[sic-en-2013-20] physical models for sic and their application to device simulations of sic...

7/27/2019 [SiC-En-2013-20] Physical Models for SiC and Their Application to Device Simulations of SiC Insulated-Gate Bipolar

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IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 2, FEBRUARY 2013 613

Physical Models for SiC and Their Application toDevice Simulations of SiC Insulated-Gate

Bipolar TransistorsTetsuo Hatakeyama, Kenji Fukuda, and Hajime Okumura

(Invited Paper)

AbstractImportant physical models for 4H silicon carbide(4H-SiC) are constructed based on the literature and experimentson the physical properties of 4H-SiC. The obtained physical mod-els are implemented into a commercial device simulator, which isused for examining the potential performance of SiC insulated-

gate bipolar transistors (IGBTs). Device simulation using thesenew physical models shows that the forward characteristics of theconventional type of planar SiC IGBTs are significantly poorerthan those of SiC p-i-n diodes, even if the carrier lifetime isimproved. It is shown that the degradation in the characteristicsof the conventional SiC IGBT is caused by the limited conductionmodulation at the cathode side of the n-base layer. We show thatthis problem can be resolved by applying device structures thatinduce a hole-barrier effect in the SiC IGBTs.

Index TermsDiodes, insulated gate bipolar transistors,MOSFET, wide band gap semiconductors, power semiconductordevices, semiconductor device modeling, 4H-SiC.

I. INTRODUCTION

SILICON carbide presents a great promise in the field of

high-power devices. Among many SiC power devices,

SiC insulated-gate bipolar transistors (IGBTs) are regarded as

promising candidates for ultrahigh-voltage (> 5 kV) electricalpower switches, owing to their good ON-state characteristics

because of conductivity modulation. However, as of now, the

low carrier lifetime of SiC has hampered the potential per-

formance of SiC IGBTs. In the recent years, several groups

have succeeded in improving the carrier lifetime by reducing

the density of deep levels denoted by Z1/2 [1], [2]. In thenear future, carrier lifetime will not be a limiting factor for the

performance of SiC IGBTs. Accordingly, the device design of

SiC IGBTs will become increasingly important. In this paper,the potential performance of SiC on IGBTs is examined using

Manuscript received July 24, 2012; revised September 14, 2012; acceptedSeptember 27, 2012. Date of publication November 30, 2012; date of currentversion January 18, 2013. This work was supported by a grant from the JapanSociety for the Promotion of Science through the Funding Program for World-Leading Innovative R&D on Science and Technology program, initiated bythe Council for Science and Technology Policy. The review of this paper wasarranged by Editor M. Miura-Mattausch.

The authors are with the Advanced Power Electronics Research Center,National Institute of Advanced Industrial Science and Technology, Tsukuba305-8561 Japan (e-mail: [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TED.2012.2226590

device simulations. Before addressing this issue, the physical

models for 4H silicon carbide (4H-SiC) are constructed based

on the literature and on experiments investigating the physical

properties of 4H-SiC, because physical modeling is important

for obtaining reliable and quantitative information about de-

vices through the use of simulations.

II. PHYSICAL MODELS FOR 4H-SiC

In this paper, we focus on the 4H polytype of SiC, because

4H-SiC is the best suited material for power devices among

the many polytypes of SiC owing to its superior material

properties. For example, it has a large bandgap, which indicates

a high breakdown field or large breakdown voltage; a high bulk

mobility, which results in low ON-state resistance; and little

anisotropy in physical properties, which reduces the efforts

of designing device structure. Further, 4H-SiC has a mature

material technology. We note that high-quality large-diameter(up to 150 mm) 4H-SiC wafers, which are an indispensable

prerequisite for the production of power devices, have been

commercialized.

A. Dielectric Constant

As far as we know, the dielectric constant for 4H-SiC has not

been reported. We use the values for 6H-SiC reported by Patrick

and Choyke [3]

= 9.66 = 10.03 (1)

for directions orthogonal (

) and parallel (

) to the hexagonal

c-axis.

