2.1 Crystal Structure

SiC is a compound semiconductor, which means that only a rigid stoichiometry, 50% silicon (Si) and 50% carbon (C), is allowed. The electronic structures of neutral Si and C atoms in their ground states are:

2.1

2.2

Both Si and C atoms are tetravalent elements and have four valence electrons in their outermost shells. Si and C atoms are tetrahedrally bonded with covalent bonds by sharing electron pairs in orbitals to form a SiC crystal. Each Si atom has exactly four C atom neighbors, and vice versa. The Si–C bond energy is very high (4.6 eV), which gives SiC a variety of outstanding properties, as described below.

From a crystallographic point of view, SiC is the best known example of polytypism [1–5]. Polytypism is the phenomenon where a material can adopt different crystal structures which vary in one dimension (that is, in stacking sequence) without changes in chemical composition. The variation in the occupied sites along the c-axis in a hexagonal close-packed system brings about different crystal structures, known as polytypes. Consider the occupied sites in the hexagonal close-packed system, shown schematically in Figure 2.1. There are three possible sites, denoted as A, B, and C. Two layers cannot successively occupy the same site; the next layer on top of an “A” layer must occupy either “B” or “C” sites (and, similarly, “A” or “C” is allowed over “B”). Though there exist, in principle, almost infinite variations of the stacking sequence when stacking a number of layers; for most materials, only one stacking structure (often either the zincblende or wurtzite structure) is usually stable. However, SiC crystallizes in surprisingly many (more than 200) polytypes.

In Ramsdell's notation, polytypes are represented by the number of Si-C bilayers in the unit cell and the crystal system (C for cubic, H for hexagonal, and R for rhombohedral). 3C-SiC is often called , and other polytypes are referred to as . The structures of popular SiC polytypes; 3C-SiC, 4H-SiC, and 6H-SiC, are shown schematically in Figure 2.2, where open and closed circles denote Si and C atoms, respectively. Here, A, B, and C are the potentially occupied sites in a hexagonal close-packed structure, as described above. These site names enable 3C-SiC to be described by the repeating sequence of ABCABC, or simply ABC. In a similar manner, 4H- and 6H-SiC can be described by ABCB (or ABAC) and ABCACB, respectively. The structures of these three SiC polytypes in a ball-stick model are shown in Figure 2.3. Because there are several popular notations to define stacking structures [1], the major SiC polytypes are described using Ramsdell's, Zhdanov's, and Jagodzinski's notations in Table 2.1. Figure 2.4 shows the primitive cells and fundamental translation vectors of (a) cubic (3C) SiC and (b) hexagonal SiC. The “3C” structure is equivalent to the zincblende structure, in which most III–V semiconductors like GaAs and InP crystallize. The wurtzite structure, also found in GaN and ZnS, can be denoted by “2H”. However, it is still not fully understood why so many SiC polytypes exist. In general, crystals with strong covalent bonding crystallize in the zincblende structure, while the wurtzite structure is more stable for crystals with high ionicity. The intermediate ionicity of SiC (11% according to Pauling's definition) may be a possible reason for the occurrence of SiC polytypism [6, 7]. The space groups are for 3C-SiC, for hexagonal polytypes, and for rhombohedral polytypes [8]. Hexagonal and rhombohedral polytypes are uniaxial, and thus these polytypes exhibit unique polarized optical properties.

Ramsdell's notation	Zhdanov's notation	Jagodzinski's notation
2H	11	h
3C		k
4H	22	hk
6H	33	hkk
15R		hkkhk

Because of the variety of ways to stack Si-C bilayers, there are several lattice sites in SiC, which differ in their structures of immediate neighbors. The lattice sites with hexagonal-structured surroundings are denoted “hexagonal sites,” and those with cubic-structured surroundings are denoted “cubic sites.” In Figure 2.2, hexagonal and cubic sites are indicated by “h” and “k,” respectively. 4H-SiC has one hexagonal and one cubic site, and 6H-SiC one hexagonal and two inequivalent cubic sites, while 3C-SiC contains only cubic sites. Hexagonal and cubic sites differ in the location of the second-nearest neighbors, leading to different crystal fields. For example, the energy levels of dopants, impurities, and point defects (such as vacancies) depend on the lattice site (hexagonal/cubic). This is called the “site effect” [9–11].

The stability and nucleation probability of SiC polytypes depend strongly on temperature [12]. For example, 3C-SiC is not stable, and is transformed into hexagonal SiC polytypes such as 6H-SiC at very high temperatures, above 1900–2000 °C [13]. This instability of 3C-SiC makes it difficult to grow large 3C-SiC ingots at a reasonable growth rate. 2H-SiC is also unstable at high temperature, and large 2H-SiC crystals have not been obtained. Thus, 4H-SiC and 6H-SiC polytypes are very popular, and have been extensively investigated to date [14–20]. 3C-SiC is another popular polytype because 3C-SiC can be grown heteroepitaxially on Si substrates [21–23]. As well as these three main polytypes, 15R-SiC is occasionally obtained, and has been studied to some extent [24, 25].

Table 2.2 shows the lattice constants of major SiC polytypes at room temperature [26]. Though the lattice constants look very different for different SiC polytypes (because of their different crystal structures), all SiC polytypes possess almost the same Si-C bond length (1.89 Å). Thus the height of the Si-C bilayer along the c-axis (unit height) is 2.52 Å, although 3C-SiC and 2H-SiC have a slightly smaller height (2.50 Å). The lattice constants vary with temperature and doping density, as is also observed for other semiconductor materials. Figure 2.5 shows the c-axis lattice constant of 4H-SiC from room temperature to 1100 °C as a function of doping density (doping with nitrogen or aluminum) [27, 28]. In general, very high nitrogen doping causes lattice contraction, and lattice expansion is induced by very high aluminum doping. This trend is more pronounced at temperatures above 1000 °C. Therefore, one should expect mismatch-induced stress at the , and interfaces, which can lead to generation of extended defects such as basal plane dislocations. The axial thermal expansion coefficients of SiC perpendicular and parallel to the c-axis have been measured [29], and the temperature dependence for 4H-SiC is expressed by:

2.3

2.4

Here is the absolute temperature. The thermal expansion coefficients of different SiC polytypes do not deviate very much from each other.

Polytype	(Å)	(Å)
3C	4.3596	—
4H	3.0798	10.0820
6H	3.0805	15.1151

Because all SiC polytypes consist of similar Si-C bonds, mechanical properties such as hardness are very similar among different SiC polytypes [30]. However, different periodic potentials in different SiC polytypes result in very different electronic band structures, and thus significant variation in optical and electronic properties. This means that, for device applications, it is crucial to grow only the single desired SiC polytype; polytype control is a vital aspect of crystal growth of SiC.

