Qualifier

Molecular Beam Epitaxy

**Figure 2.1:** Schematic illustration of the fundamental processes involved in MBE. From [49]
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In this text we propose a detailed computational study of several of the processes present in Molecular Beam Epitaxy (MBE). Before we can proceed with modeling this type of system we must understand the experimental setup involved. MBE provides us with the ability to easily grow complex multilayered semiconductor structures with tailor made characteristics

In an MBE growth experiment, we start with a flat surface of material that constitutes the substrate that we bombard with a flux of particles. In order to avoid contamination and unwanted reactions, the chamber that is used in this process is submitted to an ultrahigh vacuum ( $P<10^{-8} Pa$ ). This is represented schematically in Fig. 2.1. The particles that collide with the substrate become bound to its surface, but are still able to diffuse until they find an adequate crystallographic site where they can be incorporated in to the growing surface. Temperature fluctuations can also cause particles to evaporate and join the vapor inside the chamber. The experimental setup is described in more detail in Sec. 2.2.

Growth can occur in three different modes. The so called Frank-van der Merwe, in which growth occurs layer by layer, and where a layer only forms after the previous one is complete. The Stranski-Krastanow mode, where after the completion of a few complete layers we observe the growth of three dimensional clusters of particles. Finally, we can observe the growth of three dimensional clusters right from the beginning, in which case we are in the presence of the Volmer-Weber growth mode. These three types of growth as represented in Fig. 2.2 are due to the presence of growth instabilities as described in Sec. 2.1.

**Figure 2.2:** Surface growth modes
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Instabilities

During the growth process there are several instabilities that can affect the final characteristics of the experiment. These instabilities must be well studied, understood and controlled if we are to be able to manipulate the MBE process.

Chemical Instability If we don't choose the chemical elements used criteriously, this instability can lead to the appearance of spontaneous composition modulations, non-uniformities in the band-gap of the final material. This instability can also be used to spontaneously obtain structures that are strained on nano-scales.
Lattice Mismatch A mismatch in the lattice parameter of the material that constituted the substrate and of the depositing particles leads to the formation of a thin pseudo-morphic layer. This layer is expected to grow uniformly and commensurately with the strain energy increasing linearly with the thickness up to a threshold. It also possesses a modified band-structure which can be useful for a wide range of practical applications. The lattice mismatch results in an effective elastic interaction between the adatoms as we discuss in Chapter 3.
Ehrlich-Schwoebel This type of instability causes the behavior of the adatoms on the vicinity of steps to behave differently by introducing an energy gain (or loss) for specific events such as climbing up (or descending from) a step. This leads to the formation of the three dimensional clusters characteristic of Volmer-Weber and the Stranski-Krastanow growth modes. This instability will be carefully analyzed in Sec. 4.2

Experimental Setup

In this section we describe in some detail the growth experiments performed during the 1990s by Moison et at [15,38,39]. This description will form the basis of the system that our simulation will try to reproduce. The Molecular Beam Epitaxy experiments consisted of In and As atoms being deposited on an initially atomically flat substrate formed by the surface of GaAs in an ultra high vacuum chamber maintained at $500{}^\circ C$ . The Beam Equivalent Pressure (BEP) of As used was $6\times 10^{-6}$ Torr. The surface growth rate was calibrated using RHEED measurements on homoepitaxial growth of InAs to a constant value of ML/s, where ML corresponds to a complete coverage of the surface by an InAs lattice, $5.46\times 10^{15}$ In atoms/cm ${}^2$ . The growth front could be analyzed in situ using AES and XPS and the morphology was determined ex situ using AFM and TEM. To completely grasp the results that they obtained, it is necessary to understand the imaging and visualization techniques used. These techniques will now be briefly described.

Reflection High-Energy Electron Diffraction (RHEED)

RHEED uses an electron gun and a phosphor screen for creating pictures showing the structure and/or morphology of a crystal surface. RHEED uses high electron energies and low impact angles $(<5{}^\circ )$ . The high energy sharpens the picture, while the low angle makes the electrons interact with just a few atomic layers. This makes RHEED pictures represent the structure of the surface, not the whole crystal. The resulting lines represent the reciprocal lattice of the crystal surface. RHEED is often used for monitoring crystal growth, as it doesn't block the direction vertical to the surface of the crystal.

Atomic Force Microscopy (AFM)

An Atomic Force Microscope (AFM) is a very powerful instrument that is capable of not only imaging but is also one of the foremost tools for the manipulation of matter at the nanoscale. The AFM consists of a cantilever with a sharp tip at its end. The tip is brought into close proximity of a sample surface. The force between the tip and the sample leads to a deflection of the cantilever. Typically, the deflection is measured using a laser spot reflected from the top of the cantilever.

Transmission Electron Microscopy (TEM)

Transmission Electron Microscopy (TEM) is an imaging technique whereby a beam of electrons is focused onto a specimen causing an enlarged version to appear on a fluorescent screen or layer of photographic film, or can be detected by a CCD camera. This type of microscopy allows for much higher resolution that conventional light microscopy due to the smaller wavelength of electrons.

X-ray Photoelectron Spectroscopy (XPS)

X-ray Photoelectron Spectroscopy (XPS) is a surface sensitive analytic tool to study the surface composition and electronic state of a sample. A sample maintained under under ultra high vacuum is bombarded with X-rays . The X-rays penetrate several millimeters into the sample and excite electrons (commonly referred to as photoelectrons). The small fraction of these electrons from the top $\approx 50$ Åis detected. Since the energy of core electrons is very specific to the element that the atom belongs to, this technique gives us information on the elemental composition of the shallow surface region.

Auger Electron Spectroscopy (AES)

Auger electron spectroscopy probes the chemistry of a surface by measuring the energy of electrons emitted from that surface when it is irradiated with electron of energy in the range keV. The physical process by which these electrons are made is called the Auger effect. This technique is useful as a surface analytical technique because the emitted electrons come from no more than a few nanometers deep in the surface.

Results

The RHEED measurements indicate that growth starts by being the usual surface reconstruction process until $\Theta=0.2$ ML. Above which the growth becomes increasing less laminar up until $\Theta_{th}\equiv 1.75\pm0.1$ ML[15,38,39] when 3D clusters start to form. This is characteristic of the Stranski-Krastanow growth mode described above. AFM imaging confirms this transition by showing only step discontinuities below $\Theta_{th}$ and showing dots above it. Another crossover occurs at $\Theta_{coal}\equiv 3$ ML when the dots coalesce to form larger ones. Below $\Theta_{coal}$ all dots have similar sizes and are regularly located. Above it, many large dots exist along side several groups of coalesced dots. This transition can be observed by TEM. Only in the coverage interval $\Theta_{th}<\theta<\Theta_{coal}$ are the dot characteristics sufficiently regular ( $\approx20\mathrm{nm}\times20\mathrm{nm}\times2.5\mathrm{nm}$ [15] and mean interdot distance of $60\mathrm{nm}$ [38]) to permit their utilization as Quantum Dots. In this interval, dots with similar volumes have the same shape independently of the value of $\theta$ that originated them and no exchange of matter between dots is possible before they touch[39] each other at $\Theta_{coal}$ .