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\begin{document} 
\title{Silicon Carbide Polytypes and Thermal Pase Transformation}
\author{ J.\ Y.\ Kim\\
	 \\
         Department of Physics\\ 
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}
 
\date{ }
\maketitle
\begin{abstract}

	 
The structure of Silicon carbide (SiC) was reviewed in the lights of 
the polytypism.  It is found hexagonality can be used  
to identify the unknown SiC polytypes.    
Thermal phase 
transformations from cubic SiC to hexagonal SiC and vice versa under high 
temperature are understood with their applications in polytype-controlled 
epitaxial growth.

\end{abstract}

\pagebreak

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Silicon carbide is by far the most widely investigated among polytypic 
substances because it was, for a long time, the only substance known 
to be polytypic and it exhibits the large number of 
beautifully odered modifications. Lately 
it has also received considerable attention on account of its increasing 
technical and industrial inportance.

\section{SiC polytypes and the structures}

Silicon carbide(SiC) shows polytypism which takes a close-packed structure 
with the various atom stacking sequences. It has numerous crystaline forms 
with the same chemical composition (each silicon atom is surrounded 
tetrahedrally by four carbon atoms and vice versa). More than 200 kinds of 
SiC polytypes have been reported. The structure of SiC polytypes can be 
viewed  as stacking SiC double layers on top of one another along the $c$ 
direction. To describe SiC polytypes, Ramsdell notation has been used
which shows the number of layers in the unit cell followed by the lattice 
type H, R or C. Thus a symbol $nH$ represents a structure with an $n$-layered 
repeat period  along the $c$ direction and a primitive hexagonal lattice 
 while 3$mR$
denotes a structure with a 3$m$ layered repeat period and a primitive 
rhombohedral cell. And  3C-SiC is the ony cubic
 modification which has a zinc-blende structure. Lattice structures of 
three dominat polytypes are shown below.
% To insert a postscript figure:
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\centerline{\psfig{figure=figure1.ps,height=2.5in}}

\caption[]{\em{Structure of SiC. Si and C atoms are notated by large open 
and small solid cicles, respectively.}}
\end{figure}

\section{Semiconduting properties}

All SiC polytypes have  indirect band gaps ($E_{Gx}$) and release 
exitons which 
subsequently decay by photon and phonon emission (Fig.2). Some of the  SiC 
polytypes show superior electrical properties as wide-gap semiconductors. 
3C-SiC has a wide band gap (2.2 eV at room temperature), high electron mobility 
(1000 cm$^{2}/(V \cdot s)^{2}$) and high electron drift velocity
($2.4\sim 2.7\times 10^{7}~cm/s$ in the $1 \sim 2 \times 10^{5}~V/cm$ range)
\cite{nel}\cite{ferr} and is recognized as a promising polytype for high-temperature, 
high-power and high-frequency operation of devices. And 6H-SiC with a band-gap
 of 2.86 eV is promising in blue light-emittng diode(LED) application because 
both p- and n-type conduction can be easily obtained by impurity doping compared 
with other candidate materials such as ZnS, ZnSe and GaN.\cite{woo} But because of 
difficulty in producing large-area single crystals of 6H-SiC, the development 
of such electronic devices has been delayed.   

\begin{figure}[htb]
\centerline{\psfig{figure=figure2.ps,height=2.5in}}

\caption[]{\em{The luminescene spectrum of four-particle nitrogen-exiton 
complexes in cubic SiC at 6$^{o}K$. The phonon-free line is labeled {\bf 0} 
and its energy is used as the zero for the upper "phonon energy" scale, 
from which one may read directly the energies of TA, LA, TO, and LO 
phonons. The 10 meV between $E_{Gx}$ and {\bf 0} is a measure of the binding 
energy of the exiciton to the nitrogen atom.}}
\end{figure}

\section{Hexagonality and polytype identification} 

In the structure analysis of SiC polytypes X-ray diffraction method 
has widely been 
used, 
in which postulated possible layer sequences are compared with the 
theoretical intensities of possible structures for best fit.\cite{sha} 
But for long-period polytypes, hexagonality is used as there are relationship between the 
hexagonality and the physical quantities such as band-gap energies and phonon 
properties.\cite{choyke}\cite{naka} 
\\

\begin{tabular}{c c c c}
\hline
Polytype & Exiton energy gap, E$_{Gx}$  & hk notation & hexagonality \\
\hline
3C & 2.390 & {\em k} & 0 \\
8H & 2.80 & {\em hkkk} & 0.25 \\
21R & 2.86 & {\em hkkhkkk} & 0.29 \\
15R & 2.986 & {\em hkhkk} & 0.4 \\
33R & 3.01 & {\em hkkhkkhkkhk} & 0.36 \\
6H & 3.023 & {\em hkk} & 0.33 \\
4H & 3.263 & {\em hk} & 0.5 \\
\hline
\end{tabular}
\\