B. Band Structure

1) Bandgap Model: The bandgap for 4H-SiC is expressed

by the sum of the exitonic bandgap Egx(T) and the bindingenergy for an exiton Ex as follows:

Eg(T) = Egx(T) + Ex. (2)

The temperature dependence of the exitonic bandgap Egx(T)for 4H-SiC has been measured by Choyke [4]. Therefore,

according to (2), the temperature dependence of the bandgap

is derived from the exitonic binding energy (Ex

= 20 meV),

0018-9383/$31.00 2012 IEEE


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614 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 2, FEBRUARY 2013

Fig. 1. Comparison of the measured temperature dependence of the bandgapfor 4H-SiC [4], [5] and the result of fitting.

which is obtained by Dubrovskii and Sankin based on the

measurements of the change in power-law energy dependence

of the absorption coefficient for 4H-SiC at 4.2 K [5]. We note

that the measured values are SEg = 3.285 eV at 4.2 K andEg = 3.25 eV at 300 K. The temperature dependence of thebandgap for 4H-SiC is fitted as follows with Varshnis empirical

expression [6]:

Eg(T) = 3.285 9.06 104 T2

T+ 2.03

103

(eV). (3)

Fig. 1 shows the measured temperature dependence of the

bandgap of 4H-SiC [4], [5] as determined by the fitting results.

2) Effective Masses: In this subsection, we discuss the den-

sity of state (DOS) masses of electrons and holes. If we neglect

the temperature dependence of the mass of electrons, the effec-

tive conduction-band DOS (Nc) is expressed by

Nc = 2nv

2mdekBT/h2(3/2)

(4)

where nv and mde are the degeneracy of the minimum of

the conduction band (valleys) and the effective DOS mass of

electrons, respectively. 4H-SiC has effectively three valleys atMpoints of the Brillouin zone [7]. Therefore

nv = 3. (5)

The effective DOS mass of electrons is given by the geometric

mean around the minimum point as [7]

mdem0

= 3mMmMKmML = 0.394 (6)

where m0 is the mass of a free electron. For the definitions ofthe other mass parameters, see [7].

Next, we account for the deviation from the effective massapproximation of the temperature dependence of the effective

Fig. 2. Temperature dependence of electron thermal DOS effective mass for4H-SiC and results obtained by fitting.

DOS by introducing thermal DOS effective masses. The elec-

tron thermal DOS effective mass is defined as [7]

n(T) =

Eg

nvmde(T)

3/2

23

2Ef(E, T)dE. (7)

The electron thermal DOS effective mass was calculated by

Wellenhofer and Rssler from first principles by using (7) [7].

Their results are fitted as follows:

mde(T)

m0= 0.394 + 3.09 108T2 + 2.23 1010T3

1.65 1013T4. (8)

We note that mde(300)/m0 0.40. Fig. 2 shows the tem-perature dependence of the thermal DOS effective mass for

electrons in 4H-SiC based on the results obtained by fitting.

In the same manner, the calculated hole thermal DOS ef-

fective mass [7] is fitted using a polynomial expression [8] as

follows:

mdh(T)

m0= 1+6.92

102T+ 1.88

106T4

1+6.58 104T2+4.32 107T42/3

. (9)

Fig. 3 shows the temperature dependence of the hole thermal

DOS effective mass based on the results obtained by fitting.

We note that mdh(300)/m0 2.64. Further, we also point outthat the hole thermal DOS effective mass for 4H-SiC is almost

independent of temperature above room temperature.

3) Intrinsic Carrier Density: The intrinsic carrier density

ni(T) is obtained using the following equation:

ni(T) =Nc(T)Nv(T)exp

Eg(T)

kT

(10)

where Nc(T) and Nv(T) are the effective conduction-bandDOS and the effective valence-band DOS, respectively. Fig. 4


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HATAKEYAMA et al.: MODELS AND THEIR APPLICATION TO SIMULATIONS OF SiC IGBTs 615

Fig. 3. Temperature dependence of the hole thermal DOS effective mass for4H-SiC and the result of fitting.

Fig. 4. Temperature dependence of intrinsic carrier concentration for 4H-SiC,6H-SiC, 3C-SiC, diamond, and silicon [9].

shows the temperature dependence of the intrinsic carrier con-

centration for 4H-SiC and other semiconductors [9].

C. Low-Field-Mobility Model

Anisotropy of the low-field bulk mobility for electrons and

holes can be expressed as [10], [11]

e, = 1.2e, h, = 0.87h, (11)

where e,(h), and e,(h), are electron (hole) mobilities par-allel and perpendicular to the c-axis, respectively. It should benoted that the anisotropy of mobility is caused by the anisotropy

of effective masses.

