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Design and simulation of 4H silicon carbide power bipolar junction transistors

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Design and simulation of 4H silicon carbide power bipolar junction transistors
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Niu, Xinyue
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English
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xii, 72 leaves : ; 28 cm

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Bipolar transistors ( lcsh )
Silicon carbide ( lcsh )
Bipolar transistors ( fast )
Silicon carbide ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 68-72).
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Xinyue Niu.

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|University of Colorado Denver
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Full Text
n
DESIGN AND SIMULATION OF 4H SILICON CARBIDE POWER
BIPOLAR JUNCTION TRANSISTORS
by
Xinyue Niu
B.E., Tianjin University, 2004
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2010


This thesis for the Master of Science
degree by
Xinyue Niu
has been approved
by
Hamid Fardi
Jan Bialasiewicz
I 2-/; IJ20IO
Date


Niu, Xinyue (M.S., Electrical Engineering)
Design and Simulation of 4H Silicon Carbide Power Bipolar Junction Transistors
Thesis directed by Professor Elamid Fardi
ABSTRACT
Because of its superior physical and electrical properties, including wide
band gap, high breakdown electric field and high thermal conductivity, 4H silicon
carbide (4H-SiC) has became the most attractive alternative to silicon as well as
to other semiconductor materials, for applications for high-voltage, high-power
density, high-temperature, and high-frequency devices. The 4H-SiC bipolar
junction transistor (BJT) is a promising wide band gap semiconductor switching
device for high temperature and high power applications. It has higher current
handling capabilities due to its bipolar character, and is free of the gate oxide
interface problems. Moreover, 4H-SiC BJT easily surpasses silicon BJT, with
higher current gain and much larger safe-operating-area, while it is also free of
the thermal breakdown problem. The purpose of our project is the design,
optimization, and performance predication of 4H-SiC NPN BJT by the way of
two-dimensional device simulations that include modeling parameters from the


most recent published literatures. Our study shows that the 4H-SiC NPN BJT has
higher performance relative to silicon BJT: In particular it has a larger blocking
breakdown voltage, increased current gain, and decreased specific on-resistance.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
Hamid Fardi


DEDICATION
I dedicate this thesis to my parents and my family, who are my keystone and
taught me the value of perseverance and resolve.


ACKNOWLEDGMENT
I would like to express my sincere gratitude to:
Professor Hamid Fardi, for having been teacher and advisor that
provided knowledgeable guidance and support. Dr. Fardi gave me
excellent guidance throughout this thesis.
Dr. Titsa Papantoni and Dr. Jan Bialasiewicz, for being committee
members for my thesis. I am grateful to the members for their valuable
time and advice in evaluating my thesis.


TABLE OF CONTENTS
LIST OF FIGURES..................................................ix
LIST OF TABLES...................................................xii
1 INTRODUCTION................................................1
1.1 The Reason of Choosing Silicon Carbide......................1
1.1.1 Overview of Power Semiconductor Devices.....................1
1.1.2 Wide-bandgap Semiconductors and Their Application in Power Devices
............................................................2
1.1.3 4H Silicon Carbide and Its Application in Power Devices.....6
1.2 Review of 4H-SiC Power BJT Research........................10
2 PHYSICAL MODELS AND DESIGN ISSUES FOR 4H-SiC DEVICE
SIMULATION................................................12
2.1 ATLAS Device Simulator.....................................13
2.2 4H-SiC Physical Models and Parameters......................15
2.2.1 Relative Dielectric Constant...............................15
2.2.2 Bandgap Model..............................................16
2.2.3 Mobility Models............................................17
2.2.4 Shockley-Read-Hall Recombination and Generation............18
2.2.5 Auger Recombination........................................19
2.2.6 Impact Ionization..........................................20
vii


2.2.7 Incomplete Ionization of Acceptors and Donors...............22
2.3 Some Design Issues of 4H-SiC Power Devices..................28
2.3.1 Critical Field of 4H-SiC....................................28
2.3.2 Drift Layer Design for Non-Punch-Through Structure..........29
2.3.3 Drift Layer Design for Punch-Through Structure..............33
3 SIMULATION OF 4H-SiC NPN BIPOLAR JUNCTION TRANSISTOR
............................................................38
3.1 Bipolar Junction Transistor fundamentals....................38
3.2 Temperature Coefficient of the Common Emitter Current Gain..41
3.3 Requirement for the Base Width and Doping Concentration.....46
3.4 4H-SiC NPN BJT Cell Structure...............................47
3.5 DC Characteristics..........................................48
3.6 Emitter Design..............................................51
3.7 Effects of Base Doping Concentration and Carrier Lifetime...53
3.8 Effects of the Surface Recombination........................59
3.9 Experimental Results and Discussions........................62
3.10 Summary.....................................................64
4 CONCLUSIONS AND SUGGESTIONS.................................66
REFERENCES.........................................................68
vm


LIST OF FIGURES
Figure 1.1 Applications of power semiconductor devices [1].................2
Figure 1.2 The width of the drift region for Si, GaAs, 6H-SiC, and 4H-SiC at
different breakdown voltages......................................8
Figure 1.3 The minimum specific on-resistance of the drift region for Si, GaAs,
6H-SiC, and 4H-SiC as function of breakdown voltage.............8
Figure 2.1 Ionization rate of Nitrogen in 4H-SiC as a function of doping
concentration (a) and temperature (b)............................26
Figure 2.2 Ionization rate of Aluminum in 4H-SiC as a function of doping
concentration (a) and temperature (b)............................27
Figure 2.3 Dependence of the critical field in 4H-SiC on the doping concentration.
...................................................................29
Figure 2.4 Theoretical specific on-resistance of silicon and 4H-SiC drift layers at
different voltage ratings.......................................32
Figure 2.5 Dependence of the breakdown voltage (a) and the depletion region
width at breakdown (b) on the drift layer doping concentration of
one-sided abrupt P^N junction...................................33
Figure 2.6 Comparison of punch-through structure with non punch-through
structure.......................................................34
Figure 2.7 Dependence of the breakdown voltages in 4H-SiC punch-through
structures on the drift region doping concentration.............35
Figure 2.8 Optimization of the drift region doping concentration and thickness for
a 14kV punch-through structure in 4H-SiC at 300K................35
Figure 2.9 Optimized doping concentration and thickness for the drift region of
4H-SiC punch-through structure..................................36
Figure 2.10 Comparison of the optimized specific on-resistance of 4H-SiC
punch-through structure with that of non punch-through structure.... 36
Figure 3.1 Basic NPN BJT structure with carrier distribution and internal current
components in the active region........................................39
IX


Figure 3.2 fi/C as a function of the temperature at different base doping
concentrations (a) and the temperature at d/3/dT = 0 as a function of
the base doping concentrations (b)................................45
Figure 3.3 Schematic cross-section view of the 4H-SiC NPN BJT cell structure 48
Figure 3.4 Simulated output DC characteristics of the 4H-SiC NPN BJT at 300K
and 523K..........................................................49
Figure 3.5 Simulated output common emitter current gain as a function of the
collector current density at different temperatures...............49
Figure 3.6 Simulated blocking characteristics of the 4H-SiC NPN BJT at 300K
and 523K..........................................................50
Figure 3.7 Simulated current gain /? as a function of the collector current density
at different base electron mobilities at 300K and 523K............50
Figure 3.8 Effects of the emitter width on the current gain of the 4H-SiC NPN
BJT at 300K.......................................................52
Figure 3.9 Effects of the emitter doping concentration on the current gain of the
4H-SiC NPN BJT at 300K............................................53
Figure 3.10 Current gain as a function of the collector current density at different
temperatures and carrier lifetimes when the base doping concentration
is 3.7 x 1017cm"3.................................................55
Figure 3.11 Current gain as a function of the base current density at different
temperatures and carrier lifetimes when the base doping concentration
is 3.7x 10,7c//T3.................................................57
Figure 3.12 Dependence of the current gain on the temperature at different carrier
lifetimes when Jb = 66.7A/cm2 and NAli = 3.7 x 1017cm3..........57
Figure 3.13 Current gain as a function of the carrier lifetime at 300K and a base
current density of 66.7 A/cm2.....................................58
Figure 3.14 Temperatures at zero temperature coefficient of the current gain as a
function of the maximum carrier lifetime under different base drive
current densities and base widths............................................58
x


Figure 3.15 Effects of the surface recombination on the current gain of the
4H-SiC NPN BJT at different bulk carrier lifetimes at 300K.....61
Figure 3.16 Surface recombination rate along the Si02/4H-SiC interface on the
bottom of the emitter mesa trench at 300K when Jb =66,7 A/cm Vce
=20V, rn0 = 200ns and s = 5 x 103 cm/s The device structure is
shown in Figure 3.3...................................................61
Figure 3.17 Measured and simulated output characteristics (Ig vs. Vc£) (a) and
transfer functions (Ie vs. Ib) (b) of the fabricated 4H-SiC NPN BJT at
room temperature, all measured data is taken from [46]............................64
xi


LIST OF TABLES
Table 1.1 Comparison of Mechanical and Electrical Properties of SiC and Other
Semiconductors [2] [3] [4]...................................3
Table 1.2 Comparison of the Normalized Figures of Merit of SiC and Other
Semiconductors...............................................5
Table 1.3 Important Reported 4H-SiC Power BJTs........................10
Xll