Except for 3C-SiC, crystal planes and directions in SiC polytypes are usually expressed by using four Miller–Bravais indices [31]. A crystal plane is equivalent to a plane , defined by three Miller indices in a monoclinic system, when the following relations are satisfied:

2.5

In a similar manner, a crystalline direction is equivalent to a direction , defined by three Miller indices in a monoclinic system, when the following relations are satisfied:

2.6

Because SiC is a compound semiconductor, the valence electrons are slightly localized near C atoms, which are more electronegative than silicon (C: 2.5, Si: 1.8). In this sense, Si atoms can be referred to as cations and C atoms as anions. This ionicity gives rise to polarity in SiC, which is of academic and technological importance. Schematic illustrations of bond configurations in a hexagonal SiC polytype are shown in Figure 2.6. In a hexagonal or rhombohedral structure, the (0001) face, where one bond from a tetrahedrally-bonded Si atom is directed along the c-axis , is called the “Si face,” while the face, where one bond from a tetrahedrally-bonded C atom is directed along the c-axis, is called the “C face.” In 3C-SiC, the (111) and faces correspond to the Si face and C face, respectively; these faces are similar to the “A face” and “B face” in III–V semiconductors. The definition relies on the crystallographic orientation, and not on the terminating atoms on the surface. Figure 2.7 illustrates the definition of several major planes in a hexagonal SiC polytype. Other than the Si and C faces, the face is called the “A face (or a-face),” and the face the “M face (or m-face).” The surface energy, chemical reactivity, and electronic properties are significantly dependent on these crystal faces, details of which are described in the growth and device chapters. Standard wafers are SiC(0001) with several degrees off-axis toward [32]; these are described in detail in Section 3.5.

2.2 Electrical and Optical Properties

2.2.1 Band Structure

Figure 2.8 shows the first Brillouin zones of (a) 3C-SiC and (b) a hexagonal SiC polytype [26, 30]. Note that the height of the Brillouin zone shown in Figure 2.8b is different for different hexagonal polytypes because of their different values of the lattice parameter, c.

Figure 2.9 depicts the electronic band structures of (a) 3C-SiC, (b) 4H-SiC, and (c) 6H-SiC [33–37]. Note that the absolute values of the bandgap are underestimated in this figure, due to a limitation of the theoretical calculation (density functional theory). All the SiC polytypes have an indirect band structure, as is also the case for Si. The top of the valence band is located at the Γ point in the Brillouin zone, whereas the conduction band minima appear at the Brillouin zone boundary. The conduction band minima are located at the X point for 3C-SiC, M point for 4H-SiC, and U point (along the M–L line) for 6H-SiC. Thus, the number of conduction band minima in the first Brillouin zone is 3 for 3C-SiC, 3 for 4H-SiC, and 6 for 6H-SiC. Because Si-C covalent bonds are common to all SiC polytypes, the valence band structure is similar amongst the different polytypes, except for the splitting. The top of the valence band is doubly degenerate in 3C-SiC, as a result of its cubic symmetry, and the next valence band is shifted 10 meV from the top by the spin–orbit interaction [38]. The crystal field, which exists in all hexagonal polytypes, splits the valence band degeneracy. The magnitudes of the spin–orbit splitting and crystal-field splitting for 4H-SiC are 6.8 and 60 meV, respectively [39].

Table 2.3 summarizes the effective masses of electrons and holes in 3C-, 4H-, and 6H-SiC [40–42]. The electron effective mass and its anisotropy depend strongly on the polytype, while the hole effective mass exhibits a weak polytype dependence. The former leads to large variation of electron mobility in different polytypes, and also to anisotropic electron transport, as explained in Section 2.2.4.

Polytype	Effective mass
Electron effective mass
3C-SiC		0.667	0.68
		0.247	0.23
4H-SiC		0.33	0.31
		0.58	0.57
		0.31	0.28
		0.42	0.40
6H-SiC		2.0	1.83
		—	0.75
		—	0.24
		0.48	0.42
Hole effective mass
3C-SiC		—	0.59
		—	1.32
		—	1.64
4H-SiC		1.75	1.62
		0.66	0.61
6H-SiC		1.85	1.65
		0.66	0.60

The exciton gaps of various SiC polytypes at 2 K are plotted as a function of “hexagonality” in Figure 2.10 [35, 43]. Here, hexagonality means the ratio of the number of hexagonal sites to the total number of Si-C bilayers (hexagonal and cubic sites) in a unit cell (the hexagonality is 1 for 2H-SiC, 0 for 3C-SiC, 1/2 for 4H-SiC, and 1/3 for 6H-SiC). It is interesting that the bandgap of SiC polytypes increases monotonically with increasing hexagonality. The bandgap at room temperature is 2.36 eV for 3C-SiC, 3.26 eV for 4H-SiC, and 3.02 eV for 6H-SiC. Figure 2.11 shows the temperature dependence of the bandgap for several SiC polytypes [44]. The bandgap decreases with increasing temperature because of thermal expansion, and its temperature dependence can be semi-empirically expressed as [45]:

2.7

where is the bandgap at the absolute temperature, and and are fitting parameters . Note that the bandgap also depends on the doping density; very high impurity doping, above , causes the bandgap to shrink because of the formation of pronounced tail states near the band edges [46].

2.2.2 Optical Absorption Coefficient and Refractive Index

Figure 2.12 shows the optical absorption coefficients versus photon energy for the major SiC polytypes [47, 48]. Because of the indirect band structure of SiC, the absorption coefficient slowly increases, even when the photon energy exceeds the bandgap. Taking account of phonon absorption and emission, the absorption coefficient can be approximated as [49]:

2.8

Here is the photon energy, the energy of a phonon involved, the Boltzmann constant, and the parameter. When several different phonons are involved, the sum of those contributions must be calculated. The absorption coefficient of 4H-SiC at room temperature is at 365 nm (3.397 eV, Hg lamp), at 355 nm (3.493 eV, 3HG Nd-YAG laser), at 325 nm (3.815 eV, He-Cd laser), and at 244 nm (5.082 eV, 2HG Ar ion laser). These values should be kept in mind when SiC materials are characterized by any optical technique, or when SiC-based photodetectors are fabricated. For example, the penetration depth, as defined by , is at 365 nm, at 325 nm, and at 244 nm for 4H-SiC at room temperature.

Figure 2.13 shows the refractive index of 4H-SiC versus wavelength across a wide range, from ultraviolet to infrared, at various temperatures [50]. This dispersion of the refractive index is described by a simple Sellmeier equation given by [51]:

2.9

where are parameters. The refractive index at a wavelength of 600 nm is 2.64 for 4H-SiC. The thermo-optic coefficient, defined by , is in the visible–infrared region and increases to near the ultraviolet region, due to the shrinkage of the bandgap at elevated temperature [50]. The relative dielectric constant has also been reported for several SiC polytypes [45, 52]. The relative dielectric constants in the high-frequency (100 kHz to 1 MHz) region for 4H-SiC (6H-SiC) at room temperature are 9.76 (9.66) perpendicular to the c-axis and 10.32 (10.03) parallel to the c-axis [52]. The dielectric constant of 3C-SiC is isotropic, 9.72.

2.2.3 Impurity Doping and Carrier Density

SiC is an exceptional wide bandgap semiconductor, in the sense that control of both n- and p-type doping over a wide range is relatively easy. Nitrogen or phosphorus are employed for n-type doping and aluminum for p-type doping. Although boron was also previously employed as an acceptor, it is currently not preferred because of its large ionization energy [53], generation of a boron-related deep level (D center) [53, 54], and its abnormal diffusion [54, 55]. Gallium and arsenic work as acceptor and donor, respectively, in SiC. Their ionization energies are, however, relatively large, and their solubility limits are low. Nitrogen substitutes at the C sub-lattice site, while phosphorus, aluminum, and boron substitute at the Si sub-lattice site.

