
0
0
2.285
0
-2
0
2.361
1
1
-1
2.903
1
-1
-1
2.550
-1
1
-1
2.517
-1
-1
-1
2.511
1
-1
-1
4.258
1
1
-1
3.993
1
-1
-1
4.345
0
0
2
3.317
1
1
-1
2.313
1
-1
-1
2.291
-1
1
1
2.289
-1
-1
1
2.333
-2
0
0
2.383
0
0
2
2.392
0
2
0
3.763
1
1
1
2.290
1
-1
1
2.166
-1
1
1
2.165
-1
-1
1
2.156
0
0
0
1.748
0
-2
0
2.300
1
1
-1
2.204
1
-1
-1
2.294
-1
1
-1
2.330
-1
-1
-1
2.266
1
1
1
2.541
1
-1
1
2.537
2
0
0
2.410
-1
1
-1
2.571
-1
-1
-1
2.758
0
0
0
2.316
0
0
-2
2.423
1
1
-1
2.605
1
-1
-1
2.846
-1
1
-1
2.484
-1
-1
-1
2.905
0
2
0
2.400
1
1
1
2.850
1
-1
1
2.556
-1
1
1
2.566
-1
-1
1
2.542
0
0
0
1.612
0
2
0
3.393
1
1
-1
1.982
1
-1
-1
1.979
-1
1
1
1.911
-1
-1
1
1.938
-2
0
0
2.170
0
0
2
2.178
1
1
1
1.949
1
-1
1
1.977
-1
1
1
2.108
-1
-1
1
1.991
0
0
0
1.612
0
2
0
2.176
1
1
1
2.109
1
-1
1
1.945
-1
1
1
1.929
-1
-1
1
1.958
-2
0
0
3.849
1
1
1
2.456
1
-1
1
2.317
2
0
0
2.253
-1
1
-1
2.376
-1
-1
-1
2.352
0
0
0
2.152
0
0
-2
2.212
1
1
-1
2.288
1
-1
-1
2.388
-1
1
-1
2.301
-1
-1
-1
2.378
0
2
0
2.216
1
1
1
2.343
1
-1
1
2.299
-1
1
1
2.343
-1
-1
1
2.314
-1
1
1
3.737
0
0
0
1.472
1
1
1
1.790
1
-1
1
1.776
2
0
0
1.852
-1
1
-1
1.787
-1
-1
-1
1.715
0
0
0
1.473
0
0
-2
1.860
1
1
-1
1.717
1
-1
-1
1.717
-1
1
-1
1.760
-1
-1
-1
1.748
0
0
2
3.465
0
-2
0
1.862
1
1
-1
1.712
1
-1
-1
1.795
-1
1
-1
1.765
-1
-1
-1
1.719
1
1
1
2.047
1
-1
1
2.166
2
0
0
1.879
-1
1
-1
2.093
-1
-1
-1
2.143
0
0
0
1.883
0
0
-2
1.881
1
1
1
2.118
1
-1
1
2.198
-1
1
1
2.065
-1
-1
1
2.146
0
0
0
1.884
0
-2
0
1.892
1
1
-1
2.136
1
-1
-1
2.100
-1
1
-1
2.197
-1
-1
-1
2.077
-1
1
-1
4.036
continued on next page...
C.1. Hole Initiated Processes
v
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
v’
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
v”
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c’
5
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
v, v’, v”, c’ valleys
1
1
1
3
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
79
G = (Gx , Gy , Gz )
1
-1
1
1
1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
2
0
0
-2
0
0
0
0
2
0
0
0
0
-2
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-2
0
0
0
0
2
1
1
1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
-1
1
-1
-1
-1
-1
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
Eth [eV]
3.779
4.151
1.815
2.214
2.031
1.996
2.234
1.815
2.282
2.438
1.912
2.014
1.815
2.364
2.039
2.353
2.397
1.930
2.601
2.728
2.630
2.424
2.679
2.057
2.109
2.074
1.938
2.016
1.770
2.151
2.024
2.063
1.842
1.970
1.944
1.912
1.919
1.997
1.884
2.205
2.678
2.804
2.853
2.212
2.234
2.486
2.745
3.079
3.311
2.209
3.712
3.286
3.227
3.040
1.874
1.753
1.973
1.866
1.724
1.508
1.853
1.803
1.697
1.713
1.852
1.792
1.784
1.838
1.713
1.785
1.805
2.035
2.044
2.043
1.988
1.988
1.989
1.990
1.955
1.900
2.048
1.869
2.000
1.891
2.046
1.807
2.086
1.806
1.921
1.927
1.862
1.839
1.678
1.792
1.871
1.865
1.931
1.922
1.985
1.926
1.818
1.949
1.888
2.108
2.289
2.305
2.238
2.101
2.157
2.118
2.163
2.247
2.257
2.116
2.108
2.114
2.222
v
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
v’
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
v”
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c’
5
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
v, v’, v”, c’ valleys
1
1
1
3
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
4
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
4
1
1
1
4
1
1
1
4
G = (Gx , Gy , Gz )
Eth [eV]
-1
-1
1
4.325
1
1
-1
4.103
1
1
1
2.379
1
-1
1
2.191
2
0
0
2.024
-1
1
-1
1.950
-2
0
0
2.236
0
0
2
2.005
1
1
1
2.315
1
-1
1
2.209
-1
1
1
2.070
-1
-1
-1
2.177
0
2
0
2.407
1
1
1
2.248
1
-1
1
2.280
-1
1
1
2.236
-1
-1
1
2.187
0
0
0
2.420
-1
-1
1
3.543
0
0
0
2.425
0
0
-2
2.674
0
2
0
2.659
0
0
0
1.769
1
1
-1
2.082
1
-1
-1
2.133
-1
1
1
1.960
-1
-1
1
1.877
-2
0
0
1.973
0
0
2
2.108
1
1
1
2.120
1
-1
1
2.083
-1
1
1
1.934
-1
-1
1
1.862
0
0
0
1.769
0
-2
0
2.048
1
1
-1
2.066
1
-1
-1
1.906
-1
1
-1
2.068
-1
-1
-1
2.094
1
1
1
3.509
1
-1
1
2.809
2
0
0
2.226
-1
-1
1
3.296
0
0
0
2.210
0
0
-2
2.229
1
-1
1
2.635
-1
1
1
2.741
-1
-1
1
2.868
0
0
0
2.207
0
-2
0
2.251
1
-1
-1
3.489
-1
-1
1
2.582
0
0
0
1.508
0
2
0
2.806
1
1
-1
1.861
1
-1
-1
1.834
-1
1
1
1.812
-1
-1
1
1.827
-2
0
0
1.840
0
0
2
1.970
1
1
1
1.874
1
-1
1
1.755
-1
1
1
1.865
-1
-1
1
1.831
0
0
0
1.508
0
-2
0
1.975
1
1
-1
1.854
1
-1
-1
1.812
-1
1
-1
1.785
-1
-1
-1
1.858
1
1
1
1.909
1
-1
1
2.079
2
0
0
1.947
-1
1
-1
1.855
-1
-1
-1
1.923
0
0
0
1.830
0
0
-2
1.992
1
1
-1
2.117
1
-1
-1
2.058
-1
1
-1
2.034
-1
-1
-1
1.971
0
2
0
1.817
1
1
1
2.053
1
-1
1
2.037
-1
1
1
2.054
-1
-1
1
2.039
0
0
0
1.678
1
1
-1
1.909
1
-1
-1
1.890
-1
1
1
1.930
-1
-1
1
1.777
-2
0
0
1.898
0
0
2
1.866
1
1
1
1.949
1
-1
1
1.891
-1
1
1
1.857
-1
-1
1
1.816
0
0
0
1.678
0
-2
0
1.919
1
1
-1
1.940
1
-1
-1
1.836
-1
1
-1
2.021
-1
-1
-1
1.844
1
1
1
2.369
1
-1
1
2.083
2
0
0
2.197
-1
1
-1
2.034
-1
-1
-1
2.245
0
0
0
2.105
0
0
-2
2.122
1
1
-1
2.402
1
-1
-1
2.227
-1
1
-1
2.319
-1
-1
-1
2.266
0
2
0
2.109
1
1
1
2.125
1
-1
1
2.190
continued on next page...
