A quantum well is a potential well wherein the energy spectrum of charge carriers is discrete.
As opposed to a bulk region, wherein the carriers are free to move in three spatial directions, their motion is free only in two (planar) directions in a quantum well. The quantum size effects take place when the size of a region in at least one of the directions (the direction of growth in the case of semiconductor heterostructure quantum wells - the transverse direction) becomes comparable to the de Broglie wavelength of the carriers (electrons and holes in a semiconductor), resulting in a discrete energy spectrum for them.
Semiconductor quantum wells can be realized in double heterostructures. The double heterostructure concept was proposed in 1963 independently by Herbert Kroemer and by Zhores Alferov and Rudolf Kazarinov.Zh. I. Alferov and R.F. Kazarinov, Authors Certificate 28448 (U.S.S.R) 1963.
History In 1970, Leo Esaki and Raphael Tsu invented synthetic superlattices. They also suggested that a heterostructure made up of alternating thin layers of semiconductors with different band-gaps should exhibit interesting and useful properties. Since then, much effort and research has gone into studying the physics of quantum well systems as well as developing quantum well devices.
The development of quantum well devices is greatly attributed to the advancements in crystal growth techniques. This is because quantum well devices require structures that are of high purity with few defects. Therefore, having great control over the growth of these heterostructures allows for the development of semiconductor devices that can have very fine-tuned properties.
Semiconductor materials and devices, and particularly those based on quantum wells, have been a hot topic in the physics research. The development of high-speed and optoelectronic devices based on semiconductor heterostructures has been recognized in the Nobel Prize in Physics for Zhores Alferov and Herbert Kroemer in 2000.
The theory surrounding quantum well devices has led to significant advancements in the production and efficiency of many modern components such as light-emitting diodes, transistors for example. Today, such devices are ubiquitous in modern cell phones, computers, and many other computing devices.
Fabrication Quantum wells are formed in semiconductors by having a material, like gallium arsenide, sandwiched between two layers of a material with a wider bandgap, like aluminum arsenide. (Other examples: a layer of indium gallium nitride sandwiched between two layers of gallium nitride.) These structures can be grown by molecular beam epitaxy or chemical vapor deposition with control of the layer thickness down to monolayers.
Thin metal films can also support quantum well states, in particular, thin metallic overlayers grown in metal and semiconductor surfaces. The vacuum-metal interface confines the electron (or hole) on one side, and in general, by an absolute gap with semiconductor substrates, or by a projected band-gap with metal substrates.
There are three main approaches to growing a QW material system: lattice-matched, strain-balanced, and strained. Lattice-matched system: In a lattice-matched system, the well and the barrier have a similar lattice constant as the underlying substrate material. With this method, the bandgap difference there is minimal dislocation but also a minimal shift in the absorption spectrum. Strain-balanced system: In a strain-balanced system, the well and barrier are grown so that the increase in lattice constant of one of the layers is compensated by the decrease in lattice constant in the next compared to the substrate material. The choice of thickness and composition of the layers affect bandgap requirements and carrier transport limitations. This approach provides the most flexibility in design, offering a high number of periodic QWs with minimal strain relaxation. Strained system: A strained system is grown with wells and barriers that are not similar in lattice constant. A strained system compresses the whole structure. As a result, the structure is only able to accommodate a few quantum wells.
Description and overview
One of the simplest quantum well systems can be constructed by inserting a thin layer of one type of semiconductor material between two layers of another with a different band-gap. Consider, as an example, two layers of AlGaAs with a large bandgap surrounding a thin layer of GaAs with a smaller band-gap. Let's assume that the change in material occurs along the z-direction and therefore the potential well is along the z-direction (no confinement in the x–y plane.). Since the bandgap of the contained material is lower than the surrounding AlGaAs, a quantum well (Potential well) is created in the GaAs region. This change in band energy across the structure can be seen as the change in the potential that a carrier would feel, therefore low energy carriers can be trapped in these wells.
Within the quantum well, there are discrete energy eigenstates that carriers can have. For example, an electron in the conduction band can have lower energy within the well than it could have in the AlGaAs region of this structure. Consequently, an electron in the conduction band with low energy can be trapped within the quantum well. Similarly, holes in the valence band can also be trapped in the top of potential wells created in the valence band. The states that confined carriers can be in are particle-in-a-box-like states.