Table 2.4 shows the nonpolar covalent radii of Si, C, and major dopants for SiC [56]. The ionization energies and the solubility limits of nitrogen, phosphorus, and aluminum in major SiC polytypes are summarized in Table 2.5 [10, 11, 53, 57–62]. In SiC, the ionization energies of dopants depend on the lattice site, in particular, whether the site is hexagonal or cubic (site effect). In the case of nitrogen or phosphorus doping, the ionization energy of the donors is relatively small, and the ionization ratio of donors at room temperature is reasonably high, ranging from 50 to nearly 100%, depending on polytype and doping density. Conversely, the ionization energy of aluminum is large (200–250 meV), and incomplete ionization (5–30%) of acceptors is observed at room temperature. Note that the ionization energy decreases when the doping density is increased, as a result of bandgap shrinkage and formation of an impurity band. The dependence of dopant ionization energy, , on the dopant density is described by Efros et al. [63]:

2.10

Here is the ionization energy in lightly-doped materials, the dopant density, and a parameter . When the dopant density exceeds , the ionization energy decreases sharply. As a result, near-perfect ionization is observed in heavily aluminum-doped SiC , in spite of the relatively large ionization energy of aluminum [64].

Atom	Si	C	N	P	B	Al
Radius (Å)	1.17	0.77	0.74	1.10	0.82	1.26

	Nitrogen	Phosphorus	Aluminum	Boron (shallow)
Ionization energy (meV)
3C-SiC	55	—	250	350
4H-SiC (hexagonal/cubic)	61/126	60/120	198/201	280
6H-SiC (hexagonal/cubic)	85/140	80/130	240	350

Because the band structure (bandgap, effective mass) is known, one can calculate the effective densities of states in the conduction band and valence band as well as the intrinsic carrier density as follows [65]:

2.11

2.12

2.13

Here, is the number of conduction band minima, the density-of-state effective mass of electrons (holes), and the Planck constant. By using the density-of-state effective mass of electrons (holes) and the number of conduction band minima, the and values for 4H-SiC at room temperature are calculated as and , respectively. These values are important as they allow us to estimate whether the material will be degenerate when heavy impurity doping is performed. Figure 2.14 plots the temperature dependence of (a) the effective densities of states in the bands and (b) the intrinsic carrier density for major SiC polytypes, together with that of Si. Here, the temperature dependence of bandgaps is taken into account. The intrinsic carrier density at room temperature is extremely low in SiC, because of the wide bandgap, about for 3C-SiC, for 4H-SiC, and for 6H-SiC. This is the main reason why SiC electronic devices can operate at high temperatures with low leakage current.

Based on the Boltzmann approximation for a nondegenerate semiconductor, the neutrality equations in a semiconductor containing one type of donor or acceptor are given by [66]:

2.14

2.15

Here is the free electron (hole) density, the density of compensating acceptor (donor) levels, the donor (acceptor) density, the ionization energy of the donor (acceptor), and are the degeneracy factors for donors (acceptors), respectively. When multiple donor (or acceptor) levels exist, the sum for corresponding dopants must be considered in the right-hand term of the equation. This is the case for hexagonal SiC polytypes, because the donor (and acceptor) impurities at inequivalent lattice sites (e.g., , h for 4H-SiC) exhibit different energy levels. The Arrhenius plots of the free carrier density in (a) nitrogen-doped and (b) aluminum-doped 4H-SiC are shown in Figure 2.15. Here, the temperature dependence of the bandgap and the doping-density dependence of the ionization energies are taken into account. A compensating-level density of is assumed. As shown in Figure 2.15, incomplete ionization is significant for p-type SiC. (See Appendix A.)

The position of the Fermi level in nondegenerate semiconductors is calculated by [65]:

2.16

2.17

Here is the energy of the conduction (valence) band edge. Figure 2.16 shows the Fermi level for nitrogen- or aluminum-doped 4H-SiC as a function of temperature and impurity concentration, taking into account the temperature dependence of the bandgap and the incomplete ionization of dopants at low temperature. Because of the wide bandgap, the Fermi level does not approach the midgap (intrinsic level) even at a fairly high temperature of 700–800 K; this is as expected from the very low intrinsic carrier density shown in Figure 2.13.

2.2.4 Mobility

Figure 2.17 shows (a) the low-field electron mobility versus donor density and (b) the hole mobility versus acceptor density for 4H-SiC and 6H-SiC at room temperature. The electron mobility of 4H-SiC is almost double that of 6H-SiC at a given dopant density, and 4H-SiC exhibits a slightly higher hole mobility than 6H-SiC. The low-field electron and hole mobilities can be expressed by Caughey–Thomas equations as follows [64, 67–72]:

2.18

2.19

2.20

2.21

Here and are given in units of . The slight differences in the doping-dependence parameters between 4H- and 6H-SiC originate from the differences in ionization energies of the dopants. It should be noted that hexagonal (and rhombohedral) SiC polytypes exhibit strong anisotropy in electron mobility [67, 73]. The data shown in Figure 2.17 are mobilities perpendicular to the c-axis. The anisotropy is particularly notable in 6H-SiC, where the electron mobility along the c-axis direction is only 20–25% of that perpendicular to the c-axis (the maximum electron mobility along the c-axis is about in 6H-SiC at room temperature) [67]. The mobility anisotropy is relatively small in 4H-SiC, where the electron mobility along the c-axis direction is approximately at room temperature, which is 20% higher than that perpendicular to the c-axis. This is one of the major reasons why 4H-SiC is the most attractive polytype for vertical power devices fabricated on wafers. The bulk mobility in 3C-SiC is isotropic. The electron mobility in lightly doped 3C-SiC is in experiments [74] and is predicted to be in high-quality material [75]. In nondegenerate semiconductors, the diffusion coefficients of carriers can be obtained by using the Einstein relation [65]:

2.22

Here is the elementary charge. If the carrier lifetime is given, the diffusion length is given by .

Figure 2.18 shows (a) the low-field electron mobility versus donor density and (b) the hole mobility versus acceptor density for 4H-SiC at different temperatures [69–72]. At high temperature, the doping dependence of mobility becomes small, because the influence of impurity scattering decreases. In general, the temperature dependence of mobility is discussed by using a relationship of , where is the mobility and the absolute temperature. As seen from Figure 2.18, the value depends strongly on the doping density, since the dominant scattering mechanism varies for SiC with different doping density. For example, the value is 2.6 for lightly-doped and 1.5 for highly-doped n-type 4H-SiC [70].

Figure 2.19 shows the resistivity versus doping density at 300 K for nitrogen- or aluminum-doped 4H-SiC [64, 69–72]. In very heavily doped materials, the resistivity decreases to for n-type and for p-type. Note that the data shown in Figure 2.19 are obtained in high-quality epitaxial layers. In ion-implanted SiC, where a high density of point and extended defects is created, the resistivity is significantly higher than that shown in the figure for any given doping density. Substrates grown by sublimation (or other techniques) also show higher resistivities than those shown in Figure 2.19 because of a higher density of unwanted impurities and point defects.

The temperature dependence of electron mobility in nitrogen-doped 4H-SiC is shown in Figure 2.20, for donor densities of (a) and (b) [69]. Carrier scattering processes include acoustic-phonon scattering (ac), polar-optical-phonon scattering (pop), nonpolar-optical-phonon scattering (npo), intervalley scattering by phonons (iph), ionized-impurity scattering (ii), and neutral-impurity scattering (ni). In the figures, electron mobility determined by each scattering process is indicated, and the total mobility is approximately expressed according to Matthiessen's rule [76]:

2.23

In lightly-doped n-type SiC, the electron mobility is mainly determined by acoustic phonon scattering at low temperature (70–200 K) and by intervalley scattering at temperatures higher than 300 K, which is similar to the case of Si. In heavily-doped n-type SiC, the major scattering process is neutral impurity scattering at low temperature and intervalley scattering at high temperature.

Figure 2.21 shows the temperature dependence of hole mobility in aluminum-doped 4H-SiC with acceptor densities of (a) and (b) [72]. Mobilities determined by several scattering processes are also plotted. In moderately-doped p-type SiC, the hole mobility is mainly determined by acoustic phonon scattering at or below room temperature, and by nonpolar optical phonon scattering at high temperature . In heavily-doped p-type SiC, the major scattering process is neutral impurity scattering over a wide temperature range, since most Al acceptors remain neutral because of their large ionization energy.

2.2.5 Drift Velocity

At low electric fields, the drift velocity of carriers is proportional to the electric field strength . When the electric field is high, the accelerated carriers transfer more energy to the lattice by emitting more phonons, leading to nonlinear field dependence of drift velocity [76]. The electric field dependence of the drift velocity is expressed by [76]:

2.24

where is the sound velocity in a semiconductor and the parameter. At sufficiently high electric fields, carriers start to interact with optical phonons, and finally the drift velocity becomes saturated. The saturated drift velocity is approximately given by [65, 76]:

2.25

where is the energy of the optical phonon (LO (longitudinal optical) phonon) emitted. Figure 2.22 shows the measured drift velocity of electrons versus applied electric field for n-type (a) 4H-SiC and (b) 6H-SiC [77]. The measurements were conducted in a structure carefully designed to minimize inaccuracy in potential distribution. For 4H-SiC, a low-field mobility of was determined from the slope at low electric fields at room temperature; this agrees with the data shown in Figure 2.17 for the donor density of this particular sample. The saturated drift velocity is determined as at room temperature. This value is also in good agreement with that estimated from Equation 2.25. As indicated in Figure 2.22, the saturated drift velocity decreases with increasing temperature. Note that a so-called transferred-electron effect (Gunn effect) is not observed in SiC because of its indirect band structure. The saturated drift velocity of electrons in 6H-SiC is experimentally estimated as [77, 78]. Although the saturated drift velocity of holes in SiC has not been experimentally studied, it can be estimated at for 4H-SiC from Equation 2.25.

2.2.6 Breakdown Electric Field Strength

When a very high electric field is applied to a pn junction or Schottky barrier in the reverse-bias direction, the leakage current increases as a result of generation of electron–hole pairs, and the junction eventually breaks down. The breakdown mechanisms can be classified into (i) avalanche breakdown and (ii) Zener (tunneling) breakdown [65, 79]. For junctions with a lightly-doped region, avalanche breakdown is dominant; this is the case for most power devices. In avalanche breakdown, the carriers can gain enough energy under very high electric fields to excite electron–hole pairs by impact ionization. The generation of electron–hole pairs is multiplied inside the space-charge region of a junction, eventually leading to breakdown.

Avalanche breakdown is well described by using the impact ionization coefficients of electrons and holes [65, 79]. Breakdown can be defined as when the multiplication factor of the current approaches infinity, which has been shown to be equivalent to the following relationship [65, 79]:

2.26

Here, and are the impact ionization coefficients for electrons and holes, respectively. Integration is performed in the space charge region extending from to . The integral term of the equation is called the ionization integral. Because the impact ionization coefficients depend strongly on the electric field strength, and the field strength is not uniform inside the space-charge region, numerical calculation is required to obtain the ionization integral given by Equation 2.26. Conversely, the impact ionization coefficients can be determined by measuring the multiplication factor as a function of electric field in properly designed pn junction diodes. In the measurements, light illumination is employed to increase the current at low reverse-bias voltages, and thereby to minimize the influence of nonideal leakage current. This is important for accurate determination of the multiplication factors. In general, the impact ionization coefficients are approximately expressed by the Chynoweth equation [80]:

2.27

where and are the parameters and the electric field strength.

Figure 2.23 shows the impact ionization coefficients for electrons and holes in 4H-SiC versus the inverse of electric field strength [81–84]. Different groups have reported similar but slightly different impact ionization coefficients. The ionization coefficients for 4H-SiC are considerably lower than those for Si owing to the wide bandgap of SiC. Another striking feature of Figure 2.23 is that the ionization coefficient for holes is much larger than that for electrons in SiC, which is completely opposite to the case of Si . In 4H-SiC, the energy range of the conduction band is rather small because of the folding effect in the relationship, and the highest energy of hot electrons is limited by the upper edge of the conduction band [85, 86]. This may be the reason why the ionization coefficient for electrons is unusually low in 4H-SiC (and in 6H-SiC). Note that the data shown in Figure 2.23 are extrapolated from several experimental data sets. In particular, the ionization coefficients at relatively low electric fields need more careful investigation. The temperature dependence of the coefficients has been recently reported [84]. It should also be noted that all data in Figure 2.23 are valid along the <0001> direction because they were obtained from 4H-SiC pn diodes on off-axis substrates. Since the carrier acceleration and scattering are strongly influenced by the energy band structure, the impact ionization coefficients depend on the crystallographic orientation. In particular, hexagonal SiC polytypes exhibit strong anisotropy in impact ionization and breakdown characteristics [82, 85, 86].

A semiconductor junction breaks down when the maximum electric field strength reaches a critical value which is inherent to the material. This critical value is called the critical electric field strength or breakdown electric field strength. The critical electric field strength can be determined by calculation of the ionization integral using the impact ionization coefficients described above. Alternatively, it can be obtained experimentally from the breakdown characteristics of devices in which electric field crowding is perfectly suppressed. In n-type Schottky barrier diodes or a one-sided junction, the breakdown voltage is given by [65, 79]:

2.28

Here a non-punchthrough structure is considered. is the dielectric constant of a semiconductor.

Figure 2.24 shows the critical electric field strength versus doping density for 4H-SiC <0001>, 6H-SiC <0001>, and 3C-SiC <111> [80, 81, 87, 88]. The data for Si are also shown for comparison. 4H- and 6H-SiC exhibit approximately eight times higher critical electric field strengths than Si at a given doping density, while the field strength of 3C-SiC is only three or four times higher because this polytype has a relatively small bandgap (similar to GaP). The high critical field strength of hexagonal SiC polytypes is the main reason why SiC is very attractive for power device applications [20, 89, 90]. One must be aware of the fact that the critical field strength is strongly dependent on the doping density, as shown in Figure 2.24. When the doping density is increased, the width of the space-charge region becomes small and the distance for carriers to be accelerated becomes short. Furthermore, the mobility is reduced in highly-doped materials because of enhanced impurity scattering. These are the reasons why the critical electric field strength apparently increases with increasing doping. As shown in Figure 2.24, the critical electric field of 6H-SiC <0001> is slightly higher than that of 4H-SiC <0001>, in spite of its smaller bandgap ( eV for 6H-SiC and 3.26 eV for 4H-SiC). As described in Section 2.2.4, 6H-SiC exhibits strong anisotropy in carrier transport, and the electron mobility along the <0001> direction is unusually low, about even in a high-purity material. The narrow width of the conduction band in 6H-SiC also helps to increase the critical electric field strength of 6H-SiC <0001>. Conversely, it is known that the critical field strength of 6H-SiC is only half that of 6H-SiC <0001> [85]. The anisotropy in critical field strength of 4H-SiC is smaller, and the field strength of 4H-SiC is only 20–25% lower than that of 4H-SiC <0001> [82, 86].

The critical field strength is a convenient physical property when the ideal breakdown voltage is estimated. However, it should be noted that the critical field strength is valid only for junctions with non-punchthrough structures. When punchthrough structures are considered, the critical field strength shown in Figure 2.24 does not give the correct breakdown voltage. In this case, simulation of leakage current or calculation of the ionization integral using a device simulator is required to determine the ideal breakdown voltage. Breakdown voltage is discussed in greater detail in Chapters 7 and 10.