80
C. Thresholds of Impact Ionization Processes in Silicon
v
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
v”
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c’
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
v, v’, v”, c’ valleys
G = (Gx , Gy , Gz )
Eth [eV]
v
v’
v”
c’
v, v’, v”, c’ valleys
G = (Gx , Gy , Gz )
1
1
1
4
1
-1
-1
2.041
2
1
1
4
1
1
1
4
-1
1
1
1
1
1
4
-1
1
-1
2.247
2
1
1
4
1
1
1
4
-1
-1
1
1
1
1
4
-1
-1
-1
2.270
2
1
2
3
1
1
1
0
0
0
0
1
1
1
0
0
0
2
2.655
2
1
2
3
1
1
1
0
1
1
1
1
1
1
0
1
1
-1
1.790
2
1
2
3
1
1
1
0
1
-1
1
1
1
1
0
1
-1
-1
1.673
2
1
2
3
1
1
1
0
2
0
0
1
1
1
0
-1
1
1
1.822
2
1
2
3
1
1
1
0
-1
1
-1
1
1
1
0
-1
-1
1
1.649
2
1
2
3
1
1
1
0
-1
-1
-1
1
1
1
0
-2
0
0
1.701
2
1
2
3
1
1
1
3
0
0
0
1
1
1
3
0
0
2
1.810
2
1
2
3
1
1
1
3
0
0
-2
1
1
1
3
0
2
0
3.588
2
1
2
3
1
1
1
3
1
1
1
1
1
1
3
1
1
-1
1.674
2
1
2
3
1
1
1
3
1
-1
1
1
1
1
3
1
-1
-1
1.746
2
1
2
3
1
1
1
3
-1
1
1
1
1
1
3
-1
1
-1
1.691
2
1
2
3
1
1
1
3
-1
-1
1
1
1
1
3
-1
-1
-1
1.731
2
1
2
3
1
1
1
4
0
0
0
1
1
1
4
0
0
-2
3.541
2
1
2
3
1
1
1
4
0
2
0
1
1
1
4
0
-2
0
1.782
2
1
2
3
1
1
1
4
1
1
1
1
1
1
4
1
1
-1
1.794
2
1
2
3
1
1
1
4
1
-1
1
1
1
1
4
1
-1
-1
1.652
2
1
2
3
1
1
1
4
2
2
0
1
1
1
4
-1
1
1
1.763
2
1
2
3
1
1
1
4
-1
1
-1
1
1
1
4
-1
-1
1
1.816
2
1
2
3
1
1
1
4
-1
-1
-1
1
1
1
0
0
0
0
1.779
2
1
2
4
1
1
1
0
1
1
1
1
1
1
0
1
1
-1
1.897
2
1
2
4
1
1
1
0
1
-1
1
1
1
1
0
1
-1
-1
1.977
2
1
2
4
1
1
1
0
2
0
0
1
1
1
0
-1
1
1
1.758
2
1
2
4
1
1
1
0
-1
1
-1
1
1
1
0
-1
-1
1
1.944
2
1
2
4
1
1
1
0
-1
-1
-1
1
1
1
0
-2
0
0
1.930
2
1
2
4
1
1
1
3
0
0
0
1
1
1
3
0
0
2
1.926
2
1
2
4
1
1
1
3
0
0
-2
1
1
1
3
1
1
1
1.940
2
1
2
4
1
1
1
3
1
1
-1
1
1
1
3
1
-1
1
2.014
2
1
2
4
1
1
1
3
1
-1
-1
1
1
1
3
-1
1
1
2.001
2
1
2
4
1
1
1
3
-1
1
-1
1
1
1
3
-1
-1
1
1.819
2
1
2
4
1
1
1
3
-1
-1
-1
1
1
1
4
0
0
0
1.806
2
1
2
4
1
1
1
4
0
2
0
1
1
1
4
0
-2
0
1.930
2
1
2
4
1
1
1
4
1
1
1
1
1
1
4
1
1
-1
1.946
2
1
2
4
1
1
1
4
1
-1
1
1
1
1
4
1
-1
-1
1.946
2
1
2
4
1
1
1
4
-1
1
1
1
1
1
4
-1
1
-1
1.944
2
1
2
4
1
1
1
4
-1
-1
1
1
1
1
4
-1
-1
-1
1.947
2
2
2
3
1
1
1
0
0
0
0
1
1
1
0
0
0
-2
2.702
2
2
2
3
1
1
1
0
1
1
1
1
1
1
0
1
1
-1
1.626
2
2
2
3
1
1
1
0
1
-1
1
1
1
1
0
1
-1
-1
1.574
2
2
2
3
1
1
1
0
2
0
0
1
1
1
0
-1
1
1
1.628
2
2
2
3
1
1
1
0
-1
1
-1
1
1
1
0
-1
-1
1
1.628
2
2
2
3
1
1
1
0
-1
-1
-1
1
1
1
0
-2
0
0
1.683
2
2
2
3
1
1
1
3
0
0
0
1
1
1
3
0
0
2
1.707
2
2
2
3
1
1
1
3
0
0
-2
1
1
1
3
1
1
1
1.576
2
2
2
3
1
1
1
3
1
1
-1
1
1
1
3
1
-1
1
1.630
2
2
2
3
1
1
1
3
1
-1
-1
1
1
1
3
-1
1
1
1.556
2
2
2
3
1
1
1
3
-1
1
-1
1
1
1
3
-1
-1
1
1.576
2
2
2
3
1
1
1
3
-1
-1
-1
1
1
1
4
0
0
0
1.367
2
2
2
3
1
1
1
4
0
2
0
1
1
1
4
0
-2
0
1.691
2
2
2
3
1
1
1
4
1
1
1
1
1
1
4
1
1
-1
1.555
2
2
2
3
1
1
1
4
1
-1
1
1
1
1
4
1
-1
-1
1.512
2
2
2
3
1
1
1
4
-1
1
1
1
1
1
4
-1
1
-1
1.565
2
2
2
3
1
1
1
4
-1
-1
1
1
1
1
4
-1
-1
-1
1.493
2
2
2
4
1
1
1
0
0
0
0
1
1
1
0
1
1
1
1.763
2
2
2
4
1
1
1
0
1
1
-1
1
1
1
0
1
-1
1
1.759
2
2
2
4
1
1
1
0
1
-1
-1
1
1
1
0
2
0
0
1.700
2
2
2
4
1
1
1
0
-1
1
1
1
1
1
0
-1
1
-1
1.759
2
2
2
4
1
1
1
0
-1
-1
1
1
1
1
0
-1
-1
-1
1.753
2
2
2
4
1
1
1
0
-2
0
0
1
1
1
3
0
0
0
1.707
2
2
2
4
1
1
1
3
0
0
2
1
1
1
3
0
0
-2
1.711
2
2
2
4
1
1
1
3
1
1
1
1
1
1
3
1
1
-1
1.752
2
2
2
4
1
1
1
3
1
-1
1
1
1
1
3
1
-1
-1
1.747
2
2
2
4
1
1
1
3
-1
1
1
1
1
1
3
-1
1
-1
1.774
2
2
2
4
1
1
1
3
-1
-1
1
1
1
1
3
-1
-1
-1
1.783
2
2
2
4
1
1
1
4
0
0
0
1
1
1
4
0
2
0
1.721
2
2
2
4
1
1
1
4
0
-2
0
1
1
1
4
1
1
1
1.829
2
2
2
4
1
1
1
4
1
1
-1
1
1
1
4
1
-1
1
1.747
2
2
2
4
1
1
1
4
1
-1
-1
1
1
1
4
-1
1
1
1.752
2
2
2
4
1
1
1
4
-1
1
-1
1
1
1
4
-1
-1
1
1.747
2
2
2
4
1
1
1
4
-1
-1
-1
Table C.2: Hole initiated impact ionization processes in Si, sorted by band indices, valleys and umklapp vectors
Eth [eV]
2.240
2.246
1.474
1.807
1.642
1.902
1.676
1.660
1.474
1.892
1.670
1.791
1.668
1.704
1.474
1.830
1.773
1.654
3.755
1.708
1.744
1.965
1.944
1.928
1.917
1.943
1.801
1.923
1.951
1.992
1.868
1.975
1.944
1.885
1.815
2.002
1.915
1.367
1.599
1.568
1.688
1.575
1.615
1.367
1.681
1.529
1.616
1.630
1.569
1.707
1.508
1.506
1.617
1.575
1.698
1.738
1.766
1.760
1.750
1.701
1.714
1.756
1.751
1.767
1.763
1.706
1.708
1.698
1.783
1.747
1.765
C.2. Electron Initiated Processes
C.2
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
81
Electron Initiated Processes
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
c, c’, c”, v’ valleys
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
G = (Gx , Gy , Gz )
0
0
0
-2
0
0
1
1
1
1
-1
-1
-1
1
1
-2
0
0
0
2
0
1
-1
-1
-1
1
-1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
-1
-1
1
2
0
0
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
1
-1
-1
0
0
0
0
0
-2
0
0
2
1
1
1
1
-1
1
-1
1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
0
0
0
2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
0
0
0
0
0
-2
1
1
-1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
2
0
0
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
1
1
1
1
-1
-1
-1
1
1
Eth [eV]
1.397
1.184
2.077
1.724
2.346
1.970
2.289
2.372
1.996
1.864
1.909
1.892
2.343
2.225
1.749
1.649
1.652
2.298
1.349
1.312
1.396
1.349
1.813
1.997
2.149
1.841
1.916
1.323
1.354
1.297
1.367
1.658
2.411
1.658
1.363
2.435
2.068
1.780
2.008
1.874
1.370
1.317
1.429
1.330
1.392
1.880
2.167
1.396
1.184
1.944
1.855
1.849
1.819
2.424
1.424
1.329
1.351
1.341
2.075
1.876
1.975
1.880
1.858
1.778
1.655
1.651
1.656
1.334
1.327
1.373
1.353
1.360
1.756
1.685
1.812
1.742
2.109
1.338
1.394
1.316
1.387
1.649
1.658
1.358
1.798
2.000
2.035
2.089
1.352
1.899
1.748
2.079
2.327
1.813
1.902
2.271
2.127
1.776
2.329
1.399
1.184
1.145
1.313
2.147
1.746
2.029
1.891
1.826
2.261
2.015
1.779
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
c, c’, c”, v’ valleys
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
G = (Gx , Gy , Gz )
Eth [eV]
2
0
0
1.184
0
0
0
1.353
1
1
-1
1.784
2
0
0
1.869
-1
1
-1
1.871
0
0
0
1.354
1
-1
1
2.343
2
0
0
1.955
-1
-1
1
1.911
0
0
0
1.363
0
0
-2
1.766
1
1
-1
1.988
1
-1
-1
1.890
-1
1
-1
1.756
-1
-1
-1
2.097
0
0
2
1.658
1
-1
1
2.274
1
1
1
1.316
1
-1
1
1.404
-1
1
1
1.305
-1
-1
1
1.328
0
0
0
1.383
0
-2
0
1.816
1
-1
1
2.099
-1
1
1
2.004
-1
-1
1
2.127
1
1
1
1.305
1
-1
1
1.334
-1
1
1
1.354
-1
-1
1
1.387
0
0
0
1.650
0
-2
0
1.662
0
0
0
1.650
-2
0
0
1.655
0
0
-2
2.251
1
1
-1
1.940
1
-1
-1
2.472
-1
1
1
2.092
-1
-1
1
2.040
-2
0
0
1.845
1
1
-1
1.339
1
-1
-1
1.335
-1
1
-1
1.379
-1
-1
-1
1.386
0
0
2
1.904
1
1
-1
2.436
-1
1
-1
1.987
0
0
2
1.184
0
0
0
1.363
0
0
-2
1.898
1
1
-1
1.801
1
-1
-1
1.801
-1
-1
1
1.905
1
1
1
1.360
1
-1
1
1.307
-1
1
1
1.318
-1
-1
1
1.346
0
0
0
1.363
0
2
0
1.813
1
1
1
2.159
1
-1
1
1.889
-1
1
1
1.894
-1
-1
1
2.032
0
0
0
1.651
0
-2
0
1.658
1
1
-1
2.503
-2
0
0
1.655
1
1
-1
1.351
1
-1
-1
1.353
-1
1
-1
1.300
-1
-1
-1
1.307
0
-2
0
1.915
1
1
-1
1.839
1
-1
-1
1.940
-1
1
1
1.752
-1
-1
1
1.721
-2
0
0
1.707
1
1
-1
1.337
1
-1
-1
1.341
-1
1
-1
1.365
-1
-1
-1
1.318
0
0
2
1.655
-1
-1
-1
2.902
0
0
2
1.849
1
1
1
1.990
1
-1
1
1.812
-1
1
-1
1.705
-1
-1
-1
1.757
0
2
0
1.945
1
1
1
2.533
1
-1
1
2.199
-1
1
1
1.804
-1
-1
1
2.276
0
0
0
1.355
0
-2
0
1.874
1
1
-1
2.280
1
-1
-1
1.716
-1
1
-1
1.779
-1
-1
-1
1.932
0
2
0
1.184
0
0
0
1.366
-2
0
0
1.145
1
1
1
2.010
1
-1
1
1.976
2
0
0
1.813
-1
1
-1
1.954
-1
-1
-1
1.851
0
0
0
1.340
1
-1
1
1.961
2
0
0
1.729
-1
1
-1
1.985
continued on next page...
