2.1.1 Crystallography

SiC occurs in many different crystal structures, called polytypes. A comprehensive introduction to SiC crystallography and polytypism can be found in [30,31]. Despite the fact that all SiC polytypes chemically consist of 50% carbon atoms covalently bonded with 50% silicon atoms, each SiC polytype has its own distinct set of electrical properties. While there are over 170 known polytypes of SiC, only a few are commonly grown in a reproducible form acceptable for use as an electronic semiconductor. The most common polytypes of SiC presently being developed for electronics are the cubic 3C-SiC, the hexagonal 4H-SiC and 6H-SiC, and the rhombohedral 15R-SiC. These polytypes are characterized by the stacking sequence of the biatom layers of the SiC structure.

Fig. 2.1 shows the layer structure of SiC (the [0001] direction) with the tetrahedrally bonded carbon atoms linked to three Si atoms within the bilayer and having a single bond linked to a Si atom in the layer below. If we consider the locations of the carbon atoms within a bilayer these form a hexagonal structure, labeled "A" in the figure. The next bilayer then has the option of positioning its carbon atom in the "B" or the "C" lattice sites. It is this stacking sequence that determines the material polytype.

Figure 2.1: Site locations for C atoms in the [1100] plane.

Figure 2.2: Principal axes (a) for cubic, and (b) for hexagonal crystals.

For cubic crystal, three Miller indices, $hkl$, are used to describe directions and planes in the crystal. These are integers with the same ratio as the reciprocals of the intercepts with x-, y-, and z-axes, respectively, as shown in Fig. 2.2 (a). For hexagonal structures four principal axes are commonly used: a$_1$, a$_2$, a$_3$ and c. Only three are needed to unambiguously identify a plane or a direction, since the sum of the reciprocal intercepts with a$_1$, a$_2$ and a$_3$ is zero. The three a- vectors (with 120$^\circ$ angles between each other) are all in the close-packed plane called a-plane, whereas the c-axis is perpendicular to this plane as depicted in Fig. 2.2 (b).

3C-SiC is the only form of SiC with a cubic crystal lattice structure. Each SiC bilayer can be oriented into only three possible positions with respect to the lattice while the tetrahedral bonding is maintained. If these three layers are arbitrarily denoted A, B and C and the stacking sequence is ABCABC ..., then the crystallographic structure is cubic (zinc-blende). This arrangement is known as 3C-SiC or referred to as $\beta$-SiC, The number 3 refers to the number of layers needed for periodicity.

The non-cubic polytypes of SiC are sometimes ambiguously referred to as $\alpha$-SiC. If the stacking of the bilayer is ABAB ..., then the symmetry is hexagonal (wurtzite) and referred to as 2H-SiC. All of the other SiC polytypes are a mixture of the zinc-blende and wurtzite bonding. 4H-SiC consists of an equal number of cubic and hexagonal bonds with a stacking sequences of ABCB. 6H-SiC is composed of two-thirds cubic bonds and one-third hexagonal bonds with a stacking sequences of ABCACB. The overall symmetry is hexagonal for both polytypes, despite the cubic bonds which are present in each. Similarly, 15R-SiC is a rhombohedral crystal structure composed of three-fifth cubic bonds and two-fifth hexagonal bonds. Stacking sequence and differences among five common SiC polytypes are summarized in Table 2.1.

Table 2.1: Stacking sequences in the c-axis direction for different SiC polytypes.

	Stacking sequence	No. hexagonal (h)	No. cubic (k)
2H	AB	1	0
3C	ABC	0	1
4H	ABCB	1	1
6H	ABCACB	1	2
15R	ABCACBCABACABCB	2	3

Changing of the stacking sequence has a profound effect on the electrical properties, for example the bandgap changes from 3.2 eV for 2H to 2.4 eV for 3C. Because some important electrical device properties are non-isotropic with respect to crystal orientation, lattice site, and surface polarity, some further understanding of SiC crystal structure and terminology is necessary. As discussed much more thoroughly in [32], different polytypes of SiC are actually composed of different stacking sequences of Si-C bilayers (also called Si-C double layers), where each single Si-C bilayer can simplistically be viewed as a planar sheet of silicon atoms coupled with a planar sheet of carbon atoms. The plane formed by a bilayer sheet of Si and C atoms is known as the basal plane, while the crystallographic c-axis direction, also known as the stacking direction or the $[0001]$ direction, is defined normal to the Si-C bilayer plane as shown in Fig. 2.3.

Note that currently only the 4H- and 6H-SiC polytypes are available commercially as substrate material.

Figure 2.3: Stacking sequences for different SiC polytypes in the [1120] plane.

These polytypes require four and six Si-C bilayers, respectively, to define the unit cell repeat distance along the c-axis $[0001]$ directions. The $[1100]$ direction in Fig. 2.3 is often referred to as the a-axis direction. The silicon atoms labeled "h" or "k" denote Si-C double layers that reside in quasi-hexagonal or quasi-cubic environments with respect to their immediately neighboring above and below bilayers. In the 4H stacking sequence of ABCB, all the A sites are the cubic "k" sites and all the B and C sites are the hexagonal "h" sites. Similarly in the 6H stacking sequence of ABCACB, while all the A sites are the hexagonal "h" sites, there are two kinds of inequivalent quasi-cubic sites for B and C, denoted "$k_1$" and "$k_2$"sites [33], respectively, as depicted in Fig. 2.3.

Since the hexagonal polytypes are made up of stacked double layers, several material properties are different along the c-axis or perpendicular to the c-axis. This is called anisotropy, and the degree of anisotropy is measured by the quotient of a parameter value along and perpendicular to the c-axis. Anisotropy of 1 is the same as isotropy, and it is only 3C-SiC that is isotropic. Several of the electrical parameters are anisotropic.

Table 2.2: Mechanical properties of SiC and other semiconductors [34].

	Si	GaAs	3C-SiC	6H-SiC	4H-SiC	Diamond
Lattice $a$ [Å]	5.43	5.65	4.36	3.08	3.08	3.567
Lattice $c$ [Å]	n.a.	n.a.	n.a.	10.05	15.12	n.a.
Bond length [Å]	2.35	2.45	1.89	1.89	1.89	1.54
TEC [10$^{-6}$/K]	2.6	5.73	3.0	4.5	-	0.8
Density [gm/cm$^3$]	2.3	5.3	3.2	3.2	3.2	3.5
Ther. cond. [W/cmK]	1.5	0.5	5	5	5	20
Melting point [$^{\circ}$C]	1420	1240	2830	2830	2830	4000
Mohs hardness			9	9	9	10

TEC = thermal expansion coefficient

T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation