1. Modelling & Simulation of Semiconductor Devices
Lecture 7 & 8
Hierarchy of Semiconductor Models

2. 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.
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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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∈ ℝ 𝑑 and time t > 0 is governed by the single-particle Schrodinger equation
With some initial condition

10. Quantum Models

11. Quantum Models

12. 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 .

13. Fluid Dynamic Model
The current density is now calculated as

14. Semiconductor Crystal
A solid is made of an infinite three-dimensional array of atoms arranged according to a lattice

15. Semiconductor Crystal
The state of an electron moving in this periodic potential is described Schrodinger equation:

16. Home Work
Apply Madelung Transform on above equation to obtain Fluid dynamical model of electron moving in periodic potential in semiconductor crystal.

17. End of Lectures 7-8 To download this lecture visit
End of Lectures 7-8