Physics Quantum wells and quantum well devices are a subfield of solid-state physics that is still extensively studied and researched today. The theory used to describe such systems uses important results from the fields of quantum physics, statistical physics, and electrodynamics.
Infinite well model The simplest model of a quantum well system is the infinite well model. The walls/barriers of the potential well are assumed to be infinite in this model. In reality, the quantum wells are generally of the order of a few hundred millielectronvolts. However, as a first approximation, the infinite well model serves as a simple and useful model that provides some insight into the physics behind quantum wells.
Consider an infinite quantum well oriented in the z-direction, such that carriers in the well are confined in the z-direction but free to move in the x–y plane. we choose the quantum well to run from to . We assume that carriers experience no potential within the well and that the potential in the barrier region is infinitely high.
The Schrödinger equation for carriers in the infinite well model is:
where is the reduced Planck constant and is the effective mass of the carriers within the well region. The effective mass of a carrier is the mass that the electron "feels" in its quantum environment and generally differs between different semiconductors as the value of effective mass depends heavily on the curvature of the band. Note that can be the effective mass of electrons in a well in the conduction band or for holes in a well in the valence band.
Solutions and energy levels
The solution wave functions cannot exist in the barrier region of the well, due to the infinitely high potential. Therefore, by imposing the following boundary conditions, the allowed wave functions are obtained,
The solution wave functions take the following form:
The subscript , () denotes the integer quantum number and is the wave vector associated with each state, given above. The associated discrete energies are given by:
The simple infinite well model provides a good starting point for analyzing the physics of quantum well systems and the effects of quantum confinement. The model correctly predicts that the energies in the well are inversely proportional to the square of the length of the well. This means that precise control over the width of the semiconductor layers, i.e. the length of the well, will allow for precise control of the energy levels allowed for carriers in the wells. This is an incredibly useful property for band-gap engineering. Furthermore, the model shows that the energy levels are proportional to the inverse of the effective mass. Consequently, heavy holes and light holes will have different energy states when trapped in the well. Heavy and light holes arise when the maxima of valence bands with different curvature coincide; resulting in two different effective masses.
A drawback of the infinite well model is that it predicts many more energy states than exist, as the walls of real quantum wells, are finite. The model also neglects the fact that in reality, the wave functions do not go to zero at the boundary of the well but 'bleed' into the wall (due to quantum tunneling) and decay exponentially to zero. This property allows for the design and production of superlattices and other novel quantum well devices and is described better by the finite well model.
Finite well model The finite well model provides a more realistic model of quantum wells. Here the walls of the well in the heterostructure are modeled using a finite potential , which is the difference in the conduction band energies of the different semiconductors. Since the walls are finite and the electrons can tunnel into the barrier region. Therefore the allowed wave functions will penetrate the barrier wall.
Consider a finite quantum well oriented in the z-direction, such that carriers in the well are confined in the z-direction but free to move in the x–y plane. We choose the quantum well to run from to . We assume that the carriers experience no potential within the well and potential of in the barrier regions.
The Schrodinger equation for carriers within the well is unchanged compared to the infinite well model, except for the boundary conditions at the walls, which now demand that the wave functions and their slopes are continuous at the boundaries.
Within the barrier region, Schrodinger's equation for carriers reads:
where is the effective mass of the carrier in the barrier region, which will generally differ from its effective mass within the well.
Solutions and energy levels
Using the relevant boundary conditions and the condition that the wave function must be continuous at the edge of the well, we get solutions for the wave vector that satisfy the following transcendental equations:
and
where is the exponential decay constant in the barrier region, which is a measure of how fast the wave function decays to zero in the barrier region. The quantized energy eigenstates inside the well, which depend on the wave vector and the quantum number () are given by:
The exponential decay constant is given by:
It depends on the eigenstate of a bound carrier , the depth of the well , and the effective mass of the carrier within the barrier region, .
The solutions to the transcendental equations above can easily be found using numerical or graphical methods. There are generally only a few solutions. However, there will always be at least one solution, so one bound state in the well, regardless of how small the potential is. Similar to the infinite well, the wave functions in the well are sinusoidal-like but exponentially decay in the barrier of the well. This has the effect of reducing the bound energy states of the quantum well compared to the infinite well.
Superlattices