2.3 Thermal and Mechanical Properties

2.3.1 Thermal Conductivity

Figure 2.25 shows the temperature dependence of thermal conductivity for SiC and Si [91, 92]. SiC, with its significant contribution from phonons, has a much higher thermal conductivity ( for high-purity SiC at room temperature) than Si. It has been reported that the thermal conductivity is not sensitive to the SiC polytype, but depends on the doping density and the crystal direction [93]. In heavily-nitrogen-doped 4H-SiC substrates, which are usually employed as for vertical power devices, the thermal conductivity along <0001> is at room temperature [93].

2.3.2 Phonons

Figure 2.26 shows the phonon dispersion relationships for (a) 3C-SiC and (b) 4H-SiC [94, 95]. The basic branches consist of TA (transverse acoustic), LA (longitudinal acoustic), TO (transverse optical), and LO phonons, as in other semiconductors. Due to the large energy of bonds, the phonon frequencies in SiC are high. The unit cell length of the polytype along the c-axis is times larger than the unit length (Si-C bilayer). Thus, the Brillouin zone in the direction of is reduced to of the basic Brillouin zone [31]. The dispersion curves of the phonons propagating along the <0001> direction in such polytypes can be approximated by folding the basic dispersion curve, as shown in Figure 2.24. This zone folding provides new phonon modes at the point, which are called “folded modes.” The number of atoms in the unit cell is 2 for 3C-SiC, 8 for 4H-SiC, and 12 for 6H-SiC. Therefore, the number of phonon branches is 6 for 3C-SiC, 24 for 4H-SiC, and 36 for 6H-SiC, neglecting the degeneracy.

The major phonon energies (or wavenumber) can be directly observed by Raman scattering spectroscopy. Different phonon frequencies in different SiC polytypes enable identification of individual polytypes by Raman scattering measurements [96], detail of which is described in Section 5.1.2. It is known that the observed frequency of LO phonons increases with increasing carrier density because of a carrier–LO phonon coupling effect [97]. Phonon energies are also important in luminescence measurements. In particular, photoluminescence (PL) at low temperature is a powerful tool to characterize the purity and quality of SiC crystals [9, 10, 43, 98–102]. Because SiC has an indirect band structure, phonons are intensively involved in carrier recombination processes. As a result, strong multiple phonon replicas of a zero-phonon emission line are often observed in PL spectra of SiC. For example, the energies of major phonons which create phonon replicas in PL from are 36 (TA), 46, 51, 77 (LA), 95, 96 (TO), 104, and 107 meV (LO). Real PL spectra are described in Section 5.1.1.

2.3.3 Hardness and Mechanical Properties

The mechanical properties of SiC are also unique; SiC is one of the hardest known materials. Table 2.6 shows the major mechanical properties of SiC and Si [30, 45], where the polytype dependence is small. The hardness and Young's modulus (380–700 GPa [103]) of SiC are much higher than those of Si, while the Poisson's ratio (0.21) of SiC is very similar to that of other semiconductors. SiC retains its high hardness and elasticity, even at very high temperatures. The yield (fracture) strength of SiC is as high as 21 GPa at room temperature and is estimated to be 0.3 GPa at 1000 °C, while the yield strength of Si falls to 0.05 GPa at 500 °C [104].

Properties	4H- or 6H-SiC	Si
	3.21	2.33
Young's modulus (GPa)	390–690	160
Fracture strength (GPa)	21	7
Poisson's ratio	0.21	0.22
Elastic constant (GPa)
	501	166
	111	64
	52	—
	553	—
	163	80
Specific heat	0.69	0.7
Thermal conductivity	3.3–4.9	1.4–1.5

2.4 Summary

Table 2.7 summarizes the major physical properties of common SiC polytypes (also see Appendix C). The table includes the low-frequency Baliga's figure-of-merit (BFOM) [105], normalized with respect to the value for Si. For 4H- and 6H-SiC, vertical devices on wafers are considered in the BFOM calculation. Owing to the high critical field strength and high electron mobility along the c-axis, 4H-SiC exhibits a significantly higher BFOM than other SiC polytypes. This is the main reason why 4H-SiC has been almost exclusively employed for power device applications [15, 17, 19, 20, 106–116]. Another advantage of 4H-SiC is that it has slightly smaller donor and acceptor ionization energies compared with those of other SiC polytypes. Furthermore, the availability of single-crystalline wafers with relatively large diameters and reasonable quality has driven fabrication of 4H-SiC-based electronic devices. In fact, the characteristics of commercial 4H-SiC power devices (Schottky barrier diodes and field effect transistors) have already outperformed the theoretical limits of 3C- and 6H-SiC unipolar devices. 3C-SiC is of academic interest to clarify the polytype dependence of physical properties. 3C-SiC may be attractive for relatively low-voltage applications and high-temperature sensors. For more details of physical properties, refer to the following review papers and handbooks [30, 45, 117–125].

Properties/polytype	3C-SiC	4H-SiC	6H-SiC
Bandgap (eV)	2.36	3.26	3.02
Electron mobility
perpendicular to c-axis	1000	1020	450
parallel to c-axis	1000	1200	100
Hole mobility	100	120	100
Electron saturated drift velocity
Hole saturated drift velocity
Breakdown electric field
perpendicular to c-axis	1.4	2.2	1.7
parallel to c-axis	1.4	2.8	3.0
Relative dielectric constant
perpendicular to c-axis	9.72	9.76	9.66
parallel to c-axis	9.72	10.32	10.03
BFOM (n-type, parallel to c-axis) normalized by that of Si	61	626	63
BFOM (p-type, parallel to c-axis) normalized by that of Si, taking account of incomplete ionization of acceptors	2	25	19