Hereafter, we consider the bulk mobility perpendicular to thec-axis without loss of generality. We adopt the Arora model for

Fig. 5. Doping density dependence of bulk mobility for 4H-SiC along withthat of the previous work [14] and the experimental results [13].

the doping-dependent low-field bulk mobility b(Ndop)

b(Ndop) = min +d min

1 + (Ndop/Nref)A

(12)

with

min =Amin (T/300)m (13)

d =Ad (T/300)d (14)

Nref =AN (T/300)N (15)

A =Aa (T/300)a (16)

Ndop =ND + NA (17)

where Ndop, ND, and NA are the total doping concentration,donor concentration, and acceptor concentration, respectively.

It should be noted that b is not a function of ionized impurityconcentration but of total doping concentration. As discussed

by Koizumi et al., the low-field hole mobility is a function of

total doping concentration because hole mobility in a heavily

doped sample is also limited by neutral impurity scattering as

well as ionized impurity scattering [12]. We also note that, in

many cases, a commercial device simulator treats low-field mo-

bility as a function of the total doping concentration by default.

The parameters for electron mobility are extracted by using

the published experimental results reported by Matsunami and

Kimoto [13], wherein the doping densities are estimated from

the carrier density. The results of the fitting obtained in this

work are shown in Fig. 5, along with those obtained in the

previous work [14], in addition to the experimental results [13].

For hole mobility, the parameters extracted by Koizumi et al.

[12] are adopted. The parameters for the low-field-mobility

model are summarized in Table I.

D. Incomplete Ionization

The most common n-type dopant for SiC is nitrogen. Nitro-

gen atoms substitute for carbon atoms on the lattice sites in4H-SiC. There are two types of carbon sites in SiC: One is


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TABLE IPARAMETERS FOR DOPING-D EPENDENT MOBILITY MODEL

TABLE IIPARAMETERS FOR INCOMPLETE IONIZATION

surrounded by four silicon atoms with hexagonal symmetry (h),and the other is surrounded by those with cubic symmetry (k).

As a result, there are two ionization energies (Eh and Ek)in nitrogen-doped SiC [15]. Given these two ionization ener-

gies, the ionized donor density N+D is given by the followingequation:

N+D =0.5ND

1+gDnNc

expEhkBT

+ 0.5ND1+gD

nNc

expEkkBT

(18)where gD = 2 is a degeneracy factor. If we can neglect thedynamic effects accompanying incomplete ionization, these

two donor levels can be replaced by a single effective level

ED, as proposed by Bakowski et al. [16]

N+D =ND

1 + gDnNc

expEDkBT

. (19)It should be noted that the ionization energy (E) is effectivelyreduced when the total doping density (Ndop) is increased asfollows:

E= E0 N1/3dop (20)where E0 and are the ionization energy at the low dopingdensity limit and a fitting parameter, respectively. The param-

eters of the single-energy-level model were obtained by fitting

the activation ratio as a function of the doping density based

on the two-energy-level model [14]. For acceptors, parameters

extracted by Koizumi et al. [12] are used in this paper. It should

be noted that the degeneracy factor for holes gA is four. Theparameters for the incomplete ionization model are summarized

in Table II. Fig. 6 shows the activation ratio as a function of

doping density for donors and acceptors. For donors, results

calculated using the two-level model and the single-energy

model are shown for comparison.

E. Impact Ionization

Impact ionization coefficients are important material proper-

ties for power devices, because the avalanche breakdown of apower device is caused by the impact ionization phenomenon.

Fig. 6. Activation ratio as a function of doping density for donors andacceptors. For donors, results calculated using the two-energy-level model andthe single-energy-level model are shown for comparison.

Accordingly, a physical model of the impact ionization co-

efficient is indispensable for predicting the electrical charac-

teristics of power devices. Further, owing to the hexagonal

crystal structure of 4H-SiC, carrier transport in 4H-SiC exhibits

anisotropy, which results in an anisotropy in impact ionization.

In this subsection, we briefly describe an anisotropic impact

ionization model for 4H-SiC.