82
C. Thresholds of Impact Ionization Processes in Silicon
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c, c’, c”, v’ valleys
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
G = (Gx , Gy , Gz )
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
-1
1
1
-1
-1
1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
-1
-1
1
1
-1
-1
1
0
0
0
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
0
0
0
1
-1
-1
-1
1
-1
-2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
1
1
-1
-1
1
1
-1
-1
-1
0
0
2
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
Eth [eV]
1.808
1.313
1.972
2.028
2.008
2.112
1.681
1.879
1.631
2.713
2.396
2.228
1.329
1.271
1.297
1.306
1.320
1.838
1.960
1.804
2.003
1.657
2.059
1.308
1.268
1.281
1.271
1.627
2.235
2.289
2.687
1.620
2.087
2.112
2.075
1.628
1.720
1.812
2.253
1.783
1.942
1.787
2.019
1.255
1.264
1.296
1.266
1.321
1.763
1.956
1.857
1.994
1.962
1.145
1.342
1.717
2.180
2.215
1.872
2.004
1.296
1.285
1.318
1.274
1.323
1.839
1.863
1.745
1.816
1.927
1.620
1.630
2.217
2.009
2.041
1.620
2.386
2.122
1.634
1.301
1.272
1.315
1.300
1.335
2.206
1.868
1.840
1.994
1.890
1.749
1.312
1.278
1.286
1.311
1.621
1.627
2.907
2.084
2.184
1.783
2.133
1.855
1.955
1.734
1.743
1.299
1.760
2.162
1.980
1.844
1.805
1.798
1.948
1.942
2.108
2.065
1.368
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c, c’, c”, v’ valleys
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
G = (Gx , Gy , Gz )
Eth [eV]
-2
0
0
1.753
0
0
2
1.852
1
1
1
2.254
1
-1
1
1.766
2
0
0
1.847
-1
1
-1
1.760
-1
-1
-1
1.787
0
0
0
1.620
0
0
-2
1.629
1
-1
1
2.225
-1
1
-1
2.116
0
0
0
2.047
1
1
-1
1.344
1
-1
-1
1.276
-1
1
-1
1.285
-1
-1
-1
1.319
0
2
0
1.823
1
1
1
2.109
1
-1
1
1.949
-1
1
1
1.960
-1
-1
1
1.712
-2
0
0
1.696
1
1
1
1.290
1
-1
1
1.284
-1
1
1
1.314
-1
-1
1
1.298
0
0
0
1.622
0
-2
0
1.629
1
-1
-1
2.470
-1
1
-1
2.282
-1
-1
-1
2.572
1
1
1
2.179
2
0
0
1.626
-1
1
-1
2.634
-1
-1
-1
2.744
0
0
0
1.329
0
0
-2
2.461
1
1
-1
1.950
1
-1
-1
1.741
-1
1
1
1.841
-1
-1
1
1.700
-2
0
0
1.797
1
1
1
1.301
1
-1
1
1.356
2
0
0
2.612
-1
1
-1
1.300
-1
-1
-1
1.318
0
0
2
1.815
1
1
1
2.053
1
-1
1
2.158
-1
1
1
1.898
-1
-1
1
2.051
0
0
0
1.368
0
0
-2
1.145
0
0
2
1.617
1
1
1
1.935
1
-1
1
2.226
-1
1
1
1.883
-1
-1
1
1.760
0
0
0
2.006
1
1
-1
1.312
1
-1
-1
1.327
-1
1
-1
1.252
-1
-1
-1
1.295
0
0
2
1.894
0
-2
0
1.888
1
1
-1
1.683
1
-1
-1
2.011
-1
1
-1
2.036
-1
-1
-1
1.844
0
2
0
1.623
1
1
1
2.533
1
-1
1
2.275
-1
1
1
2.460
-1
-1
-1
2.451
1
-1
1
2.539
2
0
0
1.629
-1
-1
-1
2.478
0
0
0
2.041
1
1
-1
1.350
1
-1
-1
1.299
-1
1
-1
1.292
-1
-1
-1
1.376
0
2
0
1.981
1
1
1
1.916
1
-1
1
1.864
2
0
0
1.706
-1
1
-1
1.686
-1
-1
-1
1.853
0
0
0
2.006
1
1
-1
1.310
1
-1
-1
1.294
-1
1
-1
1.287
-1
-1
-1
1.271
0
0
2
1.633
1
1
1
2.304
1
-1
1
2.327
-1
1
-1
2.744
0
0
0
1.321
0
0
-2
1.831
0
-2
0
1.805
1
1
-1
1.824
1
-1
-1
1.977
-1
1
-1
1.766
-1
-1
-1
2.045
0
2
0
1.876
1
1
1
1.971
1
-1
1
1.748
-1
1
1
1.748
-1
-1
1
1.690
0
0
0
1.340
0
-2
0
1.826
1
1
-1
1.912
1
-1
-1
2.014
-1
1
-1
1.884
-1
-1
-1
1.711
0
2
0
1.145
continued on next page...
C.2. Electron Initiated Processes
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c, c’, c”, v’ valleys
4
4
4
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
G = (Gx , Gy , Gz )
0
-2
0
2
0
0
-2
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
83
Eth [eV]
1.145
1.142
1.143
1.943
1.788
1.833
1.958
1.740
1.445
2.128
1.717
1.959
1.753
1.943
1.344
1.683
1.719
1.875
1.792
1.911
1.750
1.490
1.460
2.125
2.181
2.436
2.005
2.235
1.228
1.182
1.215
1.207
1.729
1.905
1.723
1.795
1.829
1.695
1.961
1.220
1.227
1.201
1.203
1.460
2.350
1.983
2.299
2.115
1.491
2.351
2.065
2.184
2.392
1.459
1.787
1.851
1.909
1.725
1.831
1.788
1.953
1.193
1.220
1.214
1.199
1.413
1.894
1.765
1.781
1.788
2.331
1.142
1.259
1.433
2.120
1.811
1.834
1.937
1.183
1.237
1.199
1.204
1.269
1.944
1.686
1.761
1.867
1.819
1.745
1.459
2.046
2.082
1.991
2.155
1.491
2.015
2.046
2.134
2.139
1.459
1.229
1.213
1.244
1.206
1.280
1.963
1.743
1.780
1.763
1.786
1.745
2.471
1.216
1.214
1.203
1.221
1.459
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c, c’, c”, v’ valleys
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
G = (Gx , Gy , Gz )
Eth [eV]
0
0
0
1.348
-1
-1
1
2.154
0
0
0
1.257
1
1
1
2.320
1
-1
1
1.710
2
0
0
1.375
-1
1
-1
1.878
-1
-1
-1
1.893
0
0
0
1.266
1
1
1
1.751
1
-1
1
1.841
2
0
0
1.493
-1
1
-1
1.824
-1
-1
-1
1.787
0
0
0
1.273
0
0
-2
1.713
1
1
-1
1.873
1
-1
-1
2.014
-1
1
1
1.829
-1
-1
1
1.694
-2
0
0
1.795
0
0
2
1.460
1
1
1
2.169
1
-1
1
2.119
-1
1
1
2.055
-1
-1
1
2.222
0
0
0
1.947
1
1
1
1.198
1
-1
1
1.197
-1
1
1
1.190
-1
-1
1
1.226
0
0
0
1.263
0
-2
0
1.645
1
1
-1
1.897
1
-1
-1
2.081
-1
1
1
1.790
-1
-1
1
2.019
-2
0
0
1.716
1
1
1
1.180
1
-1
1
1.240
-1
1
1
1.214
-1
-1
1
1.214
0
0
0
1.491
0
-2
0
1.460
1
1
-1
1.828
1
-1
-1
2.059
-1
1
-1
2.022
-1
-1
-1
2.006
1
1
1
2.102
1
-1
1
2.082
2
0
0
1.460
-1
1
-1
1.998
-1
-1
-1
2.399
0
0
0
1.263
0
0
-2
1.827
1
1
-1
2.041
1
-1
-1
1.771
-1
1
1
1.967
-1
-1
1
1.748
-2
0
0
1.692
1
1
1
1.197
1
-1
1
1.201
-1
1
1
1.185
-1
-1
1
1.227
0
0
0
1.269
0
0
-2
1.523
1
1
-1
1.832
1
-1
-1
1.908
-1
1
-1
1.896
-1
-1
-1
2.109
0
0
0
1.348
0
0
-2
1.142
0
0
2
1.420
1
1
1
1.859
1
-1
1
1.909
-1
1
1
1.977
-1
-1
1
1.793
0
0
0
1.960
1
1
-1
1.222
1
-1
-1
1.218
-1
1
-1
1.216
-1
-1
-1
1.220
0
0
2
1.785
0
2
0
1.730
1
1
1
1.846
1
-1
1
1.903
-1
1
1
1.985
-1
-1
1
1.652
0
0
0
1.491
0
-2
0
1.460
1
1
-1
2.264
1
-1
-1
2.061
-1
1
-1
2.090
-1
-1
-1
2.225
1
1
1
2.033
1
-1
1
2.121
2
0
0
1.461
-1
1
-1
2.157
-1
-1
-1
2.214
0
0
0
1.959
1
1
-1
1.232
1
-1
-1
1.220
-1
1
-1
1.184
-1
-1
-1
1.199
0
2
0
1.732
1
1
1
1.761
1
-1
1
1.890
2
0
0
1.702
-1
1
-1
1.863
-1
-1
-1
1.715
0
0
0
1.956
1
1
1
1.210
1
-1
1
1.211
-1
1
1
1.198
-1
-1
1
1.215
0
0
0
1.490
0
0
-2
1.460
continued on next page...
