Epitaxy is the growth of crystalline materials from a crystalline wafer. Most microfabrication deposition methods leave amorphous or polycrystalline films on a wafer. While crystalline materials are generally preferred, epitaxial growth is difficult and expensive. However, there are some applications for which epitaxy is worth the trouble. One application is the production of a lightly-doped semiconductor layer above a heavily-doped buried layer. Another is the growth of one crystalline material on top of another (i.e. AlGaAs on top of a GaAs wafer for use in optoelectronics). There are three types of epitaxy--vapor phase epitaxy (VPE), liquid phase epitaxy (LPE), and molecular beam epitaxy (MBE).
Vapor phase epitaxy is the most common epitaxial growth method. VPE is basically chemical vapor deposition (CVD) with conditions such that the deposited film is crystalline. A common VPE reaction is SiCl4(gas) + 2H2(gas) <---->Si(solid) + 4HCl(gas) at 1200 °C. Ensuring that polysilicon does not get formed in such a reaction is an arduous task. The gases arsine, diborane, or phosphine could be added to the reactants to dope the epitaxial silicon layer in situ. Typical VPE growth rates are about 1μm/min--faster growths yield polycrystalline films.
Liquid phase epitaxy is very similar to Czochralski growth. The material to be deposited is melted into liquid form, and the wafer acts as a seed for crystalline growth. Typical LPE growth rates are between 0.1 and 1.0μm/min.
Molecular beam epitaxy became more important due to the advent of optical communication and the need for lasers. In the MBE process, a beam of atoms or molecules is directed onto the wafer. The process takes place in ultrahigh vacuum (10-8 Pascals). The wafer is heated to between 400 and 900 °C. The growth rates of MBE processes of between 1nm/min to 300nm/min are very low. The low growth rates allow precise, molecular-scale control of epitaxial film thicknesses. However, they result in very low outputs, so MBE is quite expensive. MBE is used to make gallium arsenide quantum well lasers.
I used the book Introduction to Microelectronic Fabrication Volume V by Richard Jaeger as a reference.