References

1. [1] Verma, A.R. and Krishna, P. (eds) (1966) Polymorphism and Polytypism in Crystals, John Wiley & Sons, Inc., New York.
2. [2] Schneer, C.S. (1955) Polymorphism in one dimension. Acta Crystallogr., 8, 279.
3. [3] Cheng, C., Needs, R.J. and Heine, V. (1988) Inter-layer interactions and the origin of SiC polytpes. J. Phys. C, 21, 1049.
4. [4] Denteneer, P.J.H. and van Haeringen, W. (1986) Ground-state properties of polytypes of silicon carbide. Phys. Rev. B, 33, 2831.
5. [5] Fisher, G.R. and Barnes, P. (1990) Towards a unified view of polytypism in silicon carbide. Philos. Mag. B, 61, 217.
6. [6] Heine, V., Cheng, C. and Needs, R.J. (1991) The preference of silicon carbide for growth in the metastable cubic form. J. Am. Ceram. Soc., 74, 2630.
7. [7] Yoo, W.S. and Matsunami, H. (1992) Growth simulation of SiC polytypes and application to DPB-free 3C-SiC on alpha-SiC substrates, in Amorphous and Crystalline Silicon Carbide IV, Springer-Verlag, p. 66.
8. [8] Powell, R.C. (2010) Symmetry, Group Theory, and the Physical Properties of Crystals, Springer.
9. [9] Choyke, W.J. and Patrick, L. (1962) Exciton recombination radiation and phonon spectrum of 6H SiC. Phys. Rev., 127, 1868.
10. [10] Ikeda, M., Matsunami, H. and Tanaka, T. (1980) Site effect on the impurity levels in 4H, 6H, and 15R SiC. Phys. Rev. B, 22, 2842.
11. [11] Suttrop, W., Pensl, G., Choyke, W.J. et al. (1992) Hall effect and infrared absorption measurements on nitrogen donors in 6H-silicon carbide. J. Appl. Phys., 72, 3708.
12. [12] Knippenberg, W.F. (1963) Growth phenomena in silicon carbide. Philips Res. Rep., 18, 161.
13. [13] Yoo, W.S. and Matsunami, H. (1991) Solid-state phase transformation in cubic silicon carbide. Jpn. J. Appl. Phys., 30, 545.
14. [14] Itoh, A., Akita, H., Kimoto, T. and Matsunami, H. (1994) High-quality 4H-SiC homoepitaxial layers grown by step-controlled epitaxy. Appl. Phys. Lett., 65, 1400.
15. [15] Kimoto, T., Itoh, A., Akita, H. et al. (1995) Step-controlled epitaxial growth of α-SiC and application to high-voltage Schottky rectifiers, in Compound Semiconductors—1994, IOP, p. 437.
16. [16] Raghunathan, R., Alok, D. and Baliga, B.J. (1995) High voltage 4H-SiC Schottky barrier diodes. IEEE Electron Device Lett., 16, 226.
17. [17] Itoh, A., Kimoto, T. and Matsunami, H. (1995) High-performance of high-voltage 4H-SiC Schottky barrier diodes. IEEE Electron Device Lett., 16, 280.
18. [18] Kordina, O., Bergman, J.P., Hallin, C. and Janzen, E. (1996) The minority carrier lifetime of n-type 4H- and 6H-SiC epitaxial layers. Appl. Phys. Lett., 69, 679.
19. [19] Tan, J., Cooper, J.A. Jr. and Melloch, M.R. (1998) High-voltage accumulation-layer UMOSFETs in 4H-SiC. IEEE Electron Device Lett., 19, 467.
20. [20] Cooper, J.A. Jr., Melloch, M.R., Singh, R. et al. (2002) Status and prospects for SiC power MOSFETs. IEEE Trans. Electron Devices, 49, 658.
21. [21] Nishino, S., Powell, A. and Will, H.A. (1983) Production of large-area single-crystal wafers of cubic SiC for semiconductor devices. Appl. Phys. Lett., 42, 460.
22. [22] Nishino, S., Suhara, H., Ono, H. and Matsunami, H. (1987) Epitaxial growth and electric characteristics of cubic SiC on silicon. J. Appl. Phys., 61, 4889.
23. [23] Yamanaka, M., Daimon, H., Sakuma, E. et al. (1987) Temperature dependence of electrical properties of n-and p-type 3C-SiC. J. Appl. Phys., 61, 599.
24. [24] Barrett, D.L. and Campbell, R.B. (1967) Electron mobility measurements in SiC polytypes. J. Appl. Phys., 38, 53.
25. [25] Schörner, R., Friedrichs, P., Peters, D. and Stephani, D. (1999) Significantly improved performance of MOSFETs on silicon carbide using the 15R-SiC polytype. IEEE Electron Device Lett., 20, 241.
26. [26] Levinshtein, M.E., Rumyantsev, S.L. and Shur, M.S. (2001) Properties of Advanced Semiconductor Materials: GaN, AlN, InN, BN, SiC, SiGe, John Wiley & Sons, Inc., New York.
27. [27] Sasaki, S., Suda, J. and Kimoto, T. (2012) Doping-induced lattice mismatch and misorientation in 4H-SiC crystals. Mater. Sci. Forum, 717–720, 481.
28. [28] Stockmeier, M., Müller, R., Sakwe, S.A. et al. (2009) On the lattice parameters of silicon carbide. J. Appl. Phys., 105, 033511.
29. [29] Li, Z. and Bradt, R.C. (1986) Thermal expansion of the hexagonal (4H) polytype of SiC. J. Appl. Phys., 60, 612.
30. [30] Harris, G.L. (1995) Properties of Silicon Carbide, INSPEC.
31. [31] Kittel, C. (1995) Introduction to Solid State Physics, 7th edn, John Wiley & Sons, Inc, New York.
32. [32] SEMI http://ams.semi.org/ebusiness/ (accessed 28 March 2014).
33. [33] Lambrecht, W.R.L., Limpijumnong, S., Rashkeev, S.N. and Segall, B. (1997) Electronic band structure of SiC polytypes: A discussion of theory and experiment. Phys. Status Solidi B, 202, 5.
34. [34] Käckel, P., Wenzien, R. and Bechstedt, F. (1994) Electronic properties of cubic and hexagonal SiC polytypes from ab initio calculations. Phys. Rev. B, 50, 10761.
35. [35] van Haeringen, W., Bobbert, P.A. and Backes, W.H. (1997) On the band gap variation in SiC polytypes. Phys. Status Solidi B, 202, 63.
36. [36] Wellenhofer, G. and Rössler, U. (1997) Global band structure and near-band-edge states. Phys. Status Solidi B, 202, 107.
37. [37] Persson, C. and Lindefelt, U. (1997) Relativistic band structure calculation of cubic and hexagonal SiC polytypes. J. Appl. Phys., 82, 5496.
38. [38] Humphreys, R.G., Bimberg, D. and Choyke, W.J. (1981) Wavelength modulated absorption in SiC. Solid State Commun, 39, 163.
39. [39] Sridhara, S.G., Bai, S., Shigiltchoff, O. et al. (2000) Differential absorption measurement of valence band splittings in 4H SiC. Mater. Sci. Forum, 338–342, 567.
40. [40] Chen, W.M., Son, N.T., Janzen, E. et al. (1997) Effective masses in SiC determined by cyclotron resonance experiments. Phys. Status Solidi A, 162, 79.
41. [41] Volm, D., Meyer, B.K., Hofmann, D.M. et al. (1996) Determination of the electron effective-mass tensor in 4H SiC. Phys. Rev. B, 53, 15409.
42. [42] Son, N.T., Persson, C., Lindefelt, U. et al. (2004) Cyclotron resonance studies of effective masses and band structure in SiC, in Silicon Carbide—Recent Major Advances (eds W.J. Choyke, H. Matsunami and G. Pensl), Springer, p. 437.
43. [43] Choyke, W.J. and Devaty, R.P. (2004) Optical properties of SiC: 1997-2002, in Silicon Carbide—Recent Major Advances (eds W.J. Choyke, H. Matsunami and G. Pensl), Springer, p. 413.