1 INTRODUCTION
1.1 The Reason of Choosing Silicon Carbide
1.1.1 Overview of Power Semiconductor Devices
Nowadays, power semiconductor devices play a key role in the regulation
and distribution of power and energy. Figure 1.1, where the boxes indicate the
device voltage and current ratings that a system requires, shows some of the
primary applications for power semiconductor devices [1]. It shows that the
device ratings span over a broad range of voltages and currents. Silicon-based
power devices dominate the power electronics and power system applications.
Several types of silicon devices are widely used, such as diodes (p-i-n and
Schottky rectifiers), gate turn-off thyristors (GTOs), gate-control thyristors
(GCTs), bipolar junction transistors (BJTs), insulated-gate bipolar junction
transistors (IGBTs), and so on. Although many improvements have been made in
silicon material technology and in the design of new device structures, the
silicon-based power devices are rapidly approaching their theoretical limits of
performance, such as operating at high junction temperature which is greater than
150 C, high voltage, high power densities, and high switching frequencies.
Silicon-based devices due to the inherent limitations of material properties, such
as narrow bandgap, low thermal conductivity and low breakdown field, are not
able to meet these stringent requirements. The only solution for silicon-based
devices is to design them with bulky costly cooling systems, large number of
1


devices in series and parallel, and costly active or passive snubbers. These
additions will add weight and overall size to the device. It becomes essential to
develop power devices from other materials in order to achieve further
improvement in the performance of power devices.
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DEVICE BLOCKING VOLTAGE RATING (V)
Figure l. I Applications of power semiconductor devices [ l ]
1.1.2 Wide-bandgap Semiconductors and Their Application in Power Devices
Wide-bandgap semiconductors, such as silicon carbide (SiC), gallium nitride
(GaN), semi-conducting diamond, form the most attractive alternative to silicon
because of the advantage in material properties like wider bandgap, higher
breakdown field, and higher thermal conductivity. A comparison of the properties
2


of some important semiconductors for high power devices is shown in Table 1.1
[2][3] [4], The four columns, which are on the right hand side of Table 1.1, are the
properties of wide bandgap semiconductors. When compared to silicon, these
wide bandgap semiconductors offer a lower intrinsic carrier concentration (9 to 37
orders of magnitude), a higher electric breakdown field (4 to 18 times), a higher
thermal conductivity, and a larger saturated electron drift velocity (2 to 2.7 times).
Table 1.1 Comparison of Mechanical and Electrical Properties of SiC and Other
Semiconductors [2] [3] [4]
Properties Si GaAs Wide Bandgap Semiconductors
GaN 3C-SiC 6H-SiC 4H-SiC
Bandgap, Eg (eV at 300K) 1.12 1.43 3.4 2.4 3.0 3.26
Critical field Ec (MV/cm) 0.25 0.3 3.0 2.0 2.5 2.2
Saturated electron drift velocity, Ka, (107cm/s) 1 1 2.5 2.5 2 2
Electron mobility, M (cm2/Vs) 1350 8500 400 1000 500 950
Hole mobility, lip (cm2/Vs) 480 400 30 40 80 120
Dielectric constant, 11.9 13.0 9.5 9.7 10 10
Thermal cond. X (W/cmK) 1.5 0.5 1.3 3.5-5.0 3.5-5.0 3.5-5.0
Density (g/cm3) 2.3 5.3 6.1 3.2 3.2 3.2
Melting point (C) 1420 1240 2500 2830 2830 2830
Various unipolar figures-of-merit (FOMs) have been proposed to evaluate the
performance improvement possible with wide-bandgap semiconductors. In 1965,
3


Johnson [5] derived a figure of merit, JFOM = (Ec-vsal/2nf, which assumes
the device performance of a typical discrete transistor is mainly limited by the
product of the critical breakdown field Ec and the electron saturation velocity vsat.
In 1972, Keyes [6] defined another figure of merit, KFOM X(c- vra/ /4ne^2,
which considers the thermal limitation to the switching speed of transistors for
integrated circuits. Here X is the thermal conductivity, c is the velocity of light,
and sr is the dielectric constant of the material. Later, Baliga [7] proposed a
figure of merit, BFOM = e-/u-E2, which considers the material parameters,
such as mobility ju and critical field Ec to minimize the conduction losses in
high-voltage power devices. BFOM is valid only for systems operating at low
frequencies where the conduction losses are dominant. For high frequencies,
Baliga proposed another figure of merit, BHFFOM = /u Ec2 [8], which
considers the switching losses due to charging and discharging of the device input
capacitance. The figures of merit for the materials shown in Table 1.1 are
presented in Table 1.2. Wide-bandgap semiconductors offer more than an order of
magnitude performance improvement over silicon.
Power devices based on wide-bandgap semiconductors have several
advantages, (a) Intrinsic temperature: For example, assuming a doping
concentration of 1015cT3, the intrinsic temperature, at which the intrinsic carrier
concentration reaches 2xl014cm-3, is approximately 1100 C for 4H-SiC, as
4


compared to 245 C for Si. This makes wide-bandgap semiconductors extremely
attractive for high-temperature applications. A suitable high-temperature
semiconductor technology could allow bulky aircraft hydraulics and mechanical
control systems to be replaced with heat-tolerant in situ control electronics.
On-site electronics, actuators, and sensors would reduce complexity and increase
reliability.
Table 1.2 Comparison of the Normalized Figures of Merit of SiC and Other
Semiconductors
Figures of Merit Definition Si GaAs Wide Bandgap Semiconductors
GaN 3H-SiC 6H-SiC 4H-SiC
JFOM (Ec'Vsa,/2*)2 1.0 7.1 760 65 400 260
KFOM A(CVsaJ 4^)'/2 1.0 0.45 1.6 1.74 5.02 5.1
BFOM e-M-E* 1.0 16 677 34.7 716 115
BHFFOM m-ec2 1.0 10.8 77.8 10.3 84 16.9
(b) Wide bandgap: Wide bandgap radiation does not degrade the electronic
properties of wide-bandgap semiconductors, so power devices based on wide
bandgap materials can be used in aerospace applications with reduced radiation
shielding.
(c) High electric breakdown field: The drift region can be much thinner than
that of their Si counterparts for the same voltage rating, thus a much lower
specific on-resistance could be obtained. With lower specific on-resistance,
wide-bandgap-based power devices have lower conduction losses and higher
overall efficiency.
5


(d) High saturated drift velocity: power devices based on wide-bandgap
materials could be switched at higher frequencies than their Si counterparts.
Moreover, charge in the depletion region of a diode can be removed faster if the
drift velocity is higher, and therefore, the reverse recovery time is shorter.
(e) High thermal conductivity: Higher thermal conductivity allows heat
generated in the power devices to be more easily transmitted to the case, heat sink
and then to the ambient. Thus power devices based on wide-bandgap materials
have high thermal stability.
1.1.3 4H Silicon Carbide and Its Application in Power Devices
Among the wide-bandgap materials, SiC is by far the most developed due to
the availability of high quality SiC substrates, advances in
chemical-vapor-deposition (CVD) growth of epitaxial structures and the ability to
easily dope the material n and p type.
Electronic devices made of 4H-SiC have shown operating temperature as
high as up to 600C [9]. 4H-SiC has the highest thermal conductivity among all
the popular semiconductors, better than copper. The heat generated in the
junctions could dissipate rapidly across the whole chip and pass to the heat sink.
Besides wide band gap and high thermal conductivity, the inert property also
makes 4H-SiC ideal for high temperature applications. In high temperature
applications, 4H-SiC is challenging Si today just as Si did to Ge in the 1960's, but
at a higher temperature level, although the cost of materiel growth will remain an
issue.
6


Breakdown electric field in 4H-SiC is almost one order of magnitude higher
than silicon, which makes 4H-SiC superior in high voltage applications. This high
breakdown electric field allows 4H-SiC power devices to become much thinner,
shown in Figure 1.2, where a higher-doped drift layer reduces the device specific
on-state resistance. Figure 1.3 shows the theoretical specific on-resistance versus
blocking voltage for several semiconductors [10]. For a given blocking voltage,
4H-SiC device could achieve about 400 times lower specific on-resistance
compared to its Si counterpart, reducing significantly the current conduction loss.
Because of high saturated drift velocity which is 2 times than the silicon,
4H-SiC power devices have higher current density and switches faster. Together
with its superior thermal conductivity, wide band-gap and low on-state resistance,
4H-SiC power devices could be much smaller in size while providing same
amount of power output.
it should be noted that the GaN also has some very attractive properties for
power electronics applications. Such as its wide band gap (3.4 eV vs. 4H-SiCs
3.26 eV), higher carrier mobility, higher breakdown electric field (3.0 MV/cm vs.
4H-SiCs 2.2 MV/cm) [11]. GaN technology is also developing very fast recently.
However, its processing stage is still much behind SiC technology in power
semiconductor fields. The biggest obstacle is that it is extremely difficult to obtain
native GaN substrate. So far, homogenous native GaN wafer is still not available
at production volume although 3-inch GaN substrate has been demonstrated [12].
The GaN wafer quality is also much behind SiC substrates with a dislocation
density of 106cm'2 [12]. Another problem for GaN research is that p-type GaN is
7


