Modelling & Simulation of
Semiconductor Devices
Lecture 7 & 8
Hierarchy of Semiconductor Models
Introduction
• Nowadays, semiconductor materials are contained in almost
all electronic de-vices.
• Some examples of semiconductor devices and their use are
described in the following.
• Some examples of semiconductor devices and their use
are described in the following.
– Photonic devices capture light (photons) and convert it into an
electronic signal. They are used in camcorders, solar cells, and
light-wave communication systems as optical fibers.
Introduction
– Optoelectronic emitters convert an electronic signal into light.
Examples are light-emitting diodes (LED) used in displays and
indication lambs and semiconductor lasers used in compact disk
systems, laser printers, and eye surgery.
– Flat-panel displays create an image by controlling light that either
passes through the device or is reflected off of it. They are made, for
instance, of liquid crystals (liquid-crystal displays, LCD) or of thin
semiconductor films (electroluminescent displays).
– In field-effect devices the conductivity is modulated by applying an
electric field to a gate contact on the surface of the device. The most
important field-effect device is the MOSFET (metal-oxide
semiconductor field-effect transistor), used as a switch or an
amplifier. Integrated circuits are mainly made of MOSFETs.
Introduction
– Quantum devices are based on quantum mechanical phenomena, like
tunneling of electrons through potential barriers which are
impenetrable classically. Examples are resonant tunneling diodes,
super lattices (multi-quantum-well structures), quantum wires in
which the motion of carriers is restricted to one space dimension and
confined quantum mechanically in the other two directions, and
quantum dots.
• Clearly, there are many other semiconductor devices which are not
mentioned (for instance, bipolar transistors, Schottky barrier
diodes, thyristors).
• Other new developments are, for instance, nanostructure devices
(hetero-structures) and solar cells made of amorphous silicon or
organic semiconductor materials.
Introduction
• Usually, a semiconductor device can be considered as a device which
needs an input (an electronic signal or light) and produces an output
(light or an electronic signal).
• The device is connected to the outside world by contacts at which a
voltage (potential difference) is applied.
• We are mainly interested in devices which produce an electronic signal,
for instance the macroscopically measurable electric current (electron
flow), generated by the applied bias.
• In this situation, the input parameter is the applied voltage and the
output parameter is the electric current through one contact.
Introduction
• The relation between these two physical quantities is called
current-voltage characteristic. It is a curve in the two-dimensional
current-voltage space.
• The current-voltage characteristic does not need to be a monotone
function and it does not need to be a function (but a relation).
• The main objective of this subject is to derive mathematical
models which describe the electron flow through a semiconductor
device due to the application of a voltage.
Introduction
• Depending on the device structure, the main transport
phenomena of the electrons may be very different, for instance,
due to drift, diffusion, convection, or quantum mechanical effects.
• For this reason, we have to devise different mathematical models
which are able to describe the main physical phenomena for a
particular situation or for a particular device.
• This leads to a hierarchy of semiconductor models.
Hierarchy of Semiconductor Models
• Roughly speaking, we can divide semiconductor models in
three classes:
– Quantum models
– Kinetic models
– Fluid dynamical (macroscopic) models
• In order to give some flavor of these models and the methods
used to derive them, we explain these three view-points:
quantum, kinetic and fluid dynamic in a simplified situation.
Quantum Models
• Consider a single electron of mass m and elementary
charge q moving in a vacuum under the action of an
electric field E = E(x; t).
• The motion of the electron in space 𝑥 ∈ ℝ𝑑 and time t >
0 is governed by the single-particle Schrodinger equation
• With some initial condition
Quantum Models
Quantum Models
Fluid Dynamic Model
• In order to derive fluid dynamical models, for instance,
for the evolution of the particle density n and the current
density J; we assume that the wave function can be
decomposed in its amplitude 𝑛 𝑥, 𝑡 > 0 and phase
𝑆 𝑥, 𝑡 ∈ ℝ .
Fluid Dynamic Model
• The current density is now calculated as
Semiconductor Crystal
• A solid is made of an infinite three-dimensional
array of atoms arranged according to a lattice
Semiconductor Crystal
• The state of an electron moving in this periodic potential
is described Schrodinger equation:
Home Work
• Apply Madelung Transform on above equation to
obtain Fluid dynamical model of electron moving in
periodic potential in semiconductor crystal.
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END OF LECTURES 7-8
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