44. [44] Choyke, W.J. (1969) Optical properties of polytypes of SiC: Interband absorption, and luminescence of nitrogen-exciton complexes. Mater. Res. Bull., 4, 141.
45. [45] Adachi, S. (2005) Properties of Group-IV, III-V, and II-VI Semiconductors, John Wiley & Sons, Ltd, Chichester.
46. [46] Persson, C., Lindefelt, U. and Sernelius, B.E. (1999) Band gap narrowing in n-type and p-type 3C-, 2H-, 4H-, 6H-SiC, and Si. J. Appl. Phys., 86, 4419.
47. [47] Choyke, W.J. and Patrick, L. (1960) Absorption of light in alpha silicon carbide near the band edge, in Silicon Carbide (eds J.P. O'Connor and J. Smiltens), Pergamon Press, p. 306.
48. [48] Sridhara, S.G., Devaty, R.P. and Choyke, W.J. (1998) Absorption coefficient of 4H silicon carbide from 3900 to 3250 Å. J. Appl. Phys., 84, 2963.
49. [49] Yu, P.Y. and Cardona, M. (2005) Fundamentals of Semiconductors, 3rd edn, Springer.
50. [50] Watanabe, N., Kimoto, T. and Suda, J. (2012) Thermo-optic coefficients of 4H-SiC, GaN, and AlN for ultraviolet to infrared regions up to 500 ^oC. Jpn. J. Appl. Phys., 51, 112101.
51. [51] Marple, D.T.F. (1964) Refractive index of ZnSe, ZnTe, and CdTe. J. Appl. Phys., 35, 539.
52. [52] Ninomiya, S. and Adachi, S. (1994) Optical constants of 6H–SiC single crystals. Jpn. J. Appl. Phys., 33, 2479.
53. [53] Troffer, T., Schadt, M., Frank, T. et al. (1997) Doping of SiC by implantation of boron and aluminum. Phys. Status Solidi A, 162, 277.
54. [54] Kuznetsov, N.I. and Zubrilov, A.S. (1995) Deep centers and electroluminescence in 4H-SiC diodes with a p-type base region. Mater. Sci. Eng. B, 29, 181.
55. [55] Negoro, Y., Kimoto, T., Matsunami, H. and Pensl, G. (2007) Abnormal out-diffusion of epitaxially doped boron in 4H-SiC caused by implantation and annealing. Jpn. J. Appl. Phys., 46, 5053.
56. [56] Purcell, K.F. and Kotz, J.C. (1977) Inorganic Chemistry, W.B. Saunders.
57. [57] Götz, W., Schöner, A., Pensl, G. et al. (1993) Nitrogen donors in 4H-silicon carbide. J. Appl. Phys., 73, 3332.
58. [58] Greulich-Weber, S. (1997) EPR and ENDOR investigations of shallow impurities in SiC polytypes. Phys. Status Solidi A, 162, 95.
59. [59] Laube, M., Schmid, F., Semmelroth, K. et al. (2004) Phosphorus-related centers in SiC, in Silicon Carbide—Recent Major Advances (eds W.J. Choyke, H. Matsunami and G. Pensl), Springer, p. 493.
60. [60] Schöner, A., Nordell, N., Rottner, K. et al. (1996) Dependence of the aluminium ionization energy on doping concentration and compensation in 6H-SiC. Inst. Phys. Conf. Ser., 142, 493.
61. [61] Ivanov, I.G., Henry, A. and Janzén, E. (2005) Ionization energies of phosphorus and nitrogen donors and aluminum acceptors in 4H silicon carbide from the donor-acceptor pair emission. Phys. Rev. B, 71, 241201.
62. [62] Vodakov, Y.A. and Mokhov, E.N. (1974) Diffusion and solubility of impurities in silicon carbide, in Silicon Carbide (eds R.C. Marshall, J.W. Faust Jr. and C.E. Ryan), University of South Carolina Press, p. 508.
63. [63] Efros, A.L., Lien, N.V. and Shklovskii, B.I. (1979) J. Phys. C: Solid State Phys., 12, 1869.
64. [64] Kimoto, T., Itoh, A. and Matsunami, H. (1997) Step-controlled epitaxial growth of high-quality SiC layers. Phys. Status Solidi B, 202, 247.
65. [65] Sze, S.M. and Ng, K.K. (2007) Physics of Semiconductor Devices, 3rd edn, John Wiley & Sons, Inc, Hoboken, NJ.
66. [66] Pensl, G. and Choyke, W.J. (1993) Electrical and optical characterization of SiC. Physica B, 185, 264.
67. [67] Schaffer, W.J., Negley, G.H., Irvine, K.G. and Palmour, J.W. (1994) Conductivity anisotropy in epitaxial 6H and 4H SiC. Mater. Res. Soc. Symp. Proc., 339, 595.
68. [68] Iwata, H., Itoh, K.M. and Pensl, G. (2000) Theory of the anisotropy of the electron Hall mobility in n-type 4H- and 6H-SiC. J. Appl. Phys., 88, 1956.
69. [69] Pernot, J., Zawadzki, W., Contreras, S. et al. (2001) Electrical transport in n-type 4H silicon carbide. J. Appl. Phys., 90, 1869.
70. [70] Kagamihara, S., Matsuura, H., Hatakeyama, T. et al. (2004) Parameters required to simulate electric characteristics of SiC devices for n-type 4H-SiC. J. Appl. Phys., 96, 5601.
71. [71] Matsuura, H., Komeda, M., Kagamihara, S. et al. (2004) Dependence of acceptor levels and hole mobility on acceptor density and temperature in Al-doped p-type 4H-SiC epilayers. J. Appl. Phys., 96, 2708.
72. [72] Koizumi, A., Suda, J. and Kimoto, T. (2009) Temperature and doping dependencies of electrical properties in Al-doped 4H-SiC epitaxial layers. J. Appl. Phys., 106, 013716.
73. [73] Schadt, M., Pensl, G., Devaty, R.P. et al. (1994) Anisotropy of the electron Hall mobility in 4H, 6H, and 15R silicon carbide. Appl. Phys. Lett., 65, 3120.
74. [74] Shinohara, M., Yamanaka, M., Daimon, H. et al. (1988) Growth of high-mobility 3C-SiC epilayers by chemical vapor deposition. Jpn. J. Appl. Phys., 27, L434.
75. [75] Nelson, W.E., Halden, F.A. and Rosengreen, A. (1966) Growth and properties of β-SiC single crystals. J. Appl. Phys., 37, 333.
76. [76] Hamaguchi, C. (2010) Basic Semiconductor Physics, 2nd edn, Springer.
77. [77] Khan, I.A. and Cooper, J.A. Jr. (2000) Measurement of high-field electron transport in silicon carbide. IEEE Trans. Electron Devices, 47, 269.
78. [78] von Munch, W. and Pfaffeneder, I. (1977) Saturated electron drift velocity in 6H silicon carbide. J. Appl. Phys., 48, 4823.
79. [79] Baliga, B.J. (1996) Physics of Semiconductor Power Devices, JWS Publishing.
80. [80] Chynoweth, A.G. (1958) Ionization rates for electrons and holes in silicon. Phys. Rev., 109, 1537.
81. [81] Konstantinov, A.O., Wahab, Q., Nordell, N. and Lindefelt, U. (1997) Ionization rates and critical fields in 4H silicon carbide. Appl. Phys. Lett., 71, 90.
82. [82] Hatakeyama, T. (2009) Measurements of impact ionization coefficients of electrons and holes in 4H-SiC and their application to device simulation. Phys. Status Solidi A, 206, 2284.
83. [83] Loh, W.S., Ng, B.K., Ng, J.S. et al. (2008) Impact ionization coefficients in 4H-SiC. IEEE Trans. Electron Devices, 55, 1984.
84. [84] Niwa, H., Suda, J., and Kimoto, T. (2013) International Conference on Silicon Carbide and Related Materials 2013, Miyazaki, Th-2A-1 Measurement of impact ionization coefficient in 4H-SiC toward ultrahigh-voltage power devices.
85. [85] Dmitriev, A.P., Konstantinov, A.O., Litvin, D.P. and Sankin, V.I. (1983) Impact ionization and super-lattice in 6H-SiC. Sov. Phys. Semicond., 17, 686.
