MOS transistors

Metal oxide semi-conductors use insulated gates to controlo barrier energies at PN surface junctions at source and drain.

Electron transport

Drift

An electric field causes carriers to drift

$J_{drift} = qn\nu = qn\mu(\zeta)$

where:

$J_{drift}$ is the current flux
$qn$ is the charge density ($q$ is the elementary charge)
$\nu$ is the velocity
$\mu(\zeta)$ is the mobility function of the electric field

Illustration of the concept:

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Diffusion

A density gradient causes carriers to diffuse. (When to doped types are put in contact for example)

$J_{diff} = -Dq\nabla n$

where:

$D$ is the diffusion constant
$\nabla n$ is the spatial gradient of charge density

Drift and diffusion are related by the Einstein Relation:

$D=\frac{kT}{q}\mu$

It expresses the connection between the diffusion constant $D$ which determines the forward and reverse currents, the mobility constant $\mu$ which determines the conductance of the resistor and the thermal voltage $\frac{kT}{q}$.
At $T=0$ , well... $D=0$ and there are no internal diffusion currents.

Thermal equilibrium

In thermal equilibrium (no external voltage applied), diffusion and drift currents are balanced ($J_{n}=J_{p}=0$)($J_{n}=J_{n,drift}+J_{ndrift}$ and so on for hole doped).

P-N Junction

What happens at a junction between P and N (physical contact)?

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The diffusion processes give rise to:
- A net electron flow from the n-type region to the p-type region.
- A net hole flow from p-type region to n-type region.
This results in a diffusion current desity $J_{diff}=J_{n,diff}+J_{p,diff}$ from the p-type to the n-type
The diffusing minority carriers recombine with majority carriers in the vicinity of the junction $\rightarrow$ this region is devoid of mobile charge carriers. ($np\ll n_{i}^2$) The Depletion region.
On the n-type side of the junction, the donor electrons are absent and the donor atoms have a surplus proton. (+) charged side
On the p-type side of the junction, the acceptor holes are filled by the donor atom's electrons and the acceptor atoms have a surplus electron. (-) charged side
An electric field builds up in the depletion region that points from n to p.
The electric field generates electron and hole drift currents from n to p.

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Electrostatic Potential: potential energy of positive charge (electron is the mirror image)

Band structure

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I-V Characteristics

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How do we generate a net current flow through the diode?

We change the boundary conditions for the n and p type regions through external application of potential differences.
If a positive voltage is applied to p, relatively to n the potential difference is called a forward biais
If the voltage on n is higher than p, it is a reverse biais.
Current flowing from p to n is a forward current
Current flowing from n to p is a reverse current
The electron current in the n region is transformed to a hole current in the p region.
The applied voltage changes the built in voltage and modifies the width of the depletion region and the minority carriers outside the depletion region.