Hexagonality is defined as 
$\sigma = \rho_{h} /( \rho_{h} + \rho_{k})$ where  
$\rho_{h}$ and $\rho_{k}$ are numbers of layers in a  
hexagonal or cubic 
environment, respectively, which are determined depending on whether the relative orientation 
of atoms in the two neighboring planes is that found in 
hexagonal($h$) or zinc-blende($k$) lattice. Table 1 shows the hexgonalities 
of several SiC polytypes with $E_{Gx}$ at 4$^{o}$K.

 \begin{figure}[htb]
\centerline{\psfig{figure=figure3.ps,height=2.5in}}

\caption[]{\em{A plot of $E_{Gx}$ versus hexagonal(\%).}}
\end{figure}
 

 A plot of $E_{Gx}$ against hexagonality is shown in 
Fig. 3 indicating a roughly linear relationship.\cite{choyke} And Nakashima 
{\em et al.} obtained the correlation between hexagonality  and two 
transeverse optical phonon mode ($\omega (E_{t})$ and $\omega (A_{1t})$) 
spliting frequency which is given by 

\begin{displaymath}
\Delta \omega=\omega (E_{t})- \omega (A_{1t})=29.4 \sigma (cm^{-1})
\end{displaymath}

where $\sigma$ is the hexagonality.
 

\section{Thermal phase transformation}

The most of the SiC polypypes are thermally very stable up to 2000$^{o}$C but
some of them are unstable at such high temperatures and transform to more
stable polytypes. The 2H $\rightarrow$ 3C, 2H $\rightarrow$ 3C $\rightarrow$ 6H,
2H $\rightarrow$ 6H and 3C $\rightarrow$ 6H phase transformation at high
temperatures have been proposed.\cite{kni}\cite{boot}
Two kinds of mechanisms for phase-transformation were suggested.
\cite{kni} One is the idea of transformation via the vapor phase in which 
crystals are vaporized at first and then high temperature polytypes are 
formed by recrystallization from the vapor.

The other idea is the solid-state phase transformation by rearrangement of 
atoms such as the layer displacement caused by the nucleation or propagation of 
stacking faults.\cite{boot}\cite{woo} Table 2 is showing the rate of transformation 
of 2H $\rightarrow$ 3C and 3C $\rightarrow$ 6H from Ref.\cite{boot}.
\\

\begin{tabular}{@{....................................} r|l @{...................................                                }}
 \hline
\ 2H $\rightarrow$ 3C  & 3C $\rightarrow$ 6H\\
 \hline
\end{tabular}

\begin{tabular}{c|c|c|c|c|c}
\hline
Temp. ($^{o}$C) & Time (hr)  & \% 2H & Temp. ($^{o}$C) & Time (hr)  & \% 3C\\
\hline
1600 & 1 & 74 & 1600 & 504 & 100\\
1600 & 2 & 60 & 1700 & 233 & 100\\
1600 & 3 & 48 & 1800 & 114 & 75\\
1600 & 3.5 & 38 & 1900 & 65.5 & 65\\
1600 & 6 & 8 & 2300 & 1 & 50\\
1600 & 7 & 1 & 2300 & 15.5 & 0\\
\hline
\end{tabular}
\\

If one can control atomic arrangements of the surface of 3C-SiC to 6H-SiC by 
temperature, 6H-SiC will be grown on 3C-SiC substrate which will make it 
possible to grow large size crystals of 6H-SiC.




\begin{thebibliography}{99}
\bibitem{nel} W.E.\ Nelson F.A.\ Halden and A.\ Rosengreen, J.\ Appl.\ Phys.\ {\bf 37}, 333(1966).
\bibitem{ferr} D.K.\ Ferry, Phy.\ Rev. B1 {\bf 12} 2361 (1975).
\bibitem{woo} W.S.\ Yoo and H.\ Matsunami, Jpn.\ J. Appl.\ Phys.\ {\bf 30}, 545(1991).
\bibitem{sha} P.T. B.\ Shaffer, Acta Cryst.\ {\bf B64}, 477(1969).
\bibitem{choyke} W.J.\ Choyke, D.R.\ Hamilton and Lyle Patrick. Phys.\  Rev. 4A {\bf 133} A1163 (1964).
\bibitem{naka} S.\ Nakashima, A. Wada  and Z.\ Inoue, J.\ Phys.\ Soc.\ Jpn. {\bf 56}, 3375 (1987).
\bibitem{kni} W.F.\ Knippenberg, Philips Res.\ Rep.\ {\bf 18}, 161(1963).
\bibitem{boot} G.A.\ Bootsma W.F.\ Knippenberg and G.\ Verspui,  J. Cryst.\ Growth\ {\bf 8}, 341(1971).

\end{thebibliography}
\end{document}