Figure 1.2 The width of the drift region for Si, GaAs, 6H-SiC, and 4H-SiC at
different breakdown voltages.
Figure 1.3 The minimum specific on-resistance of the drift region for Si, GaAs,
6H-SiC, and 4H-SiC as function of breakdown voltage.
very difficult to get, which significantly hampers the development of GaN power
8


devices. Although AlGaN/GaN HEMT shows promising properties [13], its
commercialization is still far behind SiC devices. For high power density
applications, homogeneous epitaxial SiC material is considered to be the best
choice for device implementation because of its commercial SiC wafer
availability and its high temperature stability [11].
The superior physical and electrical properties of SiC make it a promising
material for fabricating high voltage, high power, and high temperature devices.
Over 170 SiC polytypes have been proven in existence. The most commonly
known polytypes are 3C-SiC, 4H-SiC and 6H-SiC, the last two are commercially
available. A great progress has been made in monocrystalline SiC bulk crystal
growth. At present, commercial 4H-and 6H-SiC wafers are available at diameters
up to 75mm and 100mm for research and development samples, respectively,
from a number of manufacturers [14]. The micropipe densities are as low as
1.1cm'2 over an entire 50mm diameter 4H-SiC wafer. Increasing the wafer
diameter is crucial for reducing the cost and for commercializing SiC devices. For
SiC epitaxial growth, the growth rate is as high as 50jum/h and the doping
non-uniformity is less than 4.7% in commercial muti-wafer reactors.
4H-SiC shows even more potential for high power operation than 6H-SiC
because of its higher carrier mobility and its lower dopant ionization energy. The
power devices investigated in this thesis are based on 4H-SiC.
9


1.2 Review of 4H-SiC Power BJT Research
Table 1.3 Important Reported 4H-SiC Power BJTs
Emitter (Doping [cm3] /Thickness [tan ]) Base (Doping t cm"3 ] /Thickness [fim ]) Collector (Doping [ cm"3 ] /Thickness [tan]) P Wceo [kV] [ mSlcm1 ] Ref.
NA/0.75 2.5x1017/1 2.5xlO15 /20 20 1.8 10.8 [15]
lxlOl9/l 2x10I7/1 8x1014/50 15 3.2 78 [16]
NA/0.7 3x1017/0.8 6x10I5/12 32 1 17 [17]
lxl02O/l 8.5x10'7/1.4 7x1014/50 7 9.2 49 [18]
1 x10,9/0.8 2.3 x 1017/I 1 x 1015 /45 9 4 56 [19]
NA/1.5 2x1017 /I 4.8 x 1015/15 40 1 6 [20]
2xl019 /0.8 4.1 xl017/l 8.5xlOls/12 19 0.75 2.9 [21]
2.5 xlO19 /0.8 2.9x10I7/0.9 6.75x10I5/12 1 x1018/0.5 9 1.8 4.7 [22]
lxl019/0.8 2.3x1017/1 l.lxl 015 /45 3 6 28 [23]
3x10I9/2 4x10I7/1 4.8x10I5/14 70 1.2 6.3 [24]
5x10i9/0.1 1.5x1019/0.9 4x1017/0.7 4x1015/15 60 1.2 5.2 [25]
The first fabricated SiC BJT was demonstrated by Muench et al. in 1977.
4H-SiC BJTs started to receive more attention after the reported breakdown
voltage of 1.8 kV with a specific on-resistance of 10.8 mQcm2 in 2001 [15]. Table
1.3 lists most of the important 4H-SiC BJTs reported since 2001. Most of the
reported 4H-SiC BJTs have common emitter current gains, ft, in the range of
10-60 with a high breakdown voltage, BVCEO, in the range of 700-5000 V.
Recently, progress has been made in the development of 4H-SiC bipolar junction
10


2
transistors (BJTs) with an on-state resistance, RON, of only 2.9 mQcm for a
device blocking voltage of 0.75 kV [21]. A high breakdown voltage of 9.2 kV
y
with an on-state resistance of 49 mQcm has been reported [18].
11


2 PHYSICAL MODELS AND DESIGN ISSUES FOR 4H-SiC DEVICE
SIMULATION
Device simulation has gained more relevance for the development of
semiconductor power devices due to the increasing design complexity and need
of the reduction in the experimental batch cycles. Device simulation is just the
numerical solution of a system of coupled non-linear partial-differential equations,
which govern the charge transport in semiconductor. These equations include
Possion's equation, electron current continuity equation and hole current
continuity equation. By solving these equations, the electrical characteristics of a
semiconductor device with a specified structure at certain bias conditions can be
determined.
Device simulations are advantageous and essential for analyzing and
developing SiC power devices due to the following reasons. A principal
understanding of the performance and operation of a device can be obtained by
calculating and visualizing the internal profiles of fields, carrier concentrations
and current densities, which are not accessible with any experimental methods.
The complex design of a device can be optimized so that the cost and
experimental batch cycles can be substantially reduced. The effects of the
variations in material properties on the device performance can be predicted.
Comparison made with measured characteristics of a fabricated device, some of
the material properties can be extracted, which is a prerequisite for predictive
device simulations analysis. Finally, simulation results can be used to support the
interpretation of experimental data and to extract simplified models needed for
12


circuit simulations in design of VLSI system integration.
Today, multi-dimensional general-purpose device simulators are
commercially available, such as DESSIS from ISE Integrated Systems
Engineering, Inc, ATLAS from SILVACO International and MEDICI from
Synopsys, Inc. (formerly Technology Modeling Associates, Inc.-TMA) et al.
There are also various kinds of free software such as Stanford Universitys
PISCES, from which industrially developed versions have spawned. With the
material models and parameters in the simulators calibrated for SiC, these device
simulators can be used to simulate SiC power devices. In this thesis research, the
ATLAS device simulator is used to perform two-dimensional numerical
simulations for devices investigated in this thesis.
To obtain realistic simulation results, proper physical models and their
parameter values for 4H-SiC material must be applied in the simulator. In this
chapter, a set of physical models for 4H-SiC material is set up with the parameters
extracted from the most recent published literatures. And some design issues of
SiC power devices are discussed.
2.1 ATLAS Device Simulator
ATLAS is a multi-dimensional device simulator, which can handle ID, 2D
and true 3D structures. ATLAS solves numerically the three basic semiconductor
device equations, which govern the charge transport in semiconductors. These
equations are Poisson equation, electron and hole continuity equations. The
Poisson equation is stated as:
13


£V2if/ = -q(p-n + N+D-N-A)
(2.1)
where £ is the dielectric permittivity, y/ the electrostatic potential, q is the
elementary charge, n and p are the electron and hole densities and N+D and N~
are the ionized donor and acceptor impurity concentrations, respectively. The
electron and hole continuity equations are written as follows:
VJ,=qR + q^ (2.2)
at
V-Jp=qR + q?£. (2.3)
where R is the generation-recombination rate. Jn and Jp are the electron and
hole current densities, which are given by the following equations:
Jn = -qnii^h (2.4)
JP = -qpMPvP (2.5)
where pn and pp are the electron and hole mobilities, and electron and hole quasi-Fermi potentials.
The above partial-differential equations are solved numerically with a
numerical algorithm based on finite-element method. First, the device structure is
discretized to create a simulation mesh by applying the well known "box
discretization" method. Then, a set of coupled non-linear equations is generated
with the unknown potentials and free-carrier concentrations on each mesh nodes
as variables. Finally, the simulation results are given by solving this set of
equations with a nonlinear iteration method. In ATLAS, two iteration approaches,
14


Newton's and Gummel's methods are available. No matter which iteration method
is used, the iterations are performed over entire mesh nodes until a self-consistent
potential (y/) and free-carrier concentrations (n, p) are obtained. Once y/, n, and
p are available, the electron and hole currents, Jn and Jp, can be computed.
2.2 4H-SiC Physical Models and Parameters
The first systematic work on SiC material models and parameters for
numerical simulation was reported in 1994 [26] and 1997 [27] for 6H-and 4H-SiC,
respectively. These SiC material models and parameters are widely used in the
subsequent works on SiC device simulations. Recently, some new experimental
results on the material properties of 4H-SiC have been reported. These new
results should be considered in the simulations in order to achieve more accurate
results. In this research, the physical models pre-installed in the simulator,
ATLAS, are used in the simulations with the parameters tuned for 4H-SiC.
4H-SiC parameters are extracted from the recent published literatures. In the
following subsections, the critical physical models and material parameters of
4H-SiC will be described.
2.2.1 Relative Dielectric Constant
A search for the 4H-SiC revealed no experimentally confirmed relative
dielectric constant. The relative dielectric constant at low frequency for 6H-SiC is
reported by [28]:
15


sL = 9.76,
e = 9.98
(2.6)
Since 4H-SiC has somewhat larger bandgap than 6H-SiC, one may expect the
dielectric constants in 4H-SiC to be somewhat smaller than in 6H-SiC. In this
work, a dielectric constant of 9.8 is assumed for 4H-SiC.
2.2.2 Bandgap Model
The bandgap Eg of 4H-SiC is 3.26eV at 300K. according to data reported in
[4],. Since no reports exist on the temperature dependence of the 4H-SiC bandgap,
it is assumed that 4H-SiC has the same temperature dependence as that of 6H-SiC
where T is the lattice temperature in Kelvin.
To our knowledge, no model for doping-induced bandgap narrowing for SiC
has been reported. Thus, the Slotboom formula with parameters for Si is assumed
(2.7)
As a result, the 4H-SiC bandgap can be expressed by:
Eg (T) = 3.26 -3.3 x 10'4 (T 300K)eV
(2.8)
[29]:
(2.9)
where N is the doping concentration per cubic centimeter.
16