84
C. Thresholds of Impact Ionization Processes in Silicon
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
v’
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c, c’, c”, v’ valleys
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
G = (Gx , Gy , Gz )
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
0
0
0
-2
0
0
0
0
2
1
-1
-1
-1
1
-1
0
0
0
-1
-1
1
0
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
0
0
0
-2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
-1
1
-1
-2
0
0
0
0
2
0
0
0
1
1
-1
1
-1
-1
-1
-1
1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
0
0
-2
0
2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
0
0
0
0
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
-2
0
1
1
-1
-1
1
-1
0
0
0
0
-2
0
2
0
0
0
0
0
1
1
1
1
-1
1
2
0
0
-1
-1
1
-2
0
0
0
2
0
1
-1
1
2
0
0
0
0
0
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
0
0
0
-2
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
Eth [eV]
2.232
2.091
2.087
2.100
1.259
1.790
1.932
1.887
1.718
1.770
1.791
1.426
1.920
1.742
2.052
1.898
2.318
1.366
1.969
1.781
1.745
1.800
1.348
1.142
1.411
1.420
2.518
2.255
2.562
2.483
2.182
1.768
1.793
1.459
1.423
1.467
1.380
1.707
1.466
1.370
1.403
1.548
1.789
1.787
1.790
1.480
1.648
1.461
1.427
1.950
2.460
2.239
2.704
1.420
2.501
2.093
2.244
2.266
1.655
1.478
2.037
1.558
1.725
1.787
1.791
1.791
1.522
1.486
1.431
1.436
1.731
1.837
2.021
1.771
1.460
1.792
1.744
2.600
2.169
2.205
2.142
2.658
3.130
2.316
2.226
1.415
1.420
1.390
1.955
2.474
2.114
2.754
2.388
2.877
2.438
2.447
2.760
1.632
2.240
1.717
1.391
1.354
1.380
1.342
1.631
1.390
1.589
1.384
1.424
1.717
1.717
1.716
1.718
2.092
1.388
1.564
1.379
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
v’
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c, c’, c”, v’ valleys
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
G = (Gx , Gy , Gz )
Eth [eV]
1
1
-1
2.194
1
-1
-1
2.173
-1
1
-1
1.985
-1
-1
-1
2.023
0
0
2
1.795
0
2
2
2.615
1
1
1
1.932
1
-1
1
1.728
-1
1
1
1.878
-1
-1
1
1.845
0
0
0
1.269
0
-2
0
1.396
1
1
-1
1.860
1
-1
-1
1.869
-1
1
1
1.884
-1
-1
1
1.904
0
0
0
1.260
0
-2
0
1.485
1
1
-1
1.965
1
-1
-1
1.776
-1
1
-1
1.705
-1
-1
-1
1.763
0
2
0
1.143
1
-1
-1
2.726
2
0
0
1.420
0
0
0
2.594
1
1
-1
2.224
-1
1
1
2.130
-1
-1
1
2.294
1
-1
1
2.076
-1
-1
-1
2.140
0
0
0
1.786
0
0
-2
1.793
1
1
-1
1.459
1
-1
-1
1.456
-1
1
-1
1.391
-1
-1
-1
1.848
1
1
1
1.783
1
-1
1
1.830
-1
1
1
1.592
-1
-1
1
1.410
0
0
0
1.788
0
-2
0
1.793
2
0
0
1.795
0
0
0
1.787
1
1
-1
1.521
1
-1
-1
1.598
-1
1
-1
1.767
-1
-1
-1
1.719
1
1
1
2.368
1
-1
-1
2.275
-1
-1
-1
2.563
0
0
0
1.416
0
0
-2
1.421
1
1
1
2.433
1
-1
1
2.185
-1
1
1
2.153
-1
-1
-1
2.126
1
1
-1
2.116
1
-1
-1
1.727
-1
1
-1
1.518
-1
-1
-1
1.413
0
0
2
1.894
0
2
0
1.793
0
0
0
1.788
-2
0
0
1.790
1
1
-1
1.573
1
-1
-1
1.468
-1
1
-1
1.611
-1
-1
-1
1.662
1
1
1
1.405
1
-1
1
1.460
-1
1
1
1.559
-1
-1
1
1.432
0
0
0
1.787
0
0
-2
1.795
0
-2
0
2.335
1
1
1
2.309
1
-1
1
2.110
-1
1
1
2.435
-1
-1
1
2.271
0
0
0
2.173
1
1
1
2.101
1
-1
1
2.535
-1
-1
1
2.165
0
2
0
1.420
0
0
0
1.381
-2
0
0
1.389
0
0
-2
2.427
1
1
-1
2.312
1
-1
-1
2.503
-1
1
-1
2.209
-1
-1
-1
2.278
0
0
0
2.438
1
1
-1
2.180
1
-1
-1
2.182
-2
0
0
2.776
2
0
0
1.882
0
0
0
1.716
0
0
-2
1.718
1
1
-1
1.404
1
-1
-1
1.381
-1
1
-1
1.513
-1
-1
-1
1.533
2
0
0
2.295
1
1
-1
1.431
1
-1
-1
1.402
-1
1
-1
1.433
-1
-1
-1
1.413
0
2
0
1.718
1
1
1
2.585
2
0
0
1.718
0
0
0
1.639
1
1
1
1.496
1
-1
1
1.366
-1
1
1
1.452
-1
-1
1
1.381
continued on next page...
C.2. Electron Initiated Processes
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c, c’, c”, v’ valleys
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
G = (Gx , Gy , Gz )
-1
-1
-1
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
0
0
0
0
0
-2
1
1
1
1
-1
1
-1
1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
0
0
-2
0
0
0
0
-1
-1
1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
0
0
0
0
0
0
0
-2
0
1
1
-1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
2
0
0
0
0
0
1
1
-1
2
0
2
-1
1
-1
-2
0
0
1
1
-1
2
0
0
-2
0
0
1
1
1
2
0
0
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
-1
-1
-1
1
-2
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
-1
-1
2
0
0
0
0
0
0
0
-2
-1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
1
-1
1
-1
-1
-1
0
0
2
0
0
0
0
0
-2
1
-1
-1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
1
-1
-1
0
2
0
1
-1
1
2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
-1
1
1
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
0
0
0
0
-2
0
-1
1
-1
0
2
0
1
1
-1
-1
1
1
-1
-1
1
85
Eth [eV]
1.401
2.953
2.376
2.478
2.514
2.239
2.425
1.390
2.366
2.600
2.478
2.459
2.006
1.390
1.462
1.446
1.552
1.664
1.718
1.718
1.716
2.826
1.359
1.401
1.344
1.367
1.643
2.113
1.397
1.413
1.518
1.445
1.718
1.638
2.296
2.747
2.387
2.205
2.228
2.356
2.805
2.044
2.245
2.128
1.381
1.389
1.372
1.864
2.060
2.512
2.467
2.215
2.203
2.102
2.229
2.310
1.973
1.545
1.545
1.416
1.355
1.395
1.326
2.334
2.240
1.557
1.308
1.330
1.318
1.347
1.543
2.810
1.544
1.427
1.981
2.602
1.359
1.359
1.463
1.331
1.993
2.076
2.049
1.914
1.371
2.136
2.181
2.354
1.355
1.358
1.345
1.364
1.423
1.956
2.567
1.544
2.337
1.545
1.384
1.356
1.415
1.338
1.441
1.695
2.334
1.440
1.377
1.395
1.316
1.545
1.411
1.917
2.361
2.212
2.254
2.254
2.420
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c, c’, c”, v’ valleys
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
G = (Gx , Gy , Gz )
Eth [eV]
0
0
0
2.032
0
0
-2
2.775
1
1
-1
2.015
1
-1
-1
2.428
-1
1
-1
2.513
-1
-1
-1
2.058
0
0
0
1.381
0
0
-2
1.390
0
0
2
2.989
0
-2
0
2.571
1
1
-1
2.108
1
-1
-1
2.408
-1
-1
1
2.498
1
1
-1
1.494
1
-1
-1
1.441
-1
1
-1
1.370
-1
-1
-1
1.321
0
0
-2
2.254
0
2
0
1.718
-1
1
-1
2.610
2
0
0
1.717
-2
0
0
1.718
1
1
-1
1.428
1
-1
-1
1.514
-1
1
-1
1.419
-1
-1
-1
1.363
0
2
0
2.318
1
1
1
1.447
1
-1
1
1.359
-1
1
1
1.386
-1
-1
1
1.330
0
0
0
1.717
0
0
-2
1.718
0
-2
0
2.444
0
2
0
2.921
1
1
1
2.242
1
-1
1
2.289
-1
1
-1
2.457
-1
-1
-1
2.112
0
2
0
2.539
1
1
1
2.471
1
-1
-1
2.223
-1
1
-1
2.494
-1
-1
-1
2.268
0
2
0
1.389
0
0
0
1.363
-2
0
0
1.371
1
1
1
2.232
2
0
0
2.098
-1
1
1
1.965
-1
-1
-1
1.988
0
0
0
2.161
1
-1
1
2.400
-1
-1
-1
2.144
0
0
0
1.431
1
-1
1
2.626
-2
0
0
1.598
0
0
2
1.544
1
1
1
1.316
1
-1
1
1.344
-1
1
1
1.335
-1
-1
1
1.321
0
0
0
1.437
2
0
0
1.973
-1
-1
-1
2.265
1
1
1
1.305
1
-1
1
1.356
-1
1
1
1.411
-1
-1
1
1.365
0
0
0
1.544
0
-2
0
1.542
0
0
0
1.545
-2
0
0
1.546
0
0
2
1.979
1
1
-1
2.514
0
0
0
2.286
1
1
-1
1.357
1
-1
-1
1.338
-1
1
-1
1.395
-1
-1
-1
1.317
0
0
2
2.343
1
1
-1
2.142
-1
1
1
2.343
0
0
0
1.362
0
0
-2
1.371
0
0
2
2.171
1
1
-1
2.408
-1
1
-1
2.320
1
1
-1
1.358
1
-1
-1
1.390
-1
1
-1
1.342
-1
-1
-1
1.372
0
0
2
2.202
1
1
-1
2.353
0
0
0
1.542
0
-2
0
1.543
0
0
0
1.542
-2
0
0
1.543
1
1
-1
1.359
1
-1
-1
1.347
-1
1
-1
1.340
-1
-1
-1
1.342
0
2
0
1.944
1
1
1
2.718
1
1
1
1.378
1
-1
1
1.370
-1
1
1
1.336
-1
-1
1
1.363
0
0
0
1.544
0
0
-2
1.548
0
2
0
1.956
-1
1
1
2.611
0
0
0
2.202
0
-2
0
1.964
1
-1
1
2.203
-1
1
-1
2.092
-1
-1
-1
2.097
continued on next page...
