Intrinsic Semiconductors

Transcript

0:00Introduction

Hello everyone! Let me start with a small puzzle. Heat a copper wire, and its resistance goes up. Now heat a rod of pure germanium, and its resistance goes down! Same heating, opposite result, so how is that possible? This idea comes up often in the JEE exam, so let's find out.

0:21Breaking a bond: an electron and a hole

In the picture, energy goes up the screen. On the left is a metal, on the right an insulator; today we need only the semiconductor, in the middle. Take pure silicon or germanium, S i or G e on screen. Each atom forms four covalent bonds, and the electrons of these bonds fill the lower box, the valence band. Okay? The upper box is the conduction band, and the gap between them is E g, the band gap; the small g is a subscript. Now thermal energy breaks one bond and sets an electron free. So tell me, where does that electron go? Yes, exactly, it jumps across the gap into the conduction band; the arrow on screen means goes to. These are the black dots, free to move through the crystal. With me? It leaves behind an empty place in a bond, a vacancy in the valence band. We call it a hole: the small open circles in the lower box, one for each black dot. Right? The hole behaves as a carrier of charge plus e, where e is the size of the electron's charge, so it acts like a positive charge. Clear?

1:33Pure means equal: electrons and holes

In a pure crystal, every broken bond gives one free electron and one hole. So which are more, free electrons or holes? They are equal, because they are always made in pairs, and we call such a pure semiconductor intrinsic. n e, the number of free electrons per unit volume, equals n h, the number of holes per unit volume; the small e and h are subscripts. Both equal n i, with a subscript i, the intrinsic carrier concentration. Got it? For silicon at three hundred kelvin, with capital K for kelvin, n i is one point five times ten to the power sixteen per cubic metre, written m to the power minus three. Okay? For germanium, it is about two point five times ten to the power nineteen per cubic metre, much larger; the wavy equals sign means about. With me? Thermal energy keeps making new pairs, and when an electron falls back into a hole, the pair is lost by recombination. In equilibrium the two rates are equal.

2:40In a field: two currents that add

Clear? Now put the crystal in an electric field, E with a little arrow on top, which marks a vector. Electrons are negative, so they drift against the field, and holes, being positive, drift along it. Okay? Now, electrons go one way and holes the other way. So their currents must cancel, right? No, they do not cancel; that is the trap. Negative charge moving against the field, and positive charge moving along it, both make a current along the field. So the currents add: the total current I equals I e, the electron current, plus I h, the hole current. Got it? But wait, the hole is not a particle; it is only an empty place in a bond. So how can it move and carry current? See, an electron from a neighbouring bond jumps into the vacancy, and leaves a new one where it was. So the vacancy moves along the field, just as a positive charge would. Clear?

3:39Heat it: more carriers, less resistance

Now, n i is proportional to T to the power three over two, times e to the power minus E g over two k T, where this e is the exponential, not the charge. The stretched alpha sign means proportional to, T is the temperature in kelvin, and k is Boltzmann's constant. Okay? So tell me, when the temperature goes up, what happens to the exponential part? Yes, exactly, it grows, because E g over two k T gets smaller, so the power is less negative. So n i rises steeply with temperature; the double arrow means implies. Right? More carriers means more current for the same voltage, so the resistance falls on heating. We call this a negative temperature coefficient of resistance. With me? Now, let us go back to our puzzle. In copper, the number of free electrons is fixed, and heating only adds collisions, so R, the resistance, rises. Okay? In germanium, heating creates many more carriers, and that outweighs the extra collisions, so the resistance falls. That solves our puzzle, right?

4:49Example 1: electron hole pairs in a 1 cm cube of silicon

Now, example one, an easy one. How many electron hole pairs are present at three hundred kelvin, in a cube of pure silicon of side one centimetre, written c m? Pause the video, and try it first. Done? Okay, let's check. One centimetre is ten to the power minus two metre, so the volume is ten to the power minus two cubic metre, right? No, wait, that is the trap; the side must be cubed. So V, the volume, is ten to the power minus two metre, cubed, which is ten to the power minus six cubic metre. Clear? Each free electron belongs to one pair, so the number of pairs is n i times V, okay? One point five times ten to the power sixteen, times ten to the power minus six, gives one point five times ten to the power ten pairs. Got it?

5:45Example 2: ni rises eightfold from 300 K to 330 K

Now a harder one. Using this relation, find the factor by which n i of silicon increases when the temperature rises from three hundred to three hundred and thirty kelvin. E g is one point one electron volt, written e V, and k is eight point six two times ten to the power minus five electron volt per kelvin, written with K to the power minus one. Pause the video and try it first. Here is a hint: take the ratio. Done? Okay, let's check. Three hundred to three hundred and thirty kelvin is just ten percent hotter, so n i goes up about ten percent, right? No, that is the trap; the exponential changes much faster. Take the ratio, new n i over old, and the constant cancels. We get the temperature ratio to the power three over two, times e x p on screen, which means e to the power, of the square bracket: E g over two k, times one over three hundred, minus one over three hundred and thirty. The minus sign in the power flips the order. Okay? First, let us do the easy part. Three hundred and thirty over three hundred is one point one, and one point one to the power three over two is about one point one five four. Right? Now, let us work out E g over two k. One point one, over two times eight point six two times ten to the power minus five, gives about six thousand three hundred and eighty one kelvin. With me? And one over three hundred, minus one over three hundred and thirty, is one over three thousand three hundred. So the power is six thousand three hundred and eighty one, over three thousand three hundred, which gives about one point nine three. Clear? Finally, e to the power one point nine three is about six point nine, and multiplying by one point one five four gives about eight. So just ten percent hotter means about eight times the carriers! The exponential does almost all the work. Got it?

7:56Quick revision: holes, ne = nh = ni, resistance falls

So, quick revision. Every broken bond gives an electron in the conduction band, and a hole of charge plus e in the valence band. Intrinsic means n e equals n h equals n i, and in a field the two currents add. And n i rises steeply with temperature, so the resistance falls on heating. Now try the practice problems on this page. All the best!