2.2.3 Mobility Models
As in Si, lattice scattering, and ionized impurity scattering, along with
(anisotropic) piezoelectric scattering, are the most relevant mechanisms to limit
the mean free path of carriers at low electric field in SiC [26], The low field
electron mobility jin and hole mobility np are modeled by the Masetti model
[30] as given in following equations:
950x(r/300)
-2.4
= 40 +
Mp = 15.9 +
1 + [A,/l.94xl 017]
124 x (77300)
-24
r / 17-i0.34
1 + [A,/1.76x1017]
cm2 V 1 s 1
cm2 V 1 s 1
(2.10)
(2.11)
where Nj denotes the total concentration of ionized impurities.
In high electric fields, the carrier drift velocity is no longer proportional to
the electric field E due to increased optical phonon scattering. The velocity
saturates to a finite value vsa, under high electric fields. The Canali model for
drift-diffusion simulation is used to describe this effect [31]:
fi(E)
Vi
1 + (/a£K;,)
nvp
(2.12)
where nhw is the low field mobility. vsat is carrier saturation drift velocity and is
2 x 107 cm/s at 300K for both electron and hole [2][3][4]. Since no temperature
dependent for the exponent [3 has been reported, it is assumed that 4H-SiC has
the same temperature dependence for (3 as that of silicon shown below [30]:
17


(2.13)
/?(r) = i.ix(r/300)66
^(7) = 1.213x(r/300)66 (2.14)
2.2.4 Shockley-Read-Hall Recombination and Generation
At present time not much has been reported about the energy levels,
concentrations and capture cross-sections for carrier lifetime controlling
recombination centers in SiC. Thus it is assumed that the effect of the
recombination centers can be described by the conventional Shockley-Read-Hall
(SRH) recombination/generation formula [32]:
^SRH ~ '
np-nj
n + n{ exp
ed-e,>
kT
+ T
p + exp
ed-e,
kT
(2.15)
where Rsrh is SRH recombination rate; n and p are electron and hole densities; n,
is the intrinsic carrier density; Ed is the energy level of the recombination centers;
rn and rp are electron and hole lifetimes. The doping dependence of the carrier
lifetimes is modeled with the Scharfetter relation [26] and the temperature
dependence is modeled with a power law [33]:
Tn,p
TiO.pO X
(T/300)
f N, ^
(2.16)
1 +
3x10
17
where Nt is the total concentration of ionized impurities and rn0 p0 is the electron
or hole lifetime in the material without impurities at 300K. The parameter a is
used to describe the temperature dependence of r Experimental data on
18


minority lifetimes in n-type 4H-SiC epilayer have been reported [34][35]. Their
reports show that the bulk carrier lifetime to be about 1.9-2.2pis. In [35], the
time resolved photoluminescence method was used to measure the minority
carrier lifetime in n-type 4H-SiC epilayers. In that report a value as high as
2.\/js was observed at room temperature. In the same experiment the
temperature was increased to 200C, the carrier lifetime was observed to increase
almost linearly to around 5jus. By fitting this experimental result, the parameter
a in Eq. (2.16) is found to be 1.72. However, it was also observed that there is a
very large variation in carrier lifetime in 4H-SiC. Even in the same sample, the
carrier lifetime decreases rapidly from the center towards the edge of the sample.
Therefore, the influence of carrier lifetimes on the device characteristics will be
investigated by varying the carrier lifetime within reasonable limits. If not
specified, rn0 is assumed to be 2.5jus. The relation rn0 = 5rp0, is employed in
the simulations.
2.2.5 Auger Recombination
Being a three-particle process, Auger recombination is only important at high
carrier concentrations. The band-to-band Auger recombination rate Ra is given by,
Ra =(cn + cPp)(nP-n,) (2-17)
where C and Cp denote the Auger coefficients of electrons and holes. The
coefficients for 4H-SiC used in the simulation [36][37]:
19


(2.18)
C = 5 xlO~il cm6/s
Cp = 9.9 x 1032 cm6/s (2.19)
2.2.6 Impact Ionization
In high electric field, free carriers can obtain enough energy to cause impact
ionization (avalanche generation). This process can be understood as the inverse
process to the Auger recombination. The reciprocal of the carrier mean free path
is called the impact ionization coefficient. The impact ionization and the
generation rate G can be expressed as
G = anwn + apnvp (2.20)
where an and ap denote the impact ionization coefficients of electrons and
holes, vn and vp are the electron and hole drift velocities, receptively. an and ap
are usually modeled by Chynoweth model [38]:
r
a(E) = ya0 exp -
v
yb
~E
(2.21)
where,
\ '
P
(2.22)
{2kT )
The factor y with the optical phonon energy hcoop expresses the temperature
dependence of phonon gas against the accelerated carriers. E is the electric field,
r(r)=-
2*7;
0 /
tonn
hco.
op
20


a0 and b are fitting parameters. T and T0 are the lattice temperatures in Kelvin,
and both ya0 and yb depend on temperature. The term yb is, however,
shown experimentally to be independent of the temperature in SiC [39]. Therefore,
the empirical model suggested by Okuto and Crowell will be used instead [40]:
'&(! + <* (7-300*))
cc(E) = a-[\ + c-(T-300K)]-Er' -exp-
Yi
(2.23)
where a, b, c, d, yl and y2 are fitting parameters.
Two sets of the experimental measurements on 4H-SiC impact ionization
coefficients were reported [39][41]. The hole impact ionization coefficient ap
reported in [39] was measured by using e-beam induced current (EBIC): a thin
value is much smaller than the one reported in [41] measured by direct
measurements of avalanche photodiodes. In support of these results, Monte Carlo
simulation shows that there is a significant anisotropy in the impact ionization
coefficients in 4H-SiC [42]. The impact ionization coefficients for transport
perpendicular to c-axis are from 5 to 10 times greater than the values for transport
parallel to c-axis. This implies that the breakdown voltage is mainly determined
by the breakdown perpendicular to c-axis if the electric field strengths in the
direction parallel and perpendicular to c-axis are approximately equal, as is the
case for most practical SiC devices. The anisotropy in the impact ionization
coefficients, however, has not been implemented in the simulator. It is customary
in the simulation to select a set of impact ionization coefficients between the
21


impact ionization coefficients in c-axis and perpendicular to c-axis. The measured
hole impact ionization coefficients reported in [39] are in very good agreement
with Monte Carlo simulation results in the direction of c-axis. The measured
impact ionization coefficients reported in [41] lie between the Monte Carlo results
parallel to c-axis and perpendicular to c-axis. Therefore the measured impact
ionization coefficients reported in [41] are used to predict the blocking voltage in
this research. Since no temperature dependence of the impact ionization
coefficient reported in [41], the measured temperature dependence reported in [39]
is used for the simulation. In this way, it is assumed that the temperature
coefficients of the impact ionization coefficients are the same in the direction
parallel to c-axis and perpendicular to c-axis. By fitting the experimental results
in with Eq. (2.23), the impact ionization coefficients an and ap are expressed
as below [39] [41]:
a (EJ) = 7.26 x 106 (l -1.47 x lO'3 (T 300K ))exp
ap (Ej) = 6.85 x 106 (l -1.56 x 10'3 (T 300£))exp
f 2.34 x 107 ^
1.41x10
7 A
cm
cm
(2.24)
(2.25)
2.2.7 Incomplete Ionization of Acceptors and Donors
The donor (Ed) and acceptor (EA) energy in SiC are relatively deep compared
to the thermal energy at room temperature. This will lead to incomplete ionization
of the impurities in SiC, even at high temperatures. The concentration of ionized
impurity atoms are given by
22


(2.26)
( E E \
l + 2exp -y" - 13
l kT J
(i
,and
(2.27)
where, Ep and EPp are the quasi-Fermi levels of electron and hole, respectively.
No and NA are the doping concentrations of donor and acceptor, respectively. The
degeneracy factor is assumed to be 2 and 4 for donor and acceptor, respectively.
The most common dopant for producing n-type SiC is nitrogen (N). Doping
with nitrogen leads to a donor (substituting on C sites) who has two different
energy levels below the conduction band. There are two nonequivalent C (or Si)
sites, one with a cubic (k) surrounding and the other with a hexagonal (h)
surrounding. Nitrogen atoms substituting on these sites therefore experience
somewhat different surroundings giving rise to different ionization energies [27].
In 4H-SiC, the two nitrogen donor ionization energies are [43]:
where Ec is the conduction band minimum. Eh and Ek are the ground state energy
levels of the hexagonal donor and cubic donor, respectively. These two donor
Ec-Eh=52.\meV, Ec -Ek = 9l.8meV
(2.28)
levels can be lumped together and replaced by a single effective donor ionization
energy by requiring Eq. (2.29) at 300k [27],
23


0-5 Nc
l+2JLexpl^sA.
o-5Nd
, n (Ec-Ek
Nc V kT
n ( Ec-Ed
1 + 2exp --
Nc ( kT
(2.29)
where Nc is the effective density-of-state for electrons and Np is the concentration
of nitrogen atoms. Solving this equation, the effective donor energy level Ep is:
Er-Eo=0.065eV (2.30)
Acceptors (Al, Ga or B) in 4H-SiC should also in principle have two different
energy levels corresponding to two nonequivalent sites. This energy difference
seems to be too small to be readily detectable. According to [27], the ionization
energy of Al in 4H-SiC is 191meV.
Under electrothermal equilibrium considerations, the Fermi energy is
constant, which equals the quasi-Fermi energies of electrons and holes. Hence,
the free electron and hole concentrations can be determined by solving the
equation of neutrality condition:
n + N-=p + N*D (2.31)
where NA and N+D are ionized acceptor and donor concentrations and n and p
free electron and hole concentrations, respectively. When there are only donors in
the semiconductor Eq. (2.31) is simplified to:
n = p + Np (2.32)
Since the 4H-SiC bandgap is usually much larger than the donor ionization energy,
the intrinsic carrier generations can be neglected, which means the hole
concentration in Eq. (2.32) can be neglected. Thus Eq. (2.32) becomes
24


