Motivation
More than ten years ago, I finished my telecommunications degree at Universitat Politècnica de Catalunya. I still remember the classes from Fonaments de Micro i Nanotecnologia (or FMNT, as we usually called it). In this subject, we got an introduction to what a semiconductor is, how it works, and how the material physics enable one of the biggest industries in the world: the semiconductor industry. I remember going through the entire course and understanding most of the content, but years later, I just have a brief intuition about all the concepts explained in those fantastic classes. Now I have challenged myself to go through these concepts again, using the book Intuitive Analog Design by Marc Thompson and the help of the restless AI to answer all my questions. This is the motivation for this series of posts about PN junctions, diodes, and MOSFETs.
Insulators, conductors and semiconductors
To explain the physics of the semiconductors we can start by understanding what happens in a silicon crystal. Materials can be insulators, conductors or semiconductors. On one hand, insulators have very few free charge carriers available to flow in the conduction band, therefore they present a high electrical resistance. On the other hand, conductors have plenty of free charge carriers available to move in the conductior band and they possess a low electrical resistance. Semiconductors lay between these two categories. Typical semiconductors are silicon (Si), germanium (Ge), and gallium arsenide (GaAs). In semiconductors, the current is conveyed in different ways than in metallic conductors. In semiconductors, current can be carried by the movement of electrons (negative particles with charge $-q$) and holes (positive charge carriers with charge $+q$).
Electrons and holes
In solid-state physics, energy bands are ranges of allowed energies in which electrons can exist. An isolated silicon atom has 2 electrons in the 1st shell, 8 in the 2nd and 4 in the outer shell (valence shell). However, in a crystal the 4 electrons in the outer shell form covalent bonds with 4 neighbouring atoms. It is important to note at that if we apply a voltage to this silicon crystal, at low temperature and without additional excitation, the voltage will accelerate only those electrons that are already free in the conduction band. It will not by itself promote electrons from the valence band to the conduction band. The main mechanisms that can provide enough energy to lift electrons from the valence band to the conduction band are heat (thermal energy), light (photons), or collisions with sufficiently energetic particles.
In a piece of silicon that is not at 0 K, the thermal energy in the lattice shakes the electrons and some of them can reach the conduction band. In the conduction band these electrons can move freely and if a voltage is applied, a net current is produced. The place where the freed electron was is called hole and behaves as if it carries a positive charge of $+q$. Holes can also move in the lattice so, in essence, they can also contribute to the total net current in the lattice, which can be expressed as:
\[ J = J_e + J_h \]Where $J$ is the total current density in $A/m^2$, whereas $J_e$ is the electron current density and $J_h$ is the hole current density. One may ask: hey, if holes are just the absence of electrons, aren’t we counting twice the effective current (density) in the material? The key is that electrons and holes don’t move in the same place. And you may ask yourself again: but hold on, isn’t current the movement of electrons in the conduction band? Why are we counting the movement of holes or electrons in other bands? The answer is that this in metals (conductors), conduction is dominated by free electrons in the conduction band. However, in a intrisic semiconductor current flow simultaneously in two bands:
- Conduction band: free electrons (those that broke the covalent bond and gained enough energy as to jump the band gap) move freely. This is the familiar “free electron” current.
- Valence band: electrons that still bound hop between atoms to fill the holes. The net effect of this hopping is tracked as a hole current moving in the opposite direction.
Current is not carried exclusively by conduction-band electrons. If the valence band contains holes, bound electrons can hop sequentially toward the positive terminal. At the metal–semiconductor contact, these valence-band electrons exit the semiconductor into the metal wire and flow toward the battery’s positive terminal. The battery then re-injects electrons at the negative terminal, creating new holes and sustaining the hopping process. The net effect is a continuous current loop, even though no individual valence-band electron travels the full length of the device.
In this animation, made with P5.js, we have a piece of semiconductor. In the conduction band, we have a single electron that can move freely. In the valence band, we have a hole. On the left, we have the negative terminal of a battery that is applying an electric field and on the right, the positive terminal. The electron in the conduction band is accelerated by the electric field generated by the voltage and moves quickly towards the positive terminal of the battery. In the valence band, we see how the electrons also move towards the positive battery terminal, so effectively, the hole moves toward the left. Note that the electrons hopping in the valence band move much slower than the electron in the conduction band. When the hole in the valence band reaches the negative terminal, it is filled with an electron provided by the battery. At the positive terminal, the battery pulls the conduction-band electron out into the wire and creates a new hole in the valence band, keeping both mechanisms going continuously.
In a piece of silicon without added impurities, the number of free electrons and the number of holes are equal. This is because the free electrons come from the valence band and therefore electrons and holes are created in pairs. We call this unmodified piece of silicon an intrisic semiconductor. So we we can write this property as:
\[n = p \equiv n_i\]where $n$ is the number of free electrons, $p$ is the number of holes, and $n_i$ is the intrinsic carrier concentration. As we explained briefly before, the electron agitation varies with temperature. The higher the temperature, the more energy the electrons have and they can jump more easily the band gap to reach the conduction band.
Drift, diffusion, recombination and generation
Drift
When we have free charge carriers in a semiconductor, they can be moved or transformed in different ways. If we apply an electric field, as seen in the animation above using a battery, the carriers can drift due to the force applied by the electric field. In the animation, we can compute the velocity of the electron as:
\[ \vec{v} = -\mu_n \vec{E} \]where $\mu_n$ is a material parameter called mobility of the electron. It is defined in units of $\frac{cm^2}{V·s}$ and measures how easily electrons move through a semiconductor (or conductor) in response to an applied electric field. The reason for the minus sign is because the electrons drift in the opposite direction of the applied electric field $\vec{E}$.
Similarly, the holes move at a velocity of:
\[\vec{v} = \mu_p \vec{E}\]where $\mu_p$ is the mobility of the hole. As shown in the animation and noted before, the mobility of the holes is around 2 to 3 times lower than that of electrons in semiconductors.
If we want to compute the drift current density due to holes and electrons, we obtain the following expression:
\[\vec{J} = \vec{J_{h,drift}} + \vec{J_{e,drift}} = q \mu_p p \vec{E} + q \mu_n n \vec{E}\]where $p$ and $n$ are the densities of holes and electrons per unit volume respectively, and $\vec{E}$ is the electric field.
If the concentration of carriers is high in some areas of the material and low in others, the carriers can diffuse so that the density equalises. Imagine spraying air freshener in one corner of a closed room. Even without any fan or wind, after a while the scent spreads everywhere — the molecules move from where they are densely packed near the spray nozzle toward the rest of the room where there are fewer of them. No external force is needed; the molecules just happen to be in constant random motion, and statistically more of them wander away from the crowded region than wander back into it. In a semiconductor, carriers behave the same way: driven by thermal energy, electrons in constant random motion produce a net flow from high-concentration regions toward low-concentration ones, until the distribution evens out.