2.2. PARTICLES IN AN ATTRACTIVE POTENTIAL: BOUND STATES 29
- Electronic properties in an atom: All solids are made up of atoms and the properties of
electrons in atoms allows us to develop insight into the electronic properties of solids. We will
discuss the hydrogen atom problem since it is the simplest atom and captures the useful physics
needed to understand the theory of doping. - Electrons in a quantum well: Quantum wells, both naturally occurring and artificially cre-
ated in semiconductor structures are very important in modern technology. In devices such as
MOSFETs, lasers, modulators etc. electrons are in quantum wells of various sizes and shapes. - Electrons in free space and in crystalline materials: Most high performance semiconductor
devices are based on high quality crystals. In these periodic structures electrons have allowed
energies that form bands separated from each other (in energy) by gaps. Almost every semicon-
ductor property depends upon these bands. Once we understand thebandtheory, i.e properties
of electrons in crystalline solids we can develop the effective description mentioned above and
use simple classical concepts. - Occupation of electronic states: Quantum mechanics has very specific rules on the actual
occupation of energies allowed by Schrodinger equation. This occupation theory is central to ̈
understanding solid state physics and device behavior.
Once we have developed the basic quantum theory structure we will discuss properties of
various semiconductors and their heterostructures.
2.2 PARTICLES IN AN ATTRACTIVE POTENTIAL:
BOUND STATES
We will now examine several important quantum problems that have impact on materials and
physical phenomena useful for device applications. The Schrodinger equation for electrons can ̈
be written in as [
−^2
2 m 0∇^2 +V(r, t)]
Ψ(r, t)=EΨ(r, t)wherem 0 is the mass of the electron andV(r, t)is the potential energy. This is a differential
equation with solutionsΨ. Once the equation is solved we get a series of allowed energies and
wavefunctions. Energies are allowed while others not consistent with the equation are forbidden.
The band theory that forms the basis of all semiconductor devices is based on energy bands and
gaps.
2.2.1 Electronic levels in a hydrogen atom
The hydrogen atom problem is of great relevance in understanding dopants in semiconductors.
We will briefly summarize these findings. The hydrogen atom consists of an electron and a pro-
ton interacting with the Coulombic interaction. The problem can be solved exactly and provides
insight into how electrons behave inside atoms.
Wavefunctions in the H-atom problem have the following term:
ψnm(r, θ, φ)=Rn(r)Fm(θ)Gm(φ)