(2.33)
= K
Under equilibrium condition, the electron concentration is given by
n-Ncexp
( ec-ef'
k0T
(2.34)
where Ep is the Fermi energy in equilibrium state. Nc is the effective state density
of conduction band, which is given by
Nc =2.54x19
19
f A
m
\mvJ
3/2
C p \3/2
cm
-3
(2.35)
.3007:,
where m0 is the electron mass and w* is the electron effective mass. Since few
measured on electron effective mass in 4H-SiC are reported, the default value of
0.61m0 for SiC in the simulator is used in this research. Solving Eqs. (2.26),
(2.34) and (2.33) simultaneously, we obtain:
R/V f F -F ^
(2.36)
N
n = NJ} = pexp
f E -E ^
k0T j
ii
Nr
exp
fEc~ED'
v KT
-1
Similarly, when there are only acceptors in the semiconductor, the hole
concentration p under equilibrium condition is given by
Nn
P = N~=-^QXP
( E -E ^
k0T
|1 + 1^
Ny
exp
V KT
-1
(2.37)
where, Ny is the effective state density of valance band, which is given by
/ y/2
Nv = 2.54 x 19
19
m
\mo J
x3/2
V
300/f
cm
-3
(2.38)
m'h is the hole effective mass. In this research, the default hole effective mass of
25


(a)
Temperature T(K)
(b)
Figure 2.1 Ionization rate of Nitrogen in 4H-SiC as a function of doping
concentration (a) and temperature (b).
1.0m0 for SiC in the simulator is used in this research. From Eqs. (2.36) and
26


(a)
Figure 2.2 Ionization rate of Aluminum in 4H-SiC as a function of doping
concentration (a) and temperature (b).
(2.37), the ionization rate of donors and acceptors can be determined. The
ionization rate of the donor Nitrogen in 4H-SiC at different temperatures and
27


doping concentrations is shown in Figure 2.1. It is seen that incomplete ionization
of nitrogen becomes relevant only at low temperatures and high doping
concentrations. More than 90% of nitrogen atoms are ionized for temperatures
above 250K and ND < 1016cm3. The ionization rate of the acceptor Aluminum in
4H-SiC at different temperatures and doping concentrations is presented in Figure
2.2. When NA = 1017cm3, only 18% of A1 atoms are ionized at 300K. The A1
ionization rate decreases with increasing doping concentration and decreasing
temperature, which finally leads to the freeze-out of holes at low temperatures.
2.3 Some Design Issues of 4H-SiC Power Devices
For power devices, the high blocking voltage is supported by a thick lightly
doped drift layer. Since the drift layer is thick and low-doped, its resistance
dominates the on-resistance of the power device. Thus, the first step for the design
of power devices is to choose a proper drift layer. This section discusses some
design issues of 4H-SiC power devices.
2.3.1 Critical Field in 4H-SiC
The avalanche breakdown due to impact ionization occurs when the electric
field in the semiconductor exceeds a certain value. This electric field is called the
critical field of the semiconductor. The critical field in 4H-SiC has been reported
in [41] as follows:
28


(2.39)
E,=
2.49 xlO6

N
vlOl6cm"3
V/cm
where N is the doping concentration. As shown in Figure 2.3, the critical field in
4H-SiC is dependent on the doping concentration. The critical field in 4H-SiC is
about 3MV/cm at a doping concentration 5 x 1016cm 3 10 times higher than that
in silicon.
Figure 2.3 Dependence of the critical field in 4H-SiC on the doping
concentration.
2.3.2 Drift Layer Design for Non-Punch-Through Structure
In the blocking state, the depletion region extends into the lightly doped drift
region, where the depleted drift region supports the high blockage voltage. For a
power device with an n-type lightly doped drift layer, the blocking junction can
29


be approximated with a one-side abrupt P+N junction. The maximum electric field
in the depletion region is given by:
2gNDva
(2.40)
where Va is the applied voltage, ND is the drift layer doping concentration and ss
is the dielectric constant of the semiconductor. From this equation, it can be seen
that the maximum electric field in the depletion region increases with increasing
the applied bias. When this field approaches the critical electric field, the
ionization coefficients become large and the junction approaches the breakdown
condition. Eq. (2.40) shows that for any applied bias, the electric field is smaller
for junctions with lower doping concentration (lightly doped drift layer).The
breakdown voltage can be increased by reducing the doping concentration of the
drift layer. When the maximum electric field in the depletion region reaches the
critical electric field of the semiconductor, the avalanche breakdown occurs. The
breakdown voltage Vbr can be derived from Eq. (2.40):
£ E
V -
V BR
2qN L
(2.41)
Therefore, for the non-punch-through structure, the doping level (No) required to
support a given breakdown voltage Vbr can be determined from Eq. (2.41):
ssE2cr
Nd =
2 qVt
(2.42)
BR
The drift layer thickness should be larger than the maximum width of the
30


depletion region at breakdown, which is given by:
w= 22vm
\ qND Ecr
(2.43)
Thus, the theoretical specific on-resistance Rsp on associated with the drift layer is
RSP ON = resistance x area =
W
4V,
BR
£sMnEa
(2.44)
where /un is the drift layer electron mobility. From the above equation the value
of VgR/Rsp ON is only dependent on the material properties:
<2-45>
Thus, the value of VlRjRsp ON is often used to evaluate how close the
performance of a fabricated power device approaches the material limit. Since the
critical field is known for a 4H-SiC device, the theoretical specific on-resistance
for 4H-SiC drift layer can be derived from Eqs. (2.39) and (2.44).
For silicon, the drift layer doping concentration and thickness required to
support a given breakdown voltage are given by [44] :
Nn = 2.01 x 1018F^/3 (2.46)
W = 2.58x10'6F-/^6 (2.47)
So the theoretical specific on-resistance of silicon drift layer in Qcm is
V The theoretical specific on-resistance for silicon drift layer and 4H-SiC drift
31


layer are presented in Figure 2.4. The specific on-resistance of SiC drift layer is
about 550 times lower than that of silicon drift layer for the same voltage rating
due to the higher critical field in SiC. Figure 2.5 presents the breakdown voltage
and the depletion region width at breakdown as a function of the doping
concentration No for 4H-SiC. For a given breakdown voltage, the required drift
layer doping concentration and thickness can be determined from the Figure 2.4.
Figure 2.4 Theoretical specific on-resistance of silicon and 4H-SiC drift layers
at different voltage ratings.
32


(a)
Doping Concentration ND (cm'3)
(b)
Figure 2.5 Dependence of the breakdown voltage (a) and the depletion region
width at breakdown (b) on the drift layer doping concentration of one-sided
abrupt P*N junction.
2.3.3 Drift Layer Design for Punch-Through Structure
For most power devices, it is preferable to use a punch-through structure to
support the voltage, as shown in Figure 2.6. In general, the punch-through
33


NON
PUNCH
THROUGH
Figure 2.6 Comparison of punch-through structure with non punch-through
structure.
structure has a lower doping concentration on the lightly doped side with a high
concentration contact region, and the thickness of the lightly doped side is smaller
than that for non punch-through structure of equal breakdown voltages, changing
the electric field distribution. In punch-through structure, the electric field varies
more gradually with distance within the lightly doped region, resulting in a
rectangular electric field profile as compared to a triangular electric field profile
for the non punch-through structure, as illustrated in Figure 2.6. The breakdown
voltage for punch-through structure is given by [44]:
qN'Wl
yBR=EJtrp-\-L. (2.49)
where Wp and N are the thickness and doping concentration of the drift region
(lightly doped region), respectively. Figure 2.7 shows the breakdown voltage
calculated for the punch-through structure in 4H-SiC as a function of the drift
region doping concentration. When the doping concentration and the thickness of
34


Figure 2.7 Dependence of the breakdown voltages in 4H-SiC punch-through
structures on the drift region doping concentration.
Figure 2.8 Optimization of the drift region doping concentration and thickness
for a 14kV punch-through structure in 4H-SiC at 300K.
the drift region increase, the breakdown voltage approaches that for the non
punch-through structure. In addition, the breakdown voltage of the punch-through
structure is a weak function of the drift layer doping concentration for small
35


120
Figure 2.9 Optimized doping concentration and thickness for the drift region of
4H-SiC punch-through structure.
Figure 2.10 Comparison of the optimized specific on-resistance of 4H-SiC
punch-through structure with that of non punch-through structure.
thickness.
For a given breakdown voltage, the drift region thickness and doping
concentration of punch-through structure can be optimized to give the lowest
specific on-resistance by using Eqs. (2.39), (2.44) and (2.49). Figure 2.8
36


illustrates such an optimization scheme performed for a breakdown voltage of
14kV at 300K. The optimum drift region thickness and doping concentration are
114/im and 6.6 x 1014cm~3 respectively, which gives the lowest specific
on-resistance of 117mQcm2. The optimum drift region thickness and doping
concentration for 4H-SiC punch-through structure at different breakdown
voltages are presented in Figure 2.9. And the optimum specific on-resistance is
compared with the theoretical specific on-resistance of non punch-through
structure in Figure 2.10. It is seen from Figure 2.10 that the optimized
punch-through structure not only has a thinner drift region, but also has a slightly
lower specific on-resistance than non punch-through structure. Hence, most
power devices utilize punch-through structure.
37