86
C. Thresholds of Impact Ionization Processes in Silicon
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
c”
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
v’
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
c, c’, c”, v’ valleys
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
3
3
1
0
3
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
4
1
3
3
0
1
3
3
3
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
4
1
4
3
0
1
4
3
3
1
4
3
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
3
1
0
3
0
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
3
1
3
0
4
1
3
3
0
1
3
3
3
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
4
1
4
4
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
3
1
0
0
4
1
0
3
0
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
G = (Gx , Gy , Gz )
-2
0
0
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
0
0
0
0
-2
0
2
0
0
0
0
0
0
0
-2
0
0
0
0
-2
0
2
0
0
1
-1
-1
0
0
0
0
0
2
0
0
0
0
-2
0
2
0
0
0
0
0
1
-1
-1
0
0
0
0
0
-2
0
0
0
0
-2
0
2
0
0
0
0
0
0
0
0
0
0
2
1
1
1
1
-1
1
-1
-1
1
0
0
0
1
-1
1
-1
1
-1
0
0
0
0
-2
0
2
0
0
0
0
0
1
-1
-1
0
0
0
0
0
2
0
0
0
-1
1
1
-1
-1
1
0
0
0
0
2
0
0
0
0
-2
0
0
-1
1
-1
-1
-1
-1
1
1
1
0
0
0
0
0
-2
0
2
0
0
0
0
0
0
0
0
-2
0
2
0
0
0
0
0
2
0
0
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
-2
0
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
0
0
0
-2
0
0
0
0
2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
0
0
0
0
0
-2
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
0
0
-2
0
2
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
0
0
0
0
0
0
0
-2
0
0
2
0
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
-1
Eth [eV]
2.398
2.372
2.166
2.086
2.093
2.174
1.363
1.371
1.849
1.835
1.878
1.834
1.873
1.885
3.066
3.343
1.848
1.833
1.854
1.891
3.352
2.794
1.832
1.869
1.866
1.849
1.818
2.873
3.125
1.853
2.719
3.093
3.140
3.256
3.001
3.176
1.793
1.870
1.861
2.809
3.110
3.129
1.817
3.335
3.054
2.995
2.828
1.846
1.793
1.844
2.766
3.055
3.015
1.793
1.844
3.344
2.877
1.829
1.817
1.758
2.461
2.755
2.866
1.749
1.993
1.944
1.824
2.136
2.680
3.048
1.922
1.803
2.175
1.961
1.750
1.738
1.749
2.661
1.962
1.937
1.883
1.890
2.716
1.759
1.758
2.753
2.014
2.060
1.959
2.066
2.719
1.737
1.755
1.749
2.005
1.854
2.149
2.108
2.546
1.933
1.900
2.122
1.895
1.740
1.747
2.587
2.662
2.858
1.757
2.043
2.786
2.757
2.754
2.642
1.495
2.664
2.153
c
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
c”
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
v’
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
c, c’, c”, v’ valleys
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
0
0
0
1
0
0
0
1
0
3
3
1
0
4
3
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
4
1
3
3
3
1
3
3
3
1
3
4
4
1
4
0
0
1
4
0
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
4
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
4
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
4
1
3
0
4
1
3
3
3
1
3
3
3
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
4
1
4
3
0
1
4
3
3
1
4
3
4
1
4
4
0
1
4
4
3
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
4
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
4
1
4
4
0
1
4
4
3
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
G = (Gx , Gy , Gz )
Eth [eV]
0
0
0
2.046
0
2
0
2.157
0
-2
-2
2.426
1
-1
1
2.424
-1
1
1
2.132
-1
-1
1
2.057
0
2
0
1.371
0
0
0
1.865
-2
0
0
1.849
0
0
2
1.862
-1
-1
1
3.261
0
2
0
1.956
0
0
0
1.835
-2
0
0
1.864
-1
-1
1
3.433
0
0
0
1.865
0
0
-2
1.848
0
2
0
1.914
0
0
0
1.835
-2
0
0
1.871
1
1
-1
3.151
-1
1
1
3.268
0
0
2
1.962
0
0
0
3.614
0
2
0
1.849
0
0
0
1.829
-2
0
0
1.817
0
0
0
3.007
0
0
0
1.792
0
0
-2
1.808
1
1
-1
3.205
1
-1
-1
2.941
-1
-1
-1
2.923
1
1
-1
2.962
1
-1
-1
2.421
-1
-1
-1
3.089
0
2
0
1.832
0
0
0
1.795
-2
0
0
1.833
1
-1
1
2.508
-1
-1
-1
3.015
0
0
0
1.825
0
0
-2
1.818
1
-1
1
2.824
-1
1
-1
2.752
-1
-1
-1
3.179
0
0
0
1.793
0
-2
0
1.857
2
0
0
1.831
1
1
-1
3.293
-1
-1
1
2.609
0
0
0
3.456
-1
-1
1
2.950
0
0
2
1.823
0
0
0
3.191
0
0
0
2.876
0
2
0
3.141
0
2
0
1.818
0
0
0
1.756
-2
0
0
1.758
0
0
0
2.588
0
0
0
2.681
0
0
0
1.737
0
0
-2
1.748
1
1
-1
1.950
1
-1
-1
1.901
-1
1
-1
2.001
-1
-1
-1
2.086
2
0
0
2.707
1
1
1
2.210
1
-1
1
2.167
-1
1
1
1.929
-1
-1
1
2.092
0
0
0
1.738
0
-2
0
1.753
2
0
0
1.748
0
0
0
2.596
1
1
1
2.026
1
-1
1
1.903
-1
1
1
1.775
-1
-1
1
2.033
0
0
0
2.496
0
0
-2
3.230
0
0
2
1.758
0
0
0
2.651
0
0
-2
2.697
1
1
-1
1.980
1
-1
-1
1.871
-1
1
-1
1.911
-1
-1
-1
2.096
0
0
2
3.284
0
2
0
1.747
0
0
0
1.740
-2
0
0
1.747
1
1
-1
2.295
1
-1
-1
1.814
-1
1
-1
2.046
-1
-1
-1
2.200
0
-2
0
2.697
1
1
-1
1.931
1
-1
-1
1.815
-1
1
-1
1.838
-1
-1
-1
2.040
0
0
2
1.753
0
0
0
2.377
0
-2
0
2.685
0
2
0
2.679
0
0
0
1.756
0
-2
0
1.758
1
1
1
2.660
1
-1
1
2.765
2
0
0
1.495
-1
1
-1
2.750
-1
-1
-1
2.785
0
0
0
2.070
1
1
1
2.614
1
-1
1
2.311
continued on next page...
C.2. Electron Initiated Processes
c
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
c, c’, c”, v’ valleys
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
G = (Gx , Gy , Gz )
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
0
2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
87
Eth [eV]
2.644
2.191
2.376
2.179
2.745
2.319
2.250
2.415
2.620
2.266
2.416
2.525
2.672
2.712
2.648
2.610
1.829
2.267
2.441
2.440
2.543
2.458
2.747
1.187
1.186
1.189
1.189
1.866
2.233
2.531
2.609
2.681
2.432
2.155
2.542
1.187
1.189
1.186
1.195
2.321
2.614
2.614
2.581
2.526
1.798
2.422
2.618
2.498
2.245
2.235
2.892
2.544
2.553
2.481
2.727
2.397
1.187
1.188
2.370
1.187
1.190
2.050
2.256
2.262
2.269
2.350
2.507
2.046
1.495
2.638
2.763
2.678
2.675
2.207
2.675
2.200
2.136
2.263
2.364
3.035
1.187
1.188
1.190
1.195
1.960
2.242
2.486
2.392
2.406
2.429
2.222
2.606
2.308
2.469
2.505
1.855
2.477
2.466
2.457
2.534
2.319
1.187
1.187
2.670
1.187
1.188
1.824
2.639
2.299
2.525
2.468
2.520
2.302
2.431
1.191
1.189
1.196
c
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
c, c’, c”, v’ valleys
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
G = (Gx , Gy , Gz )
Eth [eV]
2
0
0
2.163
-1
1
-1
2.617
-1
-1
-1
2.584
0
0
0
2.085
1
1
1
2.431
1
-1
1
2.124
2
0
0
2.137
-1
1
-1
2.693
-1
-1
-1
2.104
0
0
0
1.852
0
0
-2
1.913
1
1
-1
2.384
1
-1
-1
2.523
-1
1
1
2.551
-1
-1
1
2.643
-2
0
0
2.636
0
0
2
1.948
1
1
1
2.513
1
-1
1
2.364
-1
1
1
2.604
-1
-1
1
2.418
0
0
0
2.256
0
0
-2
2.380
1
1
-1
1.188
1
-1
-1
1.197
-1
1
-1
1.194
-1
-1
-1
1.189
0
2
0
2.230
1
1
1
2.588
1
-1
1
2.390
2
0
0
2.902
-1
1
-1
2.370
-1
-1
-1
2.486
0
2
0
2.724
1
1
1
1.187
1
-1
1
1.187
-1
1
1
1.189
-1
-1
1
1.189
0
0
0
1.790
0
-2
0
2.299
1
1
-1
2.560
1
-1
-1
2.200
-1
1
-1
2.551
-1
-1
-1
2.563
1
1
1
2.616
1
-1
1
2.613
2
0
0
2.325
-1
1
-1
2.518
-1
-1
-1
2.487
0
0
0
1.761
1
1
1
2.625
1
-1
1
2.416
2
0
0
2.313
-1
1
-1
2.438
-1
-1
-1
2.323
0
0
0
2.190
1
1
-1
1.189
1
-1
-1
1.188
-1
1
1
1.193
-1
-1
1
1.188
-2
0
0
2.385
0
0
2
2.260
1
1
1
2.539
1
-1
1
2.295
-1
1
1
2.466
-1
-1
1
2.638
-2
0
0
2.763
0
0
2
1.496
1
1
1
2.653
1
-1
1
2.784
-1
1
1
2.259
-1
-1
1
2.642
0
0
0
1.945
0
0
-2
2.256
1
1
1
2.642
1
-1
1
2.154
-1
1
1
2.410
-1
-1
1
2.282
0
0
0
2.175
0
-2
0
2.428
1
1
-1
1.189
1
-1
-1
1.191
-1
1
-1
1.186
-1
-1
-1
1.187
0
2
0
2.259
1
1
1
2.609
1
-1
1
2.513
-1
1
1
2.538
-1
-1
1
2.388
0
0
0
1.805
0
-2
0
1.850
1
1
-1
2.484
1
-1
-1
2.518
-1
1
-1
2.381
-1
-1
-1
2.369
1
1
1
2.508
1
-1
1
2.187
2
0
0
2.052
-1
1
-1
2.561
-1
-1
-1
2.467
0
0
0
2.212
1
1
-1
1.188
1
-1
-1
1.189
-1
1
1
1.191
-1
-1
1
1.188
-2
0
0
2.632
0
2
0
2.646
1
1
1
2.409
1
-1
1
2.437
2
0
0
2.377
-1
1
-1
2.609
-1
-1
-1
2.492
0
0
0
2.196
0
0
-2
2.470
1
1
-1
1.193
1
-1
-1
1.195
-1
1
-1
1.195
continued on next page...