We assumed the following functional form as the model

of the impact ionization coefficients suggested by Chynoweth

[17], Capasso [18], and Crowell and Sze [19]:

=aeexp

be

F

(21)

=ahexp

bh

F

(22)

with

=tanh

op

2kBT0

tanh

op

2kBT(23)

where op and F represent the optical phonon energy andthe magnitude of the electric field, respectively. The other

parameters (ae, be, ah, and bh) are fitting parameters. Experi-ments performed using p+-n diodes showed that the anisotropy

of impact ionization coefficients in 4H-SiC leads to 81%

anisotropy in breakdown fields [20][22]. This anisotropy in

the impact ionization coefficient is expressed by two sets of

parameters corresponding to the main axes (c- and a-axes)parallel to the applied electric field. The parameters of the

electron and hole impact ionization coefficients are summarized

in Table III [21], [22].

We now present an analytical model of the anisotropic impact

ionization coefficients, which interpolates impact ionization co-efficients parallel and perpendicular to the c-axis [23]. First, it is


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TABLE IIIPARAMETERS OF IMPACT IONIZATION COEFFICIENTS

OF HOLES AND ELECTRONS IN 4H-SiC

assumed that the anisotropy of the impact ionization coefficient

in 4H-SiC has cylindrical symmetry with regard to the c-axis.This assumption is plausible because experiments showed that

the anisotropy of the bulk mobility in 4H-SiC has cylindrical

symmetry [10] and that the anisotropy of impact ionization

coefficients is closely related to that of mobility [22]. Based

on this assumption, the impact ionization coefficients perpen-

dicular to the c-axis are replaced by those parallel to the a-axis.Hereafter, we only discuss the impact ionization coefficient of

electrons at room temperature without loss of generality. Theimpact ionization coefficients parallel and perpendicular to the

c-axis are defined as follows:

(F) = a exp(b/F) for F c-axis (24)(F) = a exp(b/F) for F c-axis (25)

where a, b, a, and b are experimentally determined param-eters of the anisotropic impact ionization. F denotes the normof the electric field F. The electric field F is decomposed into

a parallel part F and a perpendicular part F as follows:

F= (F, F). (26)

Hence, F is defined as

F |F| F2 + F

2 . (27)

The interpolation formula for the impact ionization coefficients

is derived by considering the energy balance equation [22],

[23]. The analytic expression for the anisotropic impact ion-

ization coefficients in any direction of the electric field is

(F) = a exp

b

F1 A2bFFFb

b

2 (28)

where b, a, and A are defined by the following equations:

F2

b2=F2b2

+F2b2

(29)

a =a

(b2F2)(b2F2) a

b2F2

b2F2

(30)

A = ln

aa

. (31)

It should be noted that this anisotropic impact ionization modelhas been implemented in some commercial device simulators.

TABLE IVPARAMETERS FOR THE HIG H-F IELD-M OBILITY MODEL IN 4H-SiC

F. High-Field Mobility

The saturation of carrier velocity under a high electric field is

caused by the increase in elastic and nonelastic scattering owing

to the increase in carrier energy. Khan and Cooper measured

the electric field dependence of electron drift velocity perpen-

dicular to the c-axis in 4H-SiC at room temperature and at600 K [24]. The saturation phenomenon is modeled by Lades

[25] using the Canali model [26] as follows:

high =low

1 +low

F

vsat

1 (32)

with

vsat = vsat0,

T

300

(33)

= 0

T

300

(34)

where high , low , and v

sat are the high-field mobility, low-

field mobility, and saturation velocity, respectively. The other

parameters are fitting parameters. The suffix is added todesignate these physical parameters as those perpendicular to

the c-axis. The fitted parameters for the high-field-mobilitymodel are summarized in Table IV [24], [25].

To simulate the IV characteristics of SiC power devices,the high-field mobility parallel to the c-axis is more importantthan that perpendicular to the c-axis, because current flowsvertically through the epitaxial layer on a (0001) wafer in SiC

power devices. Unfortunately, to the best of the authors knowl-

edge, measurements of electron saturation velocity parallel

to the c-axis have not been reported. However, the anisotropy ofthe saturation velocity can be estimated from the anisotropy of

the impact ionization coefficients [22]. The electron saturation

velocity parallel to the c-axis is estimated to be about 60% ofthat perpendicular to the c-axis. Further, the hole saturationvelocity parallel to the c-axis is estimated to be about 80%of that perpendicular to the c-axis. Monte Carlo studies of

high-field transport in 4H-SiC have shown similar results ofthe anisotropy of the carrier velocity [27][29]. Further, to the

authors knowledge, no measurements have been reported for

the hole saturation velocity perpendicular to the c-axis. Thecalculated hole saturation velocity in Monte Carlo studies is

around 1 107 cm/s [28], [29].