3 SIMULATION OF 4H-SiC NPN BIPOLAR JUNCTION TRANSISTOR
SiC bipolar junction transistors (BJTs) overcome several disadvantages of
silicon BJTs and offer important tradeoffs compared to SiC MOSFETs. They are
easy to fabricate, do not have the problems related to gate oxide, and offer low
on-resistance due to conductivity modulation and comparable switching loss and
lower conduction loss. Thus, SiC BJT attracts more and more attention and could
become a popular power device in the future. NPN structure is preferred for SiC
BJT because of the difficulties in making heavy p-type doping in SiC substrate. In
this chapter, a 4H-SiC NPN BJT structure will be investigated systematically at
different temperatures by performing two-dimensional device simulations.
3.1 Bipolar Junction Transistor fundamentals
The schematic structure of an NPN BJT is shown in Fig.3-1. The three
regions of the BJT are the emitter, the base, and the collector. These three regions
form two interacting PN junction diodes. In the normal operation, the emitter-base
junction is forward-biased and the collector-base junction is reversed-biased. In
this state, electrons from the N+ emitter are injected into the P-base, and the holes
are injected from the P-base into the emitter. The emitter current is comprised of
both of these two components. However, the portion of the emitter current due to
the hole injection into the emitter does not contribute to the collector current. In
the base, a very small percentage of the electrons injected into the base recombine
because the base width is designed to be much smaller than the electron diffusion
38


length, and the rest of the electrons are swept into the collector by the
reversed-biased base-collector junction. In Fig.3-1, the current Ie is the emitter
current component due to electron injection at the emitter-base junction and the
current Ic is the electron current at the collector-base junction. Thus, the common
base current gain can be written as:
Figure 3.1 Basic NPN BJT structure with carrier distribution and internal
current components in the active region.
In Eq. (3.1), the first term is referred to as the emitter injection efficiency yE,
which is a measurement of the ability of the emitter to inject electrons into the
base region. The second term in Eq. (3.1) is referred to as the base transport factor
aT, which is a measurement of the ability for electrons injected from the emitter
to reach the collector-base junction. Due to recombination in the base region, the
base transport factor is always less than unity. The third term in Eq. (3.1) is called
39


the collector efficiency yc, a measure of the ability for electrons to transport
through the collector. The collector efficiency is often assumed to be unity
because the electrons are swept out without loss by a strong electric field into the
collector. The common emitter current gain (3 can be expressed as
P _ h' ^
(3.2)
I k 11: I(: ^ a
Thus, the higher common emitter current gain requires higher emitter injection
efficiency and higher base transport factor. When no recombination is assumed in
the base region, the common emitter current gain is given by [44]:
fi =
'nB'l0B^pE
DpeP^Wb
(3.3)
where Db is the electron diffusion coefficient in the base region, DpE is the hole
diffusion coefficient in the emitter, LpE is the hole diffusion length in the emitter,
noB is the equilibrium electron concentration in the base, and Poe is the
equilibrium hole concentration in the emitter.
It is desirable to have a high current gain for power BJT in order to reduce the
base drive current. It is seen from Eq. (3.3) that the high current gain can be
achieved by using a very low base doping concentration and a very high emitter
doping concentration. However, an increase in the emitter doping may cause a
reduction in the hole diffusion length in the emitter, which cancel the effect of
increasing emitter doping concentration. A decrease in the base doping
concentration leads to a low reach-through breakdown voltage and a low output
40


conductance due to depletion of the base region, causing the base width not to be
too thin. The base width, the base doping concentration, and the emitter doping
concentration should be optimized in order to achieve the best performance for a
bipolar transistor.
3.2 Temperature Coefficient of the Common Emitter Current Gain
It is seen from Eq. (3.3) that, with the exception of the physical base width,
all of other parameters are a strong function of the temperature.
According to Einstein's relationship and Eqs. (2.10), (2.11) and (2.16), we
have
= -//sxr.^ =r(,+v) (3.4)

= pr oc r Ta>,p = t (3.5)
q p
(3.6)
where, a and a are the temperature coefficients of the electron mobility
and the hole mobility, respectively. In this research, it is assumed that a^ is
equal to aMp. The parameter azp is the temperature coefficient of the hole
carrier lifetime. In Eq. (3.3), hqb and Poe can be expressed as,
2

n0B
X
(3.7)
AB
41


,and
(3.8)
where NAB is the concentration of the ionized acceptor atoms in the base and
N+df is the concentration of the ionized donor atoms in the emitter, and ni is the
intrinsic carrier concentration of the material.
Thus, the temperature dependence of the common emitter current gain can be
expressed as:
Substitute Eqs. (2.36) and (2.37) into (3.9) and considering that Nc and Ny have
the same temperature dependence, the current gain ft in Eq. (3.3) can be
expressed as
where C is a constant which is independent on the temperature. For 4H-SiC, as
discussed in Chapter 2, aw=-2.4, ajp = 1.72, Ec-Eo =65meV for nitrogen,
and Ea-Ev =191meV for aluminum. It can be seen from Eq. (3.10) that with the
increase of the temperature, the decrease of the hole mobility tends to reduce the
(3.9)
(3.10)
42


current gain and the increase of the hole lifetime tends to increase the current gain.
Since (l + + aip = 0.16 > 0, the combining effect of the hole mobility and
lifetime is causing the current gain to increase when the temperature is increased.
If the impurity energy levels are sufficiently shallow, the temperature will have
almost no effect on the terms in Eq. (3.10) that are related to incomplete
ionization of the impurity atoms, meaning the temperature coefficient of the
current gain /? is positive. This is exactly the case of a silicon BJT. In 4H-SiC,
the impurity energy levels are relatively deep, which means the impurity atoms
are not completely ionized at room temperature. As a consequence of the
incomplete ionization, the electron concentration in the emitter and the hole
concentration in the base will increase with the temperature. Eq. (3.9) shows that
the increase of the hole concentration in the base tends to reduce the current gain
and. On the contrary, the increase of the electron concentration in the emitter
tends to increase the current gain. For 4H-SiC NPN BJT, it can be seen from
Figure 2.1b and Figure 2.2b that the hole concentration in the base increases
faster than the electron in the emitter due to the larger acceptor energy level in
4H-SiC. As a result, the incomplete ionization in 4H-SiC NPN BJT causes the
current gain to decrease when the temperature increases, which makes possible
the negative temperature coefficient of the current gain possible in 4H-SiC NPN
BJT.
A plot of y6/C as a function of the temperature at different base doping
concentrations is presented in Figure 3.2a. The sign of d/3/dT changes from
43


negative to positive when the temperature increases. The reason is that most of
the acceptors are ionized when the temperature is increased to a certain value.
Further increase of the temperature will not lead to a large increment in the hole
concentration in the base, meaning the effect of the incomplete ionization has
decreased. When the effect of the incomplete ionization is weaker than the effect
of the carrier lifetime, the sign of dp/dT becomes positive. In Figure 3.2a, the
points at which the temperature coefficient of the current gain P is zero (i.e.
dp/dT = 0), are indicated by the black dots connected with a thin line. The
temperature coefficient of P is negative on the left side of the thin line and
positive on the right side of the thin line. As shown in Figure 3.2b, when the base
doping concentration increases, the temperature at dp/dT = 0 increases,
indicating an increased temperature range for the negative temperature coefficient
of the current gain. When the emitter doping density increases, the temperature
range for the negative temperature coefficient of the current gain is reduced
slightly.
In the above discussions, some effects, such as high level injection in base,
base widening at high current densities, emitter current crowding, surface
recombination, and recombination in the depletion region of the forward-biased
emitter-base junction, are excluded. These effects should be considered because
they are present in the actual devices and are very important for power SiC BJTs.
In this thesis research, two-dimensional device simulations are applied to
44


Temperature T (K)
(a)
Base Doping Concentration Nw (xio^cnrf3)
(b)
Figure 3.2 fi/C as a function of the temperature at different base doping
concentrations (a) and the temperature at dp/dT = 0 as a function of the base
doping concentrations (b).
investigate the characteristics of a 4H-SiC NPN BJT. These effects will be
45


naturally included in the device simulations. This could be one of the advantages
of the device simulation.
3.3 Requirement for the Base Width and Doping Concentration
In power BJTs, the junction between the lightly doped collector and the base
supports the high breakdown voltage. At breakdown, the maximum electric field
in the junction approaches the critical electric field, Ecr. So, the width of the
depletion region on the base side at breakdown is given by:
(3'H)
<1Na
where Na is the base doping concentration. Therefore, to prevent the
punch-through of the base region, the base width and doping concentration must
satisfy the following condition:
(3-12)
The term ssEcrlq in Eq. (3.12) is actually the total charge inside the depletion
region on the base side when the breakdown occurs. In Eq. (3.12), the depletion
region width of the emitter-base junction is not considered because the
emitter-base junction is normally forward-biased and the depletion width is small.
It is interesting to see that the value of ssEcJq decreases when the breakdown
voltage increases due to the decrease in critical field. This means, for the same
base doping concentration, the base width can be smaller for high-voltage power
46