88
C. Thresholds of Impact Ionization Processes in Silicon
c
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c, c’, c”, v’ valleys
4
3
0
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
G = (Gx , Gy , Gz )
-1
-1
1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
1
-1
-1
1
-2
0
0
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
1
1
-1
1
-1
1
1
Eth [eV]
1.189
1.836
2.061
2.475
2.407
2.377
2.420
2.254
2.365
2.573
2.435
2.424
2.090
2.226
2.283
2.441
2.351
2.120
2.690
2.255
2.617
2.624
2.337
2.078
1.496
2.391
2.295
2.631
2.718
1.993
2.431
2.659
2.770
2.206
1.464
1.937
2.155
2.581
2.066
2.521
2.197
1.987
2.272
2.278
2.096
2.579
2.132
1.661
2.253
2.607
2.321
2.377
2.398
2.141
1.868
2.392
2.523
2.520
2.164
2.217
2.390
1.149
1.154
1.149
1.151
2.143
2.526
2.289
2.178
2.526
2.388
2.166
2.664
1.149
1.148
1.155
1.148
1.884
2.394
2.445
2.568
2.566
1.847
2.403
2.445
2.425
2.603
1.978
2.282
2.391
2.441
2.171
2.366
2.485
2.172
1.158
1.149
1.152
1.150
2.617
2.125
2.212
2.232
2.778
2.155
2.481
2.752
1.990
1.464
2.209
2.612
2.164
2.640
2.105
2.120
2.211
2.217
c
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
c, c’, c”, v’ valleys
4
3
0
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
3
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
0
4
4
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
0
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
3
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
0
4
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
0
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
3
3
4
1
G = (Gx , Gy , Gz )
Eth [eV]
-1
-1
-1
1.191
0
0
2
2.035
1
1
1
2.232
1
-1
1
2.544
-1
1
1
2.413
-1
-1
1
2.371
0
0
0
1.829
0
0
-2
2.036
1
1
-1
2.445
1
-1
-1
2.612
-1
1
-1
2.586
-1
-1
-1
2.726
0
2
0
2.123
1
1
1
2.182
1
-1
1
2.309
-1
1
1
2.331
-1
-1
1
2.213
0
0
0
2.052
0
2
0
2.240
1
1
1
2.578
1
-1
1
2.404
-1
1
1
2.386
-1
-1
1
2.445
0
0
0
2.048
0
-2
0
1.496
1
1
-1
2.489
1
-1
-1
2.649
-1
1
-1
2.778
-1
-1
-1
2.704
1
1
1
2.640
1
-1
1
2.696
2
0
0
1.464
-1
1
-1
2.634
-1
-1
-1
2.382
-3
1
-1
2.964
0
0
-2
2.780
1
1
-1
2.090
1
-1
-1
2.388
-1
1
1
2.337
-1
-1
1
2.280
-2
0
0
2.126
0
-2
0
2.689
1
1
-1
2.205
1
-1
-1
2.616
-1
1
1
2.229
-1
-1
1
2.378
-2
0
0
2.124
0
0
2
1.811
1
1
1
2.553
1
-1
1
2.510
2
0
0
2.624
-1
1
-1
2.454
-1
-1
-1
2.528
0
0
0
1.862
0
0
-2
1.883
1
1
-1
2.299
1
-1
-1
2.509
-1
1
-1
2.296
-1
-1
-1
2.458
0
0
2
2.434
1
1
1
1.147
1
-1
1
1.163
-1
1
1
1.157
-1
-1
1
1.146
0
0
0
1.759
0
-2
0
2.240
1
1
-1
2.481
1
-1
-1
2.463
-1
1
1
2.431
-1
-1
1
2.478
-2
0
0
2.229
0
2
0
2.394
1
1
1
1.147
1
-1
1
1.147
-1
1
1
1.150
-1
-1
1
1.146
0
0
0
1.737
0
-2
0
2.063
1
1
-1
2.340
1
-1
-1
2.357
-1
1
-1
2.097
-1
-1
-1
2.349
1
1
1
2.458
1
-1
1
2.540
2
0
0
1.829
-1
1
-1
2.474
-1
-1
-1
2.534
0
0
0
1.787
0
0
-2
2.300
1
1
-1
2.561
1
-1
-1
2.352
-1
1
1
2.272
-1
-1
1
2.382
-2
0
0
2.242
1
1
1
1.152
1
-1
1
1.146
2
0
0
2.663
-1
1
-1
1.157
-1
-1
-1
1.152
0
0
0
2.018
0
0
-2
2.126
1
1
-1
2.613
1
-1
-1
2.189
2
0
2
2.794
-1
1
-1
2.306
-1
-1
-1
2.129
-2
0
2
2.955
0
0
2
1.463
1
1
1
2.683
1
-1
1
2.528
-1
1
1
2.397
-1
-1
1
2.283
0
0
0
2.044
0
0
-2
2.122
1
1
-1
2.381
1
-1
-1
2.390
-1
1
-1
2.170
continued on next page...
C.2. Electron Initiated Processes
c
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c, c’, c”, v’ valleys
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
G = (Gx , Gy , Gz )
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
0
2
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
0
-2
0
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
2
0
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
0
2
1
1
1
1
-1
1
2
0
0
-1
1
-1
-1
-1
-1
0
0
0
0
0
-2
1
1
-1
1
-1
-1
-1
1
-1
-1
-1
-1
0
0
2
1
1
1
1
-1
1
-1
1
1
-1
-1
1
0
0
0
0
-2
0
1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
-2
0
0
0
2
0
1
1
1
89
Eth [eV]
2.247
2.350
2.310
1.154
1.146
1.153
1.153
3.142
2.128
2.407
2.526
2.227
2.445
2.438
2.296
2.355
2.370
2.262
2.490
1.814
2.416
2.403
2.604
2.375
2.090
1.148
1.150
2.324
1.147
1.150
1.777
2.347
2.288
2.210
2.446
2.393
2.222
2.275
1.150
1.151
1.148
1.149
2.319
2.564
2.142
2.413
2.379
1.844
2.246
2.889
2.332
2.234
2.452
2.408
2.128
2.132
2.219
2.920
2.618
2.144
1.958
2.674
2.095
2.257
2.362
2.247
2.121
1.463
2.583
2.291
2.626
2.281
1.862
2.582
2.347
2.467
2.466
1.419
2.677
2.246
2.330
1.876
2.088
2.060
1.880
2.922
2.000
2.430
2.269
2.472
1.875
2.283
2.357
2.318
2.311
2.393
2.461
1.787
1.943
2.321
2.135
2.165
2.363
2.239
1.143
1.141
1.142
1.145
1.717
1.922
2.271
2.409
2.607
2.387
2.561
2.197
1.147
c
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c, c’, c”, v’ valleys
3
3
4
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
0
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
3
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
3
4
4
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
0
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
3
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
0
4
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
0
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
3
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
3
4
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
0
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
3
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
4
4
4
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
3
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
0
4
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
0
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
3
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
3
4
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
0
1
0
4
3
1
0
4
3
1
0
4
3
1
G = (Gx , Gy , Gz )
Eth [eV]
-1
-1
-1
2.251
0
2
0
2.386
1
1
1
1.147
1
-1
1
1.147
-1
1
1
1.148
-1
-1
1
1.146
0
0
0
1.794
0
0
-2
2.170
0
2
-2
3.035
1
1
1
2.489
1
-1
1
2.373
-1
1
1
2.447
-1
-1
1
2.365
0
0
0
2.032
0
-2
0
2.054
1
1
-1
2.214
1
-1
-1
2.564
-1
1
-1
2.486
-1
-1
-1
2.366
1
1
1
2.490
1
-1
1
2.506
2
0
0
2.176
-1
1
-1
2.589
-1
-1
-1
2.466
0
0
0
2.236
1
1
-1
1.147
1
-1
-1
1.150
-1
1
1
1.156
-1
-1
1
1.146
-2
0
0
2.576
0
-2
0
2.623
1
1
-1
2.387
1
-1
-1
2.326
-1
1
1
2.348
-1
-1
1
2.410
-2
0
0
2.303
0
0
2
2.635
1
1
1
1.151
1
-1
1
1.149
-1
1
1
1.157
-1
-1
1
1.149
0
0
0
1.797
0
0
-2
1.846
1
1
-1
2.438
1
-1
-1
2.473
-1
1
-1
2.410
-1
-1
-1
2.457
0
0
2
2.162
0
2
0
2.283
1
1
1
2.405
1
-1
1
2.420
-1
1
1
2.445
-1
-1
1
2.414
0
0
0
1.956
0
-2
0
2.125
1
1
-1
2.141
1
-1
-1
2.353
-1
1
1
2.495
-1
-1
1
2.433
-2
0
0
2.787
0
0
2
2.666
0
2
0
2.128
1
1
1
2.406
1
-1
1
2.287
-1
1
1
2.328
-1
-1
1
2.359
0
0
0
1.993
0
-2
0
1.464
1
1
-1
2.181
1
-1
-1
2.520
-1
1
-1
2.223
-1
-1
-1
2.315
1
1
1
2.124
1
-1
1
2.209
2
0
0
1.419
-1
1
-1
2.298
-1
-1
-1
2.298
0
0
0
1.882
0
0
-2
2.687
1
1
-1
2.264
1
-1
-1
2.305
-1
1
1
2.308
-1
-1
1
2.226
-2
0
0
1.876
0
2
0
2.693
1
1
1
2.184
1
-1
1
2.453
2
0
0
1.874
-1
1
-1
2.247
-1
-1
-1
2.371
0
0
0
1.863
0
0
-2
2.162
1
1
-1
2.361
1
-1
-1
2.214
-1
1
1
2.420
-1
-1
1
2.367
-2
0
0
2.066
0
0
2
2.142
1
1
1
2.295
1
-1
1
2.192
-1
1
1
2.324
-1
-1
1
2.439
0
0
0
2.048
0
0
-2
2.247
1
1
-1
1.146
1
-1
-1
1.143
-1
1
-1
1.141
-1
-1
-1
1.144
0
2
0
2.162
1
1
1
2.211
1
-1
1
2.360
2
0
0
2.662
-1
1
-1
2.470
-1
-1
-1
2.395
0
0
0
2.177
0
-2
0
2.094
1
1
-1
1.143
continued on next page...