86. [86] Nakamura, S., Kumagai, H., Kimoto, T. and Matsunami, H. (2002) Anisotropy in breakdown field of 4H-SiC. Appl. Phys. Lett., 80, 3355.
87. [87] Palmour, J.W., Edmond, J.A., Kong, H.S. and Carter, C.H. Jr. (1993) 6H-silicon carbide devices and applications. Physica B, 185, 461.
88. [88] Neudeck, P.G., Larkin, D.J., Starr, E. et al. (1994) Electrical properties of epitaxial 3C- and 6H-SiC p-n junction diodes produced side-by-side on 6H-SiC substrates. IEEE Trans. Electron Devices, 41, 826.
89. [89] Bhatnagar, M. and Baliga, B.J. (1993) Comparison of 6H-SiC, 3C-SiC, and Si for power devices. IEEE Trans. Electron Devices, 40, 645.
90. [90] Ruff, M., Mitlehner, H. and Helbig, R. (1994) SiC devices: physics and numerical simulation. IEEE Trans. Electron Devices, 41, 1040.
91. [91] Slack, G.A. (1964) Thermal conductivity of pure and impure silicon, silicon carbide, and diamond. J. Appl. Phys., 35, 3460.
92. [92] Morelli, D., Heremans, J., Beetz, C. et al. (1994) Carrier concentration dependence of the thermal conductivity of silicon carbide. Inst. Phys. Conf. Ser., 137, 313.
93. [93] Hobgood, H.M., Brady, M., Brixius, W. et al. (2000) Status of large diameter SiC crystal growth for electronic and optical applications. Mater. Sci. Forum, 338–342, 3.
94. [94] Feldman, D.W., Parker, J.H. Jr., Choyke, W.J. and Patrick, L. (1968) Phonon dispersion curves by Raman scattering in SiC, polytypes 3C, 4H, 6H, 15R, and 21. Phys. Rev., 173, 787.
95. [95] Bechstedt, F., Käckell, P., Zywietz, A. et al. (1997) Polytypism and properties of silicon carbide. Phys. Status Solidi B, 202, 35.
96. [96] Nakashima, S. and Harima, H. (1997) Raman investigation of SiC polytypes. Phys. Status Solidi A, 162, 39.
97. [97] Harima, H., Nakashima, S. and Uemura, T. (1995) Raman scattering from anisotropic LO-phonon–plasmon–coupled mode in n-type 4H–and 6H–SiC. J. Appl. Phys., 78, 1996.
98. [98] Dean, P.J., Choyke, W.J. and Patrick, L. (1977) The location and shape of the conduction band minima in cubic silicon carbide. J. Lumin., 10, 299.
99. [99] Devaty, R.P. and Choyke, W.J. (1997) Optical properties of silicon carbide polytypes. Phys. Status Solidi A, 162, 5.
100. [100] Egilsson, T., Ivanov, I.G., Son, N.T. et al. (2004) Exciton and defect photoluminescence from SiC, in Silicon Carbide, Materials, Processing, and Devices (eds Z.C. Feng and J.H. Zhao), Taylor & Francis, p. 81.
101. [101] Janzen, E., Gali, A., Henry, A. et al. (2008) Defects in SiC, in Defects in Microelectronic Materials and Devices, Taylor & Francis Group, p. 615.
102. [102] Steeds, J., Sullivan, W., Furkert, S. et al. (2008) Creation and identification of the two spin states of dicarbon antisite defects in 4H-SiC. Phys. Rev. B, 77, 195203.
103. [103] Zorman, C.A. and Parro, R.J. (2008) Micro- and nanomechanical structures for silicon carbide MEMS and NEMS. Phys. Status Solidi B, 245, 1404.
104. [104] Suzuki, T., Yonenaga, I. and Kirchner, H.O.K. (1995) Yield strength of diamond. Phys. Rev. Lett., 75, 3470.
105. [105] Baliga, B.J. (1989) Power semiconductor device figure of merit for high-frequency applications. IEEE Electron Device Lett., 10, 455.
106. [106] Itoh, A. and Matsunami, H. (1997) Analysis of Schottky barrier heights of metal/SiC contacts and its possible application to high-voltage rectifying devices. Phys. Status Solidi A, 162, 389.
107. [107] Powell, A.R. and Rowland, L.B. (2002) SiC materials-progress, status, and potential roadblocks. Proc. IEEE, 90, 942.
108. [108] Cooper, J.A. Jr. and Agarwal, A. (2002) SiC power-switching devices-the second electronics revolution? Proc. IEEE, 90, 956.
109. [109] Elassera, A. and Chow, T.P. (2002) Silicon carbide benefits and advantages for power electronics circuits and systems. Proc. IEEE, 90, 969.
110. [110] Neudeck, P.G., Okojie, R.S. and Chen, L.Y. (2002) High-temperature electronics-a role for wide bandgap semiconductors? Proc. IEEE, 90, 1065.
111. [111] Sugawara, Y. (2004) High voltage SiC devices, in Silicon Carbide—Recent Major Advances (eds W.J. Choyke, H. Matsunami and G. Pensl), Springer, p. 769.
112. [112] Agarwal, A., Ryu, S.H. and Palmour, J. (2004) Power MOSFETs in 4H-SiC: Device Design and Technology, in Silicon Carbide—Recent Major Advances (eds W.J. Choyke, H. Matsunami and G. Pensl), Springer, p. 785.
113. [113] Okumura, H. (2006) Present status and future prospect of widegap semiconductor high-power devices. Jpn. J. Appl. Phys., 45, 7565.
114. [114] Friedrichs, P. (2008) Silicon carbide power-device products–Status and upcoming challenges with a special attention to traditional, nonmilitary industrial applications. Phys. Status Solidi B, 245, 1232.
115. [115] Nakamura, T., Miura, M., Kawamoto, N. et al. (2009) Development of SiC diodes, power MOSFETs and intelligent power modules. Phys. Status Solidi A, 206, 2403.
116. [116] Kimoto, T. (2010) Technical Digest 2010 VLSI Technology Symposium, Honolulu, Hawaii, 2010, p. 9 SiC technologies for future energy electronics.
117. [117] Marshall, R.C., Faust, J.W. Jr. and Ryan, C.E. (eds) (1974) Silicon Carbide 1973, University of South Carolina Press.
118. [118] Davis, R.F., Kelner, G., Shur, M. et al. (1991) Thin film deposition and microelectronic and optoelectronic device fabrication and characterization in monocrystalline alpha and beta silicon carbide. Proc. IEEE, 79, 677.
119. [119] Ivanov, P.A. and Chelnokov, V.E. (1992) Recent developments in SiC single-crystal electronics. Semicond. Sci. Technol., 7, 863.
120. [120] Morkoç, H., Strite, S., Gao, G.B. et al. (1994) Large-band-gap SiC, III-V nitride, and II-VI ZnSe-based semiconductor device technologies. J. Appl. Phys., 76, 1363.
121. [121] Choyke, W.J., Matsunami, H. and Pensl, G. (eds) (1997) Silicon Carbide, A Review of Fundamental Questions and Applications to Current Device Technology, vol. 1 & 2, Akademie Verlag.
122. [122] Zetterling, C.M. (2002) Process Technology for Silicon Carbide Devices, INSPEC.
123. [123] Choyke, W.J., Matsunami, H. and Pensl, G. (eds) (2004) Silicon Carbide—Recent Major Advances, Springer.
124. [124] Pensl, G., Ciobanu, F., Frank, T. et al. (2005) SiC material properties. Int. J. High Speed Electron. Syst., 15, 705.
125. [125] Friedrichs, P., Kimoto, T., Ley, L. and Pensl, G. (eds) (2010) Silicon Carbide, Vol. 1: Growth, Defects, and Novel Applications, Vol. 2: Power Devices and Sensors, Wiley-VCH Verlag GmbH, Weinheim.