Introduction to Semiconductors

Our study of analog circuits will be built on our understanding of semiconductor physics. Semiconductors (metals which are somewhere between copper wire and a brick wall in terms of resistance) allow for some very clever devices to be built that work in strange, nonlinear ways and are nothing like our familiar resistors, capacitors and inductors. Through this topic we will come to master transistor circuits. The transistor is based on understanding two back-to-back "PN junctions", so we'll need to learn what a PN junction is and how it works. But to do that we'll need to learn what P and N are, which is where we're starting here. Although much of the physics content at the beginning few topics might seem too theoretical to be useful it will become more and more important as your learning goes on. In analog circuits more than anywhere else, your ability to "picture" what is happening in the circuit is of the utmost importance. Your ability to gain an intuitive understanding of how complex and complicated circuits will behave begins with understanding what's happening "under the hood". So work through the mathematics and theory so that you can become a master of the applications.
Charge is carried around in circuits by the movement of valence electrons into available spaces in adjacent valence shells. We can also view this as the movement of available spaces in valence shells moving in the opposite direction.
This might sound confusing but its really a straightforward idea. While current is really electrons moving around on the outside of molecules into gaps on an adjacent molecule's outtermost electron "shell" we can pretend that these "free spaces" in an electron shell are really the bits moving since if an electron leaves one molecule's shell it must leave a "free space" in its place. We call these free spaces in valence shells "holes". Now as you may remember from high school chemistry atoms with only one or two electrons in their valence shell are quite "willing" to give them up whereas atoms with an almost completely full valence shell (often 8 electrons fills a valence shell) is very "willing" to accept another electron. So elements with three to five valence electrons are somewhere between and are often referred to as "semiconductors". These elements exhibit interesting chemical properties that make them useful for building solid state devices capable of more complex functions that resistors, capacitors and inductors.
Elements with three to five valence electrons are termed "semiconductors" and are the basis of microelectronic devices like diodes and transistors.
More formally, we can calculate the energy required to dislodge an electron from an atom (which is required for a current to flow). We call this energy the "bandgap energy", having a very high bandgap energy means we have an insulator while a very low bandgap energy is a strong conductor.
The energy required to dislodge an electron from an atom is called the "bandgap energy". For silicon this is \(E_g = 1.12eV\)
Using the bandgap energy we can calculate the number of free electrons at a given temperature. The more free electrons the more conductive the material will be. A larger \(E_g\) will lead to fewer free electrons and a higher temperature will lead to a greater number of free electrons.
The density of free electrons in some material is given by: $$ n_i = 5.2\times 10^{15}T^{3/2}e^{\frac{-E_g}{2kT}} \text{electrons/cm}^3 $$ Where k is the Boltzmann constant \(k = 1.38\times 10^{-23}\)J/K and T is the temperature in Kelvins.
Semiconductors typically have an \(E_g\) between 1.0eV and 1.5eV

Find the density of free electrons in silicon at room temperature (T = 300) and T = 400 (100\(^\circ\)C)

From above we see that for silicon \(E_g = 1.12\)eV \( = 1.792\times 10^{-19}\)J so plugging things into our formula: $$ n_{i(T = 300)} = 1.08\times 10^{10}\text{electrons/cm}^3 $$ $$ n_{i(T = 400)} = 3.7126\times 10^{12}\text{electrons/cm}^3 $$
While these numbers are very large the fact that silicon has \(5\times 10^{22}\)electrons/cm\(^3\) means that only one in five trillion atoms has a free electron! For this reason we do something called "doping" where we insert some other atoms (at a very low density) into our collection of silicon atoms in order to change the number of free electrons. These free electrons (or holes) come from the donated atoms. Assuming the total number of free electrons are denoted by \(n\) and the total number of available holes is given by \(p\) we come to the very useful equation:
$$np = n_i$$ In both undoped and doped semiconductors.
When we add extra free electrons or holes these will combine with some of the existing holes or free electrons to essentially cancel each other out. So while their product is always constant their ratio (and total number) can be changed by doping (since \(n + p\) is NOT constant. We call semiconductors where \(n \gt p\) n-type semiconductors and those where \(n \lt p\) p-type semiconductors.
To create an n-type semiconductor we need to add a doping element with a small number of valence electrons (so that they will be easily dislodged). To create a p-type semiconductor we need to add a doping element with a large number of valence electrons (so that they will easily accept more electrons).
practice problems