Materials Science and Engineering/Derivations/Models of Micro and Nanoscale Processing/Vapor Phase Epitaxial Film Growth

Homoepitaxy
Homoepitaxy is a kind of epitaxy performed with only one material. In homoepitaxy, a crystalline film is grown on a substrate or film of the same material. This technology is applied to growing a more purified film than the substrate and fabricating layers with different doping levels.

Heteroepitaxy
Heteroepitaxy is a kind of epitaxy performed with materials that are different from each other. In heteroepitaxy, a crystalline film grows on a crystalline substrate or film of another material. This technology is often applied to growing crystalline films of materials of which single crystals cannot be obtained and to fabricating integrated crystalline layers of different materials. Examples include gallium nitride (GaN) on sapphire or aluminium gallium indium phosphide (AlGaInP) on gallium arsenide (GaAs).

Examples
$$GaAs$$ on $$AlAs$$

$$Ge_xSi_{1-x}$$ on $$Si$$

$$GaAs$$ on $$Ge$$

$$Au$$ on $$NaCl$$

Surface Reconstruction
Surface reconstruction refers to the process by which atoms at the surface of a crystal assume a different structure than that of the bulk. Surface reconstructions are important in that they help in the understanding of surface chemistry for various materials, especially in the case where another material is adsorbed onto surface.

Basic Principles
In an ideal infinite crystal, the equilibrium position of each individual atom is determined by the forces exerted by all the other atoms in the crystal, resulting in a periodic structure. If a surface is introduced to the system by terminating the crystal along a given plane, then these forces will be altered, changing the equilibrium positions of the remaining atoms. This is most noticeable for the atoms at or near the surface plan, as they now only experience interatomic forces from one direction. This imbalance results in the atoms near the surface assuming positions with different spacing and/or symmetry from the bulk atoms, creating a different surface structure. This change in equilibrium positions near the surface can be categorized as either a relaxation or a reconstruction.

Relaxation refers to a change in the position of entire layers of atoms relative to the bulk positions. Often this is a purely normal relaxation: that is, the surface layers move in a direction normal to the surface plane, usually resulting in a smaller-than-usual inter-layer spacing. This makes intuitive sense, as a surface layer which experiences no forces from the open region can be expected to contract towards the bulk. Most metals experience this type of relaxation. Some surfaces also experience relaxations in the lateral direction as well as the normal, so that the upper layers shift become shifted relative to layers further in to minimize the positional energy.

Reconstruction refers to a change in the two-dimensional structure of the surface layers, in addition to changes in the position of the entire layer. For example, in a cubic material the surface layer might re-structure itself to assume a smaller two-dimension spacing between the atoms as lateral forces from adjacent layers are reduced. The general symmetry of a layer might also change, as in the case of the Pt (100) surface which is reconstructed from a cubic to a hexagonal structure. A reconstruction can affect one or more layer at the surface, and can either conserve the total number of atoms in a layer (a conservative reconstruction) or have a greater or lesser number than in the bulk (a non-conservative reconstruction).

Reconstruction due to Absorption
Relaxations and reconstructions often occur for atomically clean surfaces in vacuum, in which the interaction with another medium is not considered. However, reconstructions can also be induced or altered by the adsorption of other atoms onto the surface as the interatomic forces are changed. These reconstructions can assume a variety of forms as the interactions between different types of atoms are taken into account, but some general principles and dependencies can still be identified.

The reconstruction of a surface with adsorption can be said to be dependent on the following factors:


 * The composition of the substrate and of the adsorbate
 * The coverage of the substrate surface layers and of the adsorbate, measured in monolayers
 * The ambient conditions (i.e. temperature, gas pressure, etc)

Composition plays an important role in that it determines the form that the adsorption process takes, whether by relatively weak physisorption through van der Waals interactions or stronger chemisorption through the formation of chemical bonds between the substrate and adsorbate atoms. Surfaces which undergo chemisorption generally result in larger reconstructions than those which undergo physisorption, as the breaking and formation of bonds between the surface atoms alter the interaction of the substrate atoms as well as the adsorbate.

Example 1: Gold
A very well-known example of surface reconstruction occurs in silicon, a semiconductor commonly used in a variety of computing and microelectronics applications. With a diamond-like face-centered cubic (fcc) lattice, it exhibits several different well-ordered reconstructions depending on temperature and on which crystal face is exposed.