90
C. Thresholds of Impact Ionization Processes in Silicon
c
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
c’
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
c”
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
v’
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
c, c’, c”, v’ valleys
G = (Gx , Gy , Gz )
Eth [eV]
c
c’
c”
v’
c, c’, c”, v’ valleys
G = (Gx , Gy , Gz )
0
4
3
1
1
-1
1
1.141
4
3
3
2
0
4
3
1
1
-1
-1
0
4
3
1
-1
1
1
1.143
4
3
3
2
0
4
3
1
-1
1
-1
0
4
3
1
-1
-1
1
1.144
4
3
3
2
0
4
3
1
-1
-1
-1
0
4
4
1
0
0
0
1.702
4
3
3
2
0
4
4
1
0
2
0
0
4
4
1
0
-2
0
2.114
4
3
3
2
0
4
4
1
1
1
1
0
4
4
1
1
1
-1
2.320
4
3
3
2
0
4
4
1
1
-1
1
0
4
4
1
1
-1
-1
2.194
4
3
3
2
0
4
4
1
-1
1
1
0
4
4
1
-1
1
-1
2.419
4
3
3
2
0
4
4
1
-1
-1
1
0
4
4
1
-1
-1
-1
2.293
4
3
3
2
3
0
0
1
0
0
0
3
0
0
1
1
1
1
2.454
4
3
3
2
3
0
0
1
1
1
-1
3
0
0
1
1
-1
1
2.441
4
3
3
2
3
0
0
1
1
-1
-1
3
0
0
1
2
0
0
2.046
4
3
3
2
3
0
0
1
-1
1
1
3
0
0
1
-1
1
-1
2.351
4
3
3
2
3
0
0
1
-1
-1
1
3
0
0
1
-1
-1
-1
2.337
4
3
3
2
3
0
0
1
-2
0
0
3
0
3
1
0
0
0
1.747
4
3
3
2
3
0
3
1
0
0
2
3
0
3
1
0
0
-2
1.890
4
3
3
2
3
0
3
1
1
1
1
3
0
3
1
1
1
-1
2.268
4
3
3
2
3
0
3
1
1
-1
1
3
0
3
1
1
-1
-1
2.291
4
3
3
2
3
0
3
1
2
0
0
3
0
3
1
-1
1
1
2.372
4
3
3
2
3
0
3
1
-1
1
-1
3
0
3
1
-1
-1
1
2.380
4
3
3
2
3
0
3
1
-1
-1
-1
3
0
3
1
-2
0
0
1.891
4
3
3
2
3
0
4
1
0
0
0
3
0
4
1
1
1
1
1.141
4
3
3
2
3
0
4
1
1
1
-1
3
0
4
1
1
-1
1
1.144
4
3
3
2
3
0
4
1
1
-1
-1
3
0
4
1
2
0
0
2.479
4
3
3
2
3
0
4
1
-1
1
1
3
0
4
1
-1
1
-1
1.145
4
3
3
2
3
0
4
1
-1
-1
1
3
0
4
1
-1
-1
-1
1.142
4
3
3
2
3
0
4
1
-2
0
0
3
3
0
1
0
0
0
1.879
4
3
3
2
3
3
0
1
0
0
2
3
3
0
1
0
0
-2
1.874
4
3
3
2
3
3
0
1
1
1
1
3
3
0
1
1
1
-1
2.361
4
3
3
2
3
3
0
1
1
-1
1
3
3
0
1
1
-1
-1
2.207
4
3
3
2
3
3
0
1
2
0
0
3
3
0
1
-1
1
1
2.078
4
3
3
2
3
3
0
1
-1
1
-1
3
3
0
1
-1
-1
1
2.303
4
3
3
2
3
3
0
1
-1
-1
-1
3
3
0
1
-2
0
0
2.603
4
3
3
2
3
3
3
1
0
0
0
3
3
3
1
0
0
2
1.419
4
3
3
2
3
3
3
1
0
0
-2
3
3
3
1
1
1
1
2.273
4
3
3
2
3
3
3
1
1
1
-1
3
3
3
1
1
-1
1
2.141
4
3
3
2
3
3
3
1
1
-1
-1
3
3
3
1
-1
1
1
2.420
4
3
3
2
3
3
3
1
-1
1
-1
3
3
3
1
-1
-1
1
2.574
4
3
3
2
3
3
3
1
-1
-1
-1
3
3
4
1
0
0
0
1.880
4
3
3
2
3
3
4
1
0
0
2
3
3
4
1
0
0
-2
1.874
4
3
3
2
3
3
4
1
0
2
0
3
3
4
1
0
-2
0
2.632
4
3
3
2
3
3
4
1
1
1
1
3
3
4
1
1
1
-1
2.201
4
3
3
2
3
3
4
1
1
-1
1
3
3
4
1
1
-1
-1
2.324
4
3
3
2
3
3
4
1
-1
1
1
3
3
4
1
-1
1
-1
2.287
4
3
3
2
3
3
4
1
-1
-1
1
3
3
4
1
-1
-1
-1
2.387
4
3
3
2
3
4
0
1
0
0
0
3
4
0
1
0
2
0
2.438
4
3
3
2
3
4
0
1
0
-2
0
3
4
0
1
1
1
1
1.141
4
3
3
2
3
4
0
1
1
1
-1
3
4
0
1
1
-1
1
1.142
4
3
3
2
3
4
0
1
1
-1
-1
3
4
0
1
-1
1
1
1.148
4
3
3
2
3
4
0
1
-1
1
-1
3
4
0
1
-1
-1
1
1.142
4
3
3
2
3
4
0
1
-1
-1
-1
3
4
3
1
0
0
0
1.753
4
3
3
2
3
4
3
1
0
0
2
3
4
3
1
0
0
-2
2.143
4
3
3
2
3
4
3
1
0
2
0
3
4
3
1
0
-2
0
2.105
4
3
3
2
3
4
3
1
1
1
1
3
4
3
1
1
1
-1
2.383
4
3
3
2
3
4
3
1
1
-1
1
3
4
3
1
1
-1
-1
2.162
4
3
3
2
3
4
3
1
-1
1
1
3
4
3
1
-1
1
-1
2.390
4
3
3
2
3
4
3
1
-1
-1
1
3
4
3
1
-1
-1
-1
2.194
4
3
3
2
3
4
4
1
0
0
0
3
4
4
1
0
2
0
2.090
4
3
3
2
3
4
4
1
0
-2
0
3
4
4
1
1
1
1
2.316
4
3
3
2
3
4
4
1
1
1
-1
3
4
4
1
1
-1
1
2.429
4
3
3
2
3
4
4
1
1
-1
-1
3
4
4
1
-1
1
1
2.448
4
3
3
2
3
4
4
1
-1
1
-1
3
4
4
1
-1
-1
1
2.281
4
3
3
2
3
4
4
1
-1
-1
-1
4
0
0
1
0
0
0
1.745
4
3
3
2
4
0
0
1
1
1
1
4
0
0
1
1
1
-1
2.209
4
3
3
2
4
0
0
1
1
-1
1
4
0
0
1
1
-1
-1
2.465
4
3
3
2
4
0
0
1
2
0
0
4
0
0
1
-1
1
1
2.092
4
3
3
2
4
0
0
1
-1
1
-1
4
0
0
1
-1
-1
1
2.335
4
3
3
2
4
0
0
1
-1
-1
-1
4
0
0
1
-2
0
0
1.883
4
3
3
2
4
0
3
1
0
0
0
4
0
3
1
1
1
1
1.144
4
3
3
2
4
0
3
1
1
1
-1
4
0
3
1
1
-1
1
1.141
4
3
3
2
4
0
3
1
1
-1
-1
4
0
3
1
2
0
0
2.246
4
3
3
2
4
0
3
1
-1
1
1
4
0
3
1
-1
1
-1
1.142
4
3
3
2
4
0
3
1
-1
-1
1
4
0
3
1
-1
-1
-1
1.147
4
3
3
2
4
0
3
1
-2
0
0
4
0
4
1
0
0
0
1.844
4
3
3
2
4
0
4
1
0
2
0
4
0
4
1
0
-2
0
2.347
4
3
3
2
4
0
4
1
1
1
1
4
0
4
1
1
1
-1
2.423
4
3
3
2
4
0
4
1
1
-1
1
4
0
4
1
1
-1
-1
2.325
4
3
3
2
4
0
4
1
2
0
0
4
0
4
1
-1
1
1
2.307
4
3
3
2
4
0
4
1
-1
1
-1
4
0
4
1
-1
-1
1
2.252
4
3
3
2
4
0
4
1
-1
-1
-1
4
0
4
1
-2
0
0
1.765
4
3
3
2
4
3
0
1
0
0
0
4
3
0
1
0
0
2
2.635
4
3
3
2
4
3
0
1
0
0
-2
4
3
0
1
1
1
1
1.147
4
3
3
2
4
3
0
1
1
1
-1
4
3
0
1
1
-1
1
1.150
4
3
3
2
4
3
0
1
1
-1
-1
4
3
0
1
-1
1
1
1.142
4
3
3
2
4
3
0
1
-1
1
-1
4
3
0
1
-1
-1
1
1.141
4
3
3
2
4
3
0
1
-1
-1
-1
4
3
3
1
0
0
0
1.696
4
3
3
2
4
3
3
1
0
0
2
Table C.3: Electron initiated impact ionization processes in Si, sorted by band indices, valleys and umklapp vectors
Eth [eV]
1.141
1.143
1.144
1.909
2.389
2.494
2.342
1.953
1.740
2.323
2.298
2.277
2.351
2.113
1.909
2.248
2.240
1.957
2.493
2.275
2.173
1.141
1.141
1.141
1.141
2.259
1.873
2.436
2.282
2.902
2.246
2.606
1.871
1.419
2.212
2.486
2.247
2.622
1.876
2.731
2.326
2.197
2.279
2.398
2.167
2.296
1.155
1.143
1.146
1.141
1.931
2.230
2.173
2.120
2.241
2.429
1.738
1.807
2.188
2.344
2.428
2.151
2.376
2.383
1.848
2.315
2.262
2.162
1.142
1.147
1.153
1.142
2.125
1.881
2.236
2.233
2.052
2.273
2.134
2.181
2.234
1.143
1.145
1.140
1.143
1.857
Appendix D
Derivation of the Phonon–assisted Impact
Ionization Rate (in Progress)
In our derivations, we have so far only considered linear phonon coupling. According to
[Muljarov04], the term necessary to include quadratic phonon displacement would be
Hel−ph =
∑ Mq (aq + a†−q) − ∑0 Qq q0 (aq + a†−q) (aq0 + a†−q0 )
q
(D.1)
qq
= Hel−ph(1) + Hel−ph(2) .