G. Carrier Lifetime

For determining the SRH lifetimes n and p for electronsand holes, we assumed the following doping-dependent model

which is commonly used for silicon:

n,p =

n0,p0

1 +NdopNSRHref

(35)


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Fig. 7. Forward characteristics of an n-type planar IGBT with different carrierlifetimes.

where NSRHref = 31017 cm3 and = 0.3 [9]. In general, SRHlifetimes depend on the growth condition of epitaxial layers

and the process conditions of device fabrication [1], [2], [30].

Therefore, n0 and p0 are treated as parameters in devicesimulations.

III. SIMULATION OF SiC IGBTs

In this section, we describe the potential performance of SiC

IGBTs examined using device simulations. By using a device

simulation technique, Cooper et al. obtained the ideal charac-teristics of SiC IGBTs [31]. However, they did not explicitly

examine the device structure that induces the hole-barrier effect,

which is the key to reducing ON-state resistance of IGBTs. In

this paper, we focus on device structures that induce the hole-

barrier effect and improve the forward characteristics of SiC

IGBTs. It has been claimed that the low carrier lifetime of SiC

has hampered the potential performance of SiC IGBTs. In view

of this claim, first, we examine the impact of the lifetime on

the forward characteristics of planar IGBTs. Fig. 7 shows the

calculated forward characteristics of an n-type planar IGBT

with different carrier lifetimes. Forward characteristics of a SiC

p-i-n diode (max = 10 s) are also shown for comparison. Inaddition, the schematic of the device structure of the IGBT(punchthrough type) is shown in Fig. 7. The thickness of the

n-base layer is 100 m, and the doping density is 7 1014 cm3. The estimated breakdown voltage of this deviceis more than 13 kV. It should be noted that the effect of the

resistance of a p-type substrate on the forward characteristics

is neglected in the simulation by assuming that thin p-type

layers are grown on the reverse side of the wafer. In Fig. 7, the

forward characteristics of the planar IGBT are improved as the

lifetime is improved. However, the forward characteristics of

the IGBT are significantly poorer than those of the p-i-n diode.

Accordingly, it is concluded that low carrier lifetime as well as

device design has hampered the potential performance of SiCIGBTs. The analysis performed using the carrier distribution

Fig. 8. Forward characteristics of HiGTs and a conventional planar IGBT.

under conduction shows that the low performance of a SiC

IGBT under high carrier lifetime is caused by the limited

conduction modulation at the emitter side of the n-base layer.

Enhancement of the conduction modulation in the drift layer of

the SiC IGBT is required to fully utilize the high carrier lifetime

and superior material properties of SiC.

As discussed as follows, this problem is resolved by applying

the device concepts of Si IGBTs to SiC IGBTs. An n-type

hole-barrier layer surrounding a p-body of an IGBT enhances

the conduction modulation in the case of Si IGBTs [32]. We

have applied this device concept to the device design of planarSiC IGBTs. Fig. 8 shows a comparison of the forward charac-

teristics of a SiC IGBT with a hole-barrier layer (HiGT) and

a conventional planar IGBT. In this simulation, the thickness

of the hole-barrier layer is 0.5 m, and the doping density ofthe hole-barrier layer is treated as a parameter. A hole-barrier

layer significantly improves the forward characteristics of a

SiC IGBT.

Fig. 9 shows a comparison of the hole-density distribution

profile for a conventional IGBT and HiGTs with different

doping densities of the hole-barrier layer. It can be seen that

a hole-barrier layer hinders the flow of holes into the p-body,

which results in an increase in carrier density at the emitterside of the drift layer. It should be noted that inserting a hole-

barrier layer on the top of the drift layer degrades the avalanche

breakdown voltage, because the hole-barrier layer effectively

increases the doping density of the drift layer.

Another important device, which enhances the conduction

modulation, is an IEGT [33]. An IEGT is a trench gate IGBT,

whose trench gate is designed to hinder the flow of holes into

the p-body. It should be noted that the electrical field crowding

at the deep trench corner in a SiC IEGT is a central issue in its

fabrication. However, this crowding may be avoided by using

device structures for relaxing the electrical fields, such as a

dummy deep trench with a thick gate oxide. For this reason,

we neglect the crowding effect. Fig. 10 shows the forward char-acteristics of SiC IEGTs with different geometry parameters


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Fig. 9. Comparison of hole-density distribution profile along the vertical cutat a current density of 200 A/cm2.

