## Basic Concepts (Waves, Particles, Tunneling)

### Particle - Wave - Duality

Electrons can be described by particle as well as wave properties. The point of view that is taken depndes on the physical phenomenon that needs to be explained.

For interactions with other particles (or waves described as particles) it is often advantageous to utilize the particle point of view since principles such as energy and momentum conservations easily applied in this framework. Such other interacting "particles" may be other electrons, impurities, phonons (which are "particles" that describe the collective motion / vibration of the crystal lattice), etc.

For the propagation of a single particle the wave point of view typically proves to be useful. Electron waves are described by a wave equation similar to the set of electro magnetic wave equations for the electric and magnetic field. These waves can be shown mathemetically as well as experimentally to interfere, diffract, and penetrate evanescently into heterogeneous materials.

### Wave Specifications (Amplitude, frequency, momentum and phase)

A wave is typically described by an amplitude (A, intensity), a frequency (f, energy), a phase (p, zero point), and a momentum (k, wave propagator) - The mathematical description of a plane wave reads as:

A*exp(i*(2*pi*f*t+p - k*x)).

With a specified A, f, p, and k, a wave is completely specified for all space (x) and time (t).

### Dispersion, Effective Mass, Bands

The dispersion of a material describes the relation between the energy/frequency of a wave and the momentum of a wave. For photons (electromagnetic waves) the relationship is linear (E~k) while for electrons the dispersion can be quite complicated. However, in many cases it can be helpful to assume the electron dispersion to be parabolic (E~k*k/m) with a particular effective mass m. Such an assumption can be good for a particular band. Bands are formed by the interaction of neighboring electrons in the crystalline lattice.

### Phase Coherence Length

One critical issue whether or not a wave point of view can be taken for the understanding of the propagation of an electron through a particular material is typically characterized by the key word phase coherence. Electrons will interact with other particles in any realistic semiconduction or metallic structure. All interactions with particles that have a degree of freedom of their own change the phase p (the starting point of the wave). Elastic interactions may only change the momentum (k) while inelastic interaction may change the momentum (k) and the energy (frequency f) of the wave. It makes sense to utilize the wave point of view, if the mean time between phase breaking interactions is large enough to allow the propagation of the wave by a couple of wavelengths. The lengh traveled between phase breaking events is called the phase breaking length. Interference effects will only occur if the phase breaking length is significantly larger than the critical device / structure dimension over which the interefence could take place. I high mobility semiconductors such coherence lengths can be of the order of micro meters.

### Standing Waves

Two counter-propagating waves of equal amplitude and frequency can form a standing wave (sin(kx) ~ exp(ikx)-exp(-kx)). Such a standing wave can be formed if the appropriate confinement can be provided for the electrons.

### Tunneling

Waves can penetrate into materials that do not allow propagation (since the energy of the wave is too low) in an evanescent form. Mathematically the momentum vector turns imaginary where k~i*kappa, where kappa is the evanescent decay length. Electrons can be transmitted through / tunnel through barrier material, if the barrier material is sandwiched between two materials that allow electron propagation and if the barrier material is thin enough. Electron tunneling of electrons is a pure quantum mechanical effect that cannot be explained by a pure classical particle picture. The quantum mechanical wave model of the electron is essential in the understanding of this effect.

### Resonant Tunneling

Two tunneling barriers sandwiched by low energy propagating propagating material can form a resonance cavity similar to an optical Fabry-Perot filter. A set of standing waves can form in the center between the two barriers. Transmission through the whole structure can be as large as unity for electron energies at the resonance energies of such standing waves although the transmission through the individual barriers can be vanishingly small. This is the basic operating principle of the resonant tunneling diode.