(D.2)
Including this quadratic term, the idea is to obtain an expression for the phonon–assisted
impact ionization rate using a similar procedure as outlined in section 3.2.1.
First, we were able to derive an additional term for the electron–phonon scattering rate
as a byproduct. Setting up the Heisenberg equation for the quadratic part of the Hamiltonian
Hel−ph(2)
h
i
d †
(D.3)
ck ck |el−ph(2) = c†k ck , Hel−ph(2) ,
dt
we arrive at four–point operator terms. Applying the Heisenberg equation to these expressions
i h̄
again yields an equation of the form (3.6), which is solved by (3.7). The final result amounts to
d
fk |
(2)
dt el−ph
=
2π
h̄
∑0
0
|Qq q0 |2 δ E(k) − E(k0 ) + h̄(ωq − ωq0 ) · · ·
(D.4)
k ,q,q ,G
· · · fk nq nq0 + 1 − fk0 nq0 (nq + 1) + fk fk0 nq0 − nq δk,k0 −q+q0 +G
2π
+
|Qq q0 |2 δ E(k0 ) − E(k) + h̄(ωq − ωq0 ) · · ·
∑
h̄ k0 ,q,q0 ,G
· · · fk0 nq nq0 + 1 − fk nq0 (nq + 1) + fk0 fk nq0 − nq δk0 ,k−q+q0 +G ,
91
92
D. Derivation of the Phonon–assisted Impact Ionization Rate (in Progress)
where fk = hc†k ck i is the Fermi–Dirac distribution of the electrons and nq = ha†q aq i is the
Bose–Einstein distribution of the phonons. After this small correction to the electron–phonon
scattering rate, the effect of the quadratic coupling on the impact ionization rate shall be observed.
Setting up Heisenberg equations for c†k ck with Hel−ph(2) (equation (D.1)) and Wii (equation
(3.3)) however does not lead to a simple result where the odd part of δ− (Ω) cancels as it does
in section 3.2.1.
If a phonon–assisted impact ionization term in the Hamiltonian existed in addition to the
direct impact ionization term (D.3), the procedure presented in section 3.2.1 could be applied
in an analogous way. A possible additional term could look like
Hii,p =
∑
k0c ,k00c ,k0v ,kc ,q
† † †
†
∗
†
0
0
00
Mii,p ck00 ck0 dk0 ckc aq + Mii,p ckc dkv ckc ckc aq δkc ,k0c +k00c +k0v +q
c
c
v
(D.5)
with a matrix element Mii,p . The result of the calculation then leads to three expressions, which
could be interpreted as generation, inscattering and outscattering terms
d
fk |ii,p =
dt
2π
h̄
∑
k0c ,k0v ,kc ,q
|Mii,p |2 δ(E(kc ) − E(k0c ) − E(k) − Ev (k0v ) − Eg + h̄ωq ) (D.6)
δkc ,k0c +k+k0v +q fk0c fk
+ 2π
h̄
∑
k00c ,k0v ,kc ,q
2
|Mii,p |
∑
k0c ,k00c ,k0v ,q
(h)
fkc nq −
(h)
fk0 (1 + nq )
v
|Mii,p |2 δ(E(k) − E(k0c ) − E(k00c ) − Ev (k0v ) − Eg + h̄ωq )
δk,k0c +k00c +k0v +q fk0c fk00c
where fk and fk
v
δ(E(kc ) − E(k) − E(k00c ) − Ev (k0v ) − Eg + h̄ωq )
δkc ,k+k00c +k0v +q fk fk00c
− 2π
h̄
(h)
fkc nq − fk0 (1 + nq )
(h)
fk nq − fk0 (1 + nq ) ,
v
represent the electron and hole distribution functions respectively. Apart
from these absorption terms, there are obviously also emission terms which are omitted here for
brevity.
List of Figures
2.1
FCC Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2
Conventional cubic cell of the diamond lattice. . . . . . . . . . . . . . . . . . .
14
2.3
The first Brillouin zone for Silicon . . . . . . . . . . . . . . . . . . . . . . . .
17
2.4
Silicon band structure (empirical pseudo–potential method) along Λ and ∆. . .
18
4.1
Density of states for the first conduction band in bulk Si calculated with different integration routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.2
Random-k approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.3
Unit cube for interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.4
Algorithm Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.1
Electron Impact Ionization Rate in Silicon . . . . . . . . . . . . . . . . . . . .
43
5.2
Number of processes times density of states in 100 meV intervals . . . . . . . .
44
5.3
Impact ionization coefficient for electrons in Silicon at 300K . . . . . . . . . .
44
5.4
Impact ionization coefficient for electrons in Silicon at 300K, zoomed . . . . .
45
5.5
Hole Impact Ionization Rate in Silicon . . . . . . . . . . . . . . . . . . . . . .
46
5.6
Phonon dispersion in Silicon along (100). Fitted by [Pop04] to neutron scattering data from [Dolling63]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7
51
Phonon energy (a) in terms of temperature and mean modulus of the phonon
wave vector (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
52
94
LIST OF FIGURES
5.8
Phonon Scattering Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.9
Distribution function comparison between the 1- and 2 acoustic branches model
54
5.10 Hole drift velocities in Silicon using the 2 acoustic branches model . . . . . . .
54
6.1
Gap Energy in Six Ge1−x along Γ − X and Γ − L lines as a function of x . . . . .
58
6.2
Hole initiated impact ionization rate in strained Si . . . . . . . . . . . . . . . .
60
6.3
Electron initiated impact ionization rate in strained Si . . . . . . . . . . . . . .
60
B.1 Parabolic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
List of Tables
2.1
Silicon and Germanium Lattice Constants . . . . . . . . . . . . . . . . . . . .
14
3.1
Threshold energies for impact ionization in Si . . . . . . . . . . . . . . . . . .
27
5.1
Fitting parameters for the electron impact ionization rate in Si . . . . . . . . .
42
5.2
Matrix element, electron initiated impact ionization . . . . . . . . . . . . . . .
43
5.3
Fitting parameters for the hole impact ionization rate in Si . . . . . . . . . . .
46
5.4
Parameters describing the temperature dependence of the Si band gap . . . . .
48
5.5
Coupling constants and averaged phonon energies at 300K . . . . . . . . . . .
53
6.1
Band gap energies and impact ionization thresholds in strained Si . . . . . . . .
59
6.2
Fitting parameters for the hole impact ionization rate in strained Si . . . . . . .
59
A.1 Keywords: Data Types and Default Values . . . . . . . . . . . . . . . . . . . .
65
A.2 Keywords: Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
A.3 File Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
C.1 Band and valley index nomenclature . . . . . . . . . . . . . . . . . . . . . . .
73
C.2 Hole initiated impact ionization processes in Si, sorted by band indices, valleys
and umklapp vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
C.3 Electron initiated impact ionization processes in Si, sorted by band indices,
valleys and umklapp vectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
90
96
LIST OF TABLES
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Index
X valley, 27
Diamond lattice, 13
∆-axis, 15
Diamond structure, 15
Γ valley, 27
Distribution function, 34, 51
Λ-axis, 15
Divonne algorithm, 35, 38
Downhill simplex method, 26
Annihilation operator, 20
Drift velocity, 47, 52
Auger recombination, 19
Effective mass, 15
Avalanche photodiode, 19
Electron–phonon scattering, 51
Band gap, 27
Empirical pseudopotential method, 30
Temperature dependence, 47
Energy conservation, 25, 27
Band structure, 25
Face–centered cubic (fcc), 11
Silicon, 15
Fermi–Dirac distribution, 92
Basis vector, 15
Biaxial tensile stress, 58
Hartree decoupling, 21
Body–centered cubic (bcc), 15
Heisenberg equation, 21
Bose–Einstein distribution, 50, 92
Hole
Bravais lattice, 11
heavy, 15
Brillouin zone, 15, 26
light, 15
Hole–phonon scattering, 50
Conduction band, 15
Conventional unit cell, 13
Impact ionization, 19
Coordination number, 12
Coefficient, 28
Creation operator, 20
Phonon–assisted, 55
Rate, 24
Debye temperature, 47
Delta distribution, 29
Direct, 55
Density matrix, 21
Electron initiated, 41
Density of states, 23, 32, 49, 50
Hole initiated, 46
Importance sampling, 34
DESSIS, 63
101
102
INDEX
Integration routine, 37
SPARTA, 16
Intermediate states, 56
Strained Silicon, 58
Stratified sampling, 34, 35
Keldysh formula, 41
Threshold energy, 25, 27
Lattice constant, 13, 15
Transition probability, 48
Lattice temperature, 50
Least square fit, 42
Lorentz profile, 30
Umklapp vector, 27, 42
Valence band, 15
Matrix element, 47
Variance reduction, 35
Mobility, 52
Vegard’s rule, 57
Momentum conservation, 22, 27
Monte–Carlo integration, 32
Neutron scattering, 50
Normal distribution, 70
Numerical optimization, 35
Parabolic bands, 25
Phonon
Coupling constants, 50
Phonon dispersion, 50
Phonon energy, 50
Primitive unit cell, 12
Quadratic coupling, 92
Random-k, 22
Reciprocal lattice, 14
Scattering rate
Phonon, 48
Second order perturbation theory, 56
Silicon, 25, 41
Silicon–Germanium, 57
Simplex, 26
Weighting function, 48
Wigner–Seitz primitive cell, 13