Fig. 10. Forward characteristics of IEGTs. Those of a p-i-n diode are shownfor comparison.

for the trench gate. The forward characteristics of a SiC p-i-ndiode are also shown for comparison. Upon reducing the source

width and increasing the gate depth, the forward characteristics

of an IEGT are found to approach those of a p-i-n diode. Fig. 11

shows a comparison of the hole-density distribution profile for

IEGTs with different geometry parameters for the trench gate. It

can be seen that the hole density at the emitter side increases as

the gate depth (D) increases. It should be noted that an almostflat hole-density distribution over the drift layer is realized in

the case ofW = 0.5 m and D = 7.5 m.This IEGT is expected to correspond to a practical material-

limit IGBT that is characterized by a flat carrier distribution

over the n-base layer in the conduction state, which was dis-

cussed by Nakagawa [34], [35]. Fig. 12 shows a comparisonof the forward characteristics of a practical SiC-limit IGBT, an

Fig. 11. Comparison of hole distribution profile along the vertical cut at acurrent density of 200 A/cm2.

Fig. 12. Comparison of forward characteristics of a practical SiC-limit IGBT,an IEGT, an HiGT, and a planar IGBT.

IEGT, an HiGT, and a planar IGBT. In this figure, the forwardcharacteristics of a practical material-limit IGBT are calculated

according to the formula discussed in [35]. It can be seen that

the forward characteristics of the IEGT (W = 0.5 m) arealmost the same as those of a practical SiC-limit IGBT. Thus, it

can be concluded that a practical SiC-limit IGBT is realized by

reducing the mesa width of a SiC IEGT by less than 0.5 m. Itshould be noted that the hole-barrier mechanism of a material-

limit IGBT is different from that of an IEGT [34]. In the case of

an IEGT, the narrow mesa per se acts as a hole barrier, whereas

in the case of a material-limit IGBT, elevated electron density

in the narrow mesa, which is induced by the MOS gate, acts as

a hole barrier. Fig. 13 shows the electron density at the center

of the mesa as a function of the mesa width (W). It can be seenthat the electron density increases as the parameterWdecreases


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Fig. 13. Electron density at the center of the mesa as a function of mesawidth (W).

Fig. 14. Forward voltage at 200 A/cm2 as a function of the length of then-channel (D).

when the parameter W is less than 0.5 m. This means thata SiC-limit IGBT is realized when the parameter W is lessthan 0.5 m. Fig. 14 shows the forward voltage at 200 A/cm2

as a function of the gate depth (D). The dependence of theforward voltage on the parameter D changes when the pa-rameter W is smaller than 0.5 m. In the case of IEGTs, theforward voltage decreases as the parameter D increases. Whenthe parameter W is less than 0.5 m, the forward voltageincreases as the parameter D increases. This D dependence ofthe forward voltage is also a feature of SiC-limit IGBTs.

IV. SUMMARY

Physical models for 4H-SiC have been constructed according

to the literature and experiments investigating the physicalproperties of 4H-SiC. The obtained physical models are imple-

mented in a commercial device simulator. We have examined

the potential performance of SiC using the device simulator. We

have shown that the device concepts of Si IGBTs are effective

for improving the performance of SiC IGBTs. We have also

shown that the SiC-limit IGBT is realized when the mesa width

is less than 0.5 m.

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Tetsuo Hatakeyama received the Ph.D. degree fromThe University of Tokyo, Tokyo, Japan.

He is with the Advanced Power Electronics Re-search Center, National Institute of Advanced Indus-trial Science and Technology, Tsukuba, Japan.

Kenji Fukuda received the Ph.D. degree from Hokkaido University, Sapporo,Japan.

He is with the Advanced Power Electronics Research Center, NationalInstitute of Advanced Industrial Science and Technology, Tsukuba, Japan.

Hajime Okumura received the Ph.D. degree fromOsaka University, Osaka, Japan, in 1990.

He is with Power Electronics Research Center,National Institute of Advanced Industrial Scienceand Technology, Tsukuba, Japan.

[sic-en-2013-20] physical models for sic and their application to device simulations of sic...

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