BJT than for low-voltage power BJT.
3.4 4H-SiC NPN BJT Cell Structure
Figure 3.3 shows the schematic cross sectional view of the 4H-SiC NPN BJT
cell structure, which is studied in this research. This structure consists of three
epilayers. The top N+ layer is the emitter. The middle p-type epilayer is the base.
The N' layer (drift layer) between the N+ collector and the P base is used to
support the high breakdown voltage. The emitter mesa is etched into the P base
layer by 0.2/mi A thin, highly doped P+ region can be formed by ion
implantation under the base contact to reduce contact resistance. This structure is
designed to be able to block near 2000V under the optimum reach-through
condition when the emitter is open. Thus, according to Figure 2.9, a 12jum,
7x1015cwj3 doped n-type epilayer is chosen for the drift layer. To prevent the
punch-through of the base, the initial base doping concentration is 3.7 x\0llcm~3
and the initial base width is 0.8fun. The emitter has a doping concentration of
2 x 1019 cm 3 and a thickness of 1.5fim. If not specified, the carrier mobilities
depicted by Eqs. (2.10) and (2.11) and carrier lifetime depicted by Eq. (2.16) with
a rn0 of 200ns are used in the simulations. In the following discussions, the
common emitter current gain is computed at a collector voltage of 20V.
47


1.5 pm
1 pm
5 pm
M--------------r*-
Emitter contact
Epitaxially grown emitter
Phase Na =3xl0'7cm"3 P* implantation
0.8 pm
N~ drift Nd = 7.5 x 10l5c/w 3
12pm
M* 4H SiC Substrate Collector contact
Figure 3.3 Schematic cross-section view of the 4H-SiC NPN BJT cell structure
3.5 DC Characteristics
The DC characteristics of the device shown in Figure 3.3 are presented in
Figure 3.4a, where the base current density is normalized to the base area. It is
seen that, in the active region, the collector current is saturated and almost does
not change with the increase of the collector voltage. This indicates the effect of
the base-width modulation (also called Early effect), while is very small for
high-voltage power SiC BJTs. In power SiC BJTs usually have a heavy-doped
thick base to prevent the peach-through of the base when the breakdown occurs.
At Jb = 66.7 A/cm and 300K, the specific on-resistance is 1.5mQcm which is
higher than the theoretical drift layer specific on-resistance of 1.2mDcm at 300K,
meaning the conductivity modulation is not present. At Jb = 66.7 A/cm and at
523K, the specific on-resistance is 4mQcm2. The positive temperature coefficient
of the specific on-resistance allows power BJT switch to parallel without thermal
48


Figure 3.4 Simulated output DC characteristics of the 4H-SiC NPN BJT at
300K and 523 K.
Collector Current Density Jc (A/cm2)
Figure 3.5 Simulated output common emitter current gain as a function of the
collector current density at different temperatures.
runaway. It is also seen from Figure 3.4 that the current gain /? is smaller at
523K than at 300K. Figure 3.5 shows the current gain as a function of the
49


Figure 3.6 Simulated blocking characteristics of the 4H-SiC NPN BJT at 300K
and 523 K.
Figure 3.7 Simulated current gain p as a function of the collector current
density at different base electron mobilities at 300K and 523K.
collector current density (Jc) at different temperatures, showing a negative
temperature coefficient for ft The negative temperature coefficient of the
50


current gain allows the paralleled power BJTs to work in parallel in the active
>y
region without thermal runaway. When Jc is greater than 200 A/cm /?
decreases quickly due to emitter current crowding and high level injection in the
emitter-base junction.
The blocking characteristics of the 4H-SiC NPN BJT are shown in Figure 3.6.
The device is able to block 1941V and 2094V at 300K and 523K, respectively.
When the emitter is terminal open, the blocking voltage in this Collector-Base
connection is called BVcbo When the base terminal is open, the device can block
1631V and 2033V at 300K and 523K, respectively. The blocking voltage in this
Collector-Emitter connection is called BVceo BVceo is smaller than BVcbo due to
the current-amplifying properties of the common-emitter connection.
From Eq. (3.3), it can be seen that the current gain is directly proportional to
the base electron mobility. Figure 3.7 shows the current gain [5 versus collector
current density at different base electron mobilities. At 300K, when the base
electron mobility is decreased from 411cm /Vs to 88cm /Vs, the maximum
current gain is decreased from 76 to 11. Thus, good quality epilayer for the base
with high electron mobility should be used for 4H-SiC power BJTs.
3.6 Emitter Design
Eq. (3.3) for the common emitter current gain is derived under the
assumption that the emitter width is much larger than the hole diffusion length in
the emitter. If not, the holes injected from the base can diffuse through the emitter
51


and reach the emitter contact and the injection efficiency is reduced. The effects
of the emitter width on the current gain are presented in Figure 3.8. When the
emitter width is larger than 1.5/nm, the current gain changes slightly. So, an
emitter width of 1.5/urn is chosen for the device studied in this research.
Figure 3.8 Effects of the emitter width on the current gain of the 4H-SiC NPN
BJT at 300K.
From Eq. (3.3), it is concluded that the high current gain can be obtained by
using heavy doped emitter. However, an increase in the emitter doping may lead
to a reduction in the hole diffusion length in the emitter, which cancel the effect of
increasing emitter doping concentration. Figure 3.9 shows the effects of the
emitter doping concentration on the current gain. The current gain does not
change much with the increase of the emitter doping concentration. When the
emitter doping concentration Nde is increased from lxlOl9ctff3 to 3xlOl9cm"3,
the current gain increases first and then decreases. The optimum emitter doping
52


level is around 2 x 1019cm 3, which gives the highest current gain for the device.
Figure 3.9 Effects of the emitter doping concentration on the current gain of the
4H-SiC NPN BJT at 300K.
3.7 Effects of Base Doping Concentration and Carrier Lifetime
As discussed in the Section 3.2, the base doping concentration has a great
effect on the current gain of power 4H-SiC NPN BJTs. However, the discussions
in the Section 3.2 are very fundamental and exclude many effects, such as low
level injection, high level injection, emitter current crowding, etc. which exist in
the actual devices. The base current consists of two main components. One is the
hole diffusion current, which causes a large number of electrons to be injected
from the emitter to the base. The other one is the recombination current due to the
recombination of the carriers in the depletion region of the forward-biased
emitter-base junction. The recombination current has no effect on the collector
current. In the low level injection region, the recombination current is significant
53


when compared to the hole diffusion current and can no longer be ignored. Thus,
for the same collector current, a larger base current is required where the current
gain is reduced. In the high-level injection region, increasing electron
concentration in the base causes an equal increase in the hole concentration,
which leads to a reduction in the injection efficiency and hence reduces the
current gain. These effects are naturally included in the two-dimension numerical
simulations. Thus, in this section, two-dimension numerical simulations are
performed to study the current gain of the 4H -SiC NPN BJTs under different
base doping concentrations and carrier lifetimes.
For the Base Doping Concentration of 3.7 x 1017ca/i3, Figure 3.10 shows the
current gain j3 as a function of the collector current density under different
carrier lifetimes and temperatures. The dashed lines with circle and triangle
symbols correspond to a base current density of 66.7 A/cm (IB = XfiA) and
13.3A/cm2 (1^ 0.2/uA), respectively. The carrier lifetime has a strong effect on
the current gain. For example, the maximum current gain increases substantially
from 19 to 54 when rn0 increased from 20ns to 100ns at 300K. When
Tn0 = 40ns, the current gain at Jc < 1000A/cm2 decreases when the temperature
elevated from 300K to 523K. The same trend can be seen when rn0 = 100m'. But
for rn0 = 20m', the current gain around Jc = 100A/cm2 decreased first when the
temperature increased from 300K to 450K, then increases when the temperature
54


(a)
(b)
Figure 3.10 Current gain as a function of the collector current density at
different temperatures and carrier lifetimes when the base doping concentration
is 3.7x 10l7cw 3
increases from 450K to 523K. This means that the temperature coefficient of the
current gain changes from negative to positive after a certain temperature. In
55


actual applications, the base drive current and the collector-emitter voltage are
usually fixed when the device is in on-state. Hence it is important to investigate
the properties of the current gain at a given base current. Figure 3.11 shows the
current gain /? as a function of the base current density under different carrier
lifetimes and temperatures when the base doping concentration (Nab) is
3.7x1017cw3. The maximum current gain appears at a base current density of
around 66.7A/cm2. Figure 3.12 presents the dependence of the current gains on
the temperature at different carrier lifetimes when Jb = 66.7 A/cm2 (the dashed
lines with circle symbols in Figure 3.10a and Figure 3.11). It is seen from Figure
3.12 that the current gain monotonously decreases as the elevation of the
temperature from 300K to 600K when rn{] = 100ns and 40m, indicating a
negative temperature coefficient in the current gain. For rn0 = 20m the
temperature coefficient of the current gain is zero at the temperature equal to
476K and is positive when the temperature is higher than 476K. The positive
temperature coefficient in the current gain appears at low carrier lifetime. The
reason for this is that the effects of the carrier lifetime become stronger when the
carrier lifetime decreases, where the current gain at 300K and Jb = 66.7 A/cm is
plotted as a function of rn0, shown in Figure 3.13. The rate at which the current
gain increases occurs in examples where the carrier lifetime is shortest.
56


Figure 3.11 Current gain as a function of the base current density at different
temperatures and carrier lifetimes when the base doping concentration is
3.7x10'1cm~i
Figure 3.12 Dependence of the current gain on the temperature at different
carrier lifetimes when JB = 66.7A/cm2 and NM = 3.7 x 10l7cw-3.
Figure 3.14 shows the zero temperature coefficient of the current gain,
57