When Si is cleaved along the (100) surface, the ideal diamond-like structure is interrupted and results in a 1x1 square array of surface Si atoms. Each of these has two dangling bonds remaining from the diamond structure, creating a surface which can obviously be reconstructed into a lower-energy structure. The observed reconstruction is a 2x1 periodicity, explained by the formation of dimers which consist of paired surface atoms, decreasing the number of dangling bonds by a factor of two. These dimers reconstruct in rows with a high long-range order, resulting in a surface of filled and empty rows. LEED studies and calculations also indicate that relaxations as deep as five layers into the bulk are also likely to occur.

The Si (111) structure, by comparison, exhibits a much more complex reconstruction. Cleavage along the (111) surface at low-temperatures results in another 2x1 reconstruction, differing from the (100) surface by forming long pi-bonded chains in the first and second surface layers. However, when heated above 400 C this structure converts irreversibly to the more complicated 7x7 reconstruction. In addition, a disordered 1x1 structure is regained at temperatures above 850 C which can be converted back to the 7x7 reconstruction by slow cooling.

The 7x7 reconstruction is modeled according to a dimer-adatom-stacking fault (DAS) model which was constructed by many research groups over a period of 25 years. Extending through the five top layers of the surface, the unit cell of the reconstruction contains 12 adatoms as well as two triangular subunits, nine dimers and a deep corner hole which extends to the fourth and fifth layers. This structure was gradually inferred from LEED and RHEED measurements as well as calculation, and was finally resolved in real space by Binnig, Rohrer, Gerber and Weibel as a demonstration of the STM which was developed by Binnig and Rohrer at IBM's Zurich Research Laboratory. The full structure with positions of all reconstructed atoms has also been confimed by massively parallel computation.

A number of similar DAS reconstructions have also been observed on Si (111) in non-equilibrium conditions in a (2n+1)x(2n+1) pattern, and include 3x3, 5x5 and 9x9 reconstructions. The preference for the 7x7 reconstruction is attributed to an optimal balance of charge transfer and stress, but the other DAS-type reconstructions can be obtained under conditions such as rapid quenching from the disordered 1x1 structure.

Low Energy Electron Diffraction
Low-energy electron diffraction (LEED) is a technique used to characterize the structures of surfaces.

Instrumentation
Usually, LEED experiments are performed in an Ultra high vacuum environment and are often supplemented by Auger Electron Spectroscopy for identifying surface constituents. An ion Gun is often used for surface cleaning. A LEED instrument usually consists of an Electron Gun, a Detector System and Data acquisition system.

Reflection High Energy Electron Diffraction
Reflection high-energy electron diffraction (RHEED) is a technique used to characterize the surface of crystalline materials. RHEED systems gather information only from the surface layer of the sample, which distinguishes RHEED from other materials characterization methods that rely on diffraction of high-energy electrons. Transmission electron microscopy, another common electron diffraction method samples the bulk of the sample due to the geometry of the system. Low energy electron diffraction (LEED) is also surface sensitive, but LEED achieves surface sensitivity through the use of low energy electrons.

Atomistics of Epitaxial Growth

 * Absorb
 * Desorb
 * Diffuse
 * Attach
 * Cluster
 * Reflect

Adatom Diffusion, Absorption, and Desorption
$$D_{SD} = D_0 \exp \left ( \frac{- \Delta G_{SD}^*}{kT_{sub}} \right )$$

$$J = -D \frac{\partial C}{\partial x}$$

Adatom residence time

$$\tau = \tau_0 \exp \left ( \frac{\Delta G_{a/d}^*}{kT_{sub}} \right )$$

Desorption-Limited Regime

$$D_{SD} = D_0 \exp \left ( \frac{-\Delta G_{SD}^*}{kT_{sub}} \right )$$

$$\tau = \tau_0 \exp \left ( \frac{\Delta G_{a/d}^*}{kT_{sub}} \right )$$

$$\delta_{de} \approx (\tau_0 D_0 )^{1/2} \exp \left ( \frac{ \Delta G_{a/d}^* - \Delta G_{SD}^*}{2 kT_{sub}} \right )$$

Adatom Diffusion Distance: Adatom-Clustering-Limited Regime

$$\delta_{ad} \approx \sqrt {D_{SD} \tau_{ad} }$$

$$\tau_{ad} = N_0 / F_{dep}\;$$

$$\delta_{ad} = \frac{N_0^{1/2} D_0^{1/2} \exp \left ( \frac{-\Delta G_{SD}^*}{2kT_{sub}} \right )}{F_{dep}^{1/2}}$$