Band Gap of Semiconductor

Heat a copper wire and it conducts worse. Heat a piece of silicon and it conducts better. That reversal is not a curiosity. It is the clearest everyday fingerprint of the band gap, and this experiment measures the gap by exploiting it.

Why solids have bands at all

A single atom has sharp energy levels. Bring two atoms close and each level splits in two. Bring 10²³ of them together in a crystal and each level smears into a band of levels packed so closely that you may as well treat them as continuous.

Between the bands there are ranges of energy that no electron in the crystal can have. The important pair is the valence band, which is full at absolute zero, and the conduction band above it, which is empty. The forbidden range between them is the band gap, written E_g.

How wide that gap is decides almost everything about the material. In a metal the two bands overlap, so electrons are free to move with no encouragement at all. In an insulator like diamond the gap is around 5.5 eV, far more than anything room temperature can supply. Silicon sits at about 1.1 eV, which is the interesting middle: too wide for the material to be a good conductor, narrow enough that a modest amount of heat gets some electrons across.

Reverse bias, and the current that should not be there

Bias a PN junction the wrong way round and the depletion region widens. Majority carriers are pushed away from the junction and the diode is supposed to block. What actually flows is a very small current, of the order of microamps, called the reverse saturation current, I_s.

That current is carried by minority carriers: electrons that happen to be on the p side, holes on the n side. Every one of them was created by an electron being thermally lifted across the band gap. So the reverse current is a direct headcount of how many electrons have made the jump, which is exactly the quantity that depends on E_g and T.

Statistical mechanics gives the population of that thermally excited minority as an exponential in the gap over the thermal energy:

I_s = C · e^{−E_g / kT}

Notice what is missing from the right hand side: the applied voltage. Past a volt or so of reverse bias, the current stops depending on the bias at all. That is why it is called a saturation current, and it is why you set the supply near 5 V and then leave it alone. Anything that changes in your readings after that is temperature doing the work.

Turning an exponential into a straight line

Exponentials are awkward to read off a graph. Take the natural logarithm of both sides and the awkwardness disappears:

ln I_s = ln C − (E_g / k) · (1 / T)

Compare that with y = mx + c. Plot ln(I_s) on the vertical axis and 1/T on the horizontal, and you should get a straight line whose gradient is −E_g/k. The intercept contains C, which is a property of the particular diode you happen to be holding and is of no interest here. The slope is the physics.

E_g = −slope × (k / e)

Every reading you take is one point on that line. Ten of them, spread over sixty degrees or so, pin the gradient down well enough to get silicon's 1.1 eV out of a microammeter and a thermometer.