# How Charge Moves In Semiconductors: Drift and Diffusion

In a semiconductor, just like a regular conductor, a voltage will cause electrons to be attracted to the positively charged end, the electric field that is created accelerates the electrons in the direction of current flow; however these electrons also bang into other parts of the semiconductor. These two opposing forces lead to the electrons in a semiconductor moving at a constant speed when a voltage is applied. In analog circuits we often talk of both electrons and holes moving. Sometimes it helps to picture the electrons as the ones moving, sometimes it helps to picture the holes as the ones moving. Neither is more correct than the other, just two ways of thinking about the same thing. The second way charge moves in a semiconductor is called diffusion. It's the same diffusion that you've met many times in your life. When a drop of ink slowly spreads through still water or a candle's sweet smell slowly fills a room that's diffusion. In a semiconductor electrons will move from places of high concentration to places of low concentration. This is separate to any electric field that may or may not be applied.derivation

We start by considering that the bigger the difference in high local concentration to low outside concentration the larger the current that will develop.
We'll write this as \(I \propto \frac{dn}{dx}\) where \(n\) is the carrier concentration at position \(x\). We call \(\frac{dn}{dx}\) the concentration gradient along \(x\).
Now our current is also going to be larger if each carrier holds a larger charge (since current is how much charge moves each second). And our current is going to be larger if our conductor has a bigger cross sectional area since it'll create less of a bottleneck.
So we will now say that \(I = AqD_n\frac{dn}{dx}\) where \(D_n\) is a proportionality factor called the diffusion constant and has units cm\(^2\)/s.

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The diffusion current is given by:
\(I = AqD_n\frac{dn}{dx}\)
Where \(A\) is the semiconductor's cross sectional area, \(q\) is the charge per carrier, \(D_n\) is a proportionality factor called the diffusion constant and has units cm\(^2\)/s and \(\frac{dn}{dx}\) is the concentration gradient.

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The diffusion current for holes is given similarly.
\(I = AqD_p\frac{dp}{dx}\)

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The diffusion constant is usually different for holes and electrons in the same material.

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Just like with drift current we normally write the diffusion current as a density given by:
\(J_n = qD_n\frac{dn}{dx}\)

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The electrons and hole diffusion current densities together are given by:
\(J_{tot} = q(D_n\frac{dn}{dx} - D_p\frac{dp}{dx})\)

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The constants for drift \(\mu_n\) and \(\mu_p\) are related to the constants for diffusion \(D_n\) and \(D_p\) by the

**Einstein Relation**: \(\frac{D}{\mu} = \frac{kT}{q}\) Where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvins and \(q\) is, as usual, the charge on an electron. \(\frac{kT}{q} \approx 26\)mV at \(T = 300\) (room temperature).
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