Figure 3.13 Current gain as a function of the carrier lifetime at 300K and a base
current density of 66.7 A/cm2.
Figure 3.14 Temperatures at zero temperature coefficient of the current gain as
a function of the maximum carrier lifetime under different base drive current
densities and base widths.
(i.e. dfi/dT = 0), as a function of the carrier lifetime under different base driving
current densities and different base widths. For each line in Figure 3.14, dfi/dT
58


is positive in the top-left region of the line and is negative in the bottom-right
region of the line. For the base width equal to 12jim, the maximum current gain
appears at a base current density of about 200 A/cm2 As shown in Figure 3.14
that the temperature at dfi/dT = 0 near maximum current gain is slightly lower
when the base width is increased from O.S/um to 1.2jj.ni. For the same rn0,
when the base driving current density increases, the temperature at dfi/dT = 0
increases due to the emitter current crowding becomes dominant. When the
temperature increases, the decrease in the carrier mobility increases the role of the
emitter current crowding effect, which causes the current gain to decrease. As
shown in Figure 3.14, for a given base current density, the longer is the carrier
lifetime, the higher is the temperature at d^/dT = 0. For the case with a Jb of
66.7 A/cm2 and a base width of 0.8/um, the temperature at dfi/dT = 0 is higher
than 600K when rn0 is longer than 30ns, meaning the temperature coefficient in
the current gain is negative for temperature up to 600K.
3.8 Effects of the Surface Recombination
The recombination at semiconductor surfaces and interfaces can have a
significant impact on the behavior of the power 4H-SiC BJTs. This is because the
surfaces and interfaces typically contain a large number of recombination centers
due to the abrupt termination of the semiconductor crystal, which leaves a large
59


number of electrically active dangling bonds. In addition, the surfaces and
interfaces are more like to contain impurities since they are exposed during the
device fabrication process. The surface recombination is characterized by the
surface recombination velocity s. For minority carriers in a quasi-neutral region,
the surface recombination rate Rs is the product of the surface recombination
velocity and the minority concentration at the interface. The surface
recombination velocity is strongly dependent on the processing technology. A
typical value for 4H-SiC is estimated to be about 1 x 104 cm/s [45], For the
4H-SiC NPN BJT, the most important interface is the SiC>2/4H-SiC interface
between the emitter contact and the base contact. Large surface recombination
reduces the emitter injection efficiency and increases the recombination current in
the base, hence degrades the current gain.
Figure 3.15 presents the effects of the surface recombination on the current
gain for the BJT structure simulated. When the surface recombination velocity s
increases from 0 (without surface recombination) to 2 x 104 cm/ s, the maximum
current gain is decreased about 36.8% from 76 to 48 for rn0 = 200m and 22.6%
from 31 to 24 for rn0=40m This indicates the effect of the surface
recombination is more significant when the bulk carrier lifetime is longer.
Therefore, the improvement in the 4H-SiC material quality should be
accompanied by the improvement in the processing technology in order to obtain
high device performance. Figure 3.16 shows the surface recombination rate along
60


Figure 3.15 Effects of the surface recombination on the current gain of the
4H-SiC NPN BJT at different bulk carrier lifetimes at 300K.
Figure 3.16 Surface recombination rate along the Si02/4H-SiC interface on the
bottom of the emitter mesa trench at 300K when JB =66,7 A/cm2, VCe =20 V,
r0 = 200ms and s = 5x 103 cm/s The device structure is shown in Figure 3.3.
the SiC>2/4H-SiC interface on the bottom of the emitter mesa trench when Jb =
66.7 A/cm2, Vce = 20V, rn0 = 200w.v and .v = 5 x 103 cm/s It is seen that the
61


surface recombination mainly takes place near the emitter mesa edge at
X -1 /um where the emitter current crowds. At X = 9jum the surface
recombination rate is decreased to l x 10l* cm~3/s with only 1% of the surface
recombination rate at the emitter mesa edge.
3.9 Experimental Results and Discussions
The experimental data in this section is taken from Luo et al. [46]. The cell
structure of the fabricated device is the same as the one shown in Figure 3.3
except the emitter layer thickness is 0.7[im. The device active area is 0.012cm2.
The experimental data and simulated output characteristics of the fabricated
device at room temperature (25C) are shown in Figure 3.17a. The collector
current (Ic) is for 4.41A {Jc = 368 A/cm2) at a base current (Ib) of 140mA,
corresponding to a common emitter current gain of 31 at Vce = 8 V. The maximum
current gain is 32 at Jc = 319 A/cm The specific on-resistance for this device is
17 mQcm2 at Vce =5V and /g = 140mA [46]. The open-base blocking voltage
(BVceo) of 1100V is reported, where the leakage current is only 1.2mA. The
leakage current in Figure 3.17a has been amplified by a factor of 1000 in order to
show the details. This result represents state of the art for 4H-SiC NPN BJTs with
both high blocking capability and high current gain at high current density.
Device simulations have been performed to evaluate the carrier lifetimes in the
emitter and the base by fitting the measured Ic~Vce curves at room temperature.
After several trail simulations, an excellent fitting, as shown in Figure 3.17a, is
62


achieved when an electron lifetime of 22ns and an electron mobility of
347cm /Vs are used in the base and a hole lifetime of 5.7ns and a hole mobility of
34.5cm /Vs in the emitter. The electron lifetime used in the simulation seems to
be reasonable. The theoretical specific on-resistance of the fabricated device is
about 1.5 mQcm (assuming the maximum electron mobility is 947cm/Vs),
which is about 11 times lower than the measured specific on-resistance. The high
measured specific on-resistance may not be due to the low electron mobility and
series resistance. The fitting to the measured Ic-Vce curves can not be achieved by
using low electron mobility even when the maximum carrier lifetimes are as high
as 5/us. At present, it is actually not well understood why the exprimental
specific on-resistance is so high. Thus, a resistor of 1.24Q is connected to the
collector in order to fit the specific on-resistance of the device. Another important
parameter for the fitting is the base leakage current. It can be seen from Figure
3.17b that the experimental transfer function (Ic vs. Ig) is almost linear. The
extrapolation of the transfer function intersects x-axis at ho = 7mA. This indicates
there is 7mA in h that did not take part in the BJT operation, ho is the base
leakage current possibly through JTE region or the surface between the emitter
and the base. Without considering this base leakage current, the fitting can not be
made to the whole set of the measured Ic-Vce curves.
63


(a)
(b)
Figure 3.17 Measured and simulated output characteristics (4 vs. VCE) (a) and
transfer functions (4 vs. 4) (b) of the fabricated 4H-SiC NPN BJT at room
temperature, all measured data is taken from [46],
3.10 Summary
In this chapter, a 4H-SiC NPN BJT is studied systematically by performing
64


two-dimensional numerical simulations. Several design issues are discussed.
Depending on the base doping concentration and the carrier lifetimes, both
positive and negative temperature coefficients in the common emitter current gain
could exist in 4H-SiC NPN BJTs with aluminum-doped base. The temperature
coefficients of the current gain at different base doping concentrations and
different carrier lifetimes are determined. A high base doping concentration
reduces the requirement for the carrier lifetime in order to obtain negative
temperature coefficient in current gain. Device simulations are performed to
evaluate the role of carrier lifetimes on output Ic-Vce characteristics. An excellent
fitting is obtained and the base electron lifetime and the emitter hole lifetime are
estimated to be about 22ns and 5.7ns, respectively.
65


4 CONCLUSIONS AND SUGGESTIONS
This thesis presents the design, optimization, and performance prediction of a
4H-SiC NPN BJT by way of two-dimensional device simulations using the
physical models of 4H-SiC semiconductor material described in chapter 2. The
SiC BJT has the potential and performance characteristics to become the popular
device structures in the future 4H-SiC industry.
The major contributions of this thesis research in the following. A systematic
analyses of the temperature coefficients of the common-emitter current gain of
4H-SiC NPN BJT is performed. Unlike Si BJT, 4H-SiC NPN BJT could have a
negative temperature coefficient in current gain, which is an essential condition
for power BJTs to avoid thermal runaway when connected in parallel. The
conditions for the negative temperature coefficient of the current gain have been
determined. The effects of the surface recombination on the current gain are
discussed. It is found that the surface recombination is significant only in a region
near the emitter-mesa edge. The experimental results from published literatures
are discussed and compared with the simulation results.
Some suggestions for the future research should aim toward more accurate
physical models for 4H-SiC in simulations. In 4H-SiC, there are significant
anisotropy in electron mobility model and in impact ionization coefficients. The
anisotropy, however, has not been implemented in the TCAD simulator. In order
to predict accurately the forward and blocking performance of a 4H-SiC power
device, the anisotropy must be considered. There need to have more accurate
66


physical parameters derived from measured data for 4H-SiC, such as the
temperature coefficient in the carrier mobility, the temperature coefficient in the
carrier lifetime, and the temperature coefficient in the impact ionization
coefficients. One of the advantages of 4H-SiC material is being able to operate
the device at high temperature. These parameters are essential for operating the
device performance at high temperature. Development of a SPICE model for
4H-SiC power devices studied in this thesis is also important, so that the
performance and the advantages of these 4H-SiC devices can be evaluated in a
VLSI system. Simulation results presented show that SiC BJT have excellent
performance at both room temperature and 523K. Further research is definitely
needed in the area of device processing and experimental verification.
67


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