Energy Bands in Solids

Transcript

0:00Introduction

Hello everyone! Let us begin with a small puzzle. Diamond, silicon and germanium have the same crystal structure, and every atom has four valence electrons. And yet diamond is a transparent insulator, while silicon and germanium are semiconductors. So what makes diamond so different? The whole answer is just one number. This band picture comes up often in JEE, so let's begin.

0:28From sharp levels to energy bands

So, an isolated atom has sharp energy levels. Now bring together capital N atoms to build a crystal, N being a huge number. Let me ask you this: what happens to each sharp level now? It spreads into N closely spaced levels, so close that together we call them a band. The arrow on screen means becomes. Clear? Now, the valence band is the highest band that is filled at zero kelvin, the zero K on screen. It holds the electrons of the bonds. Right? The next band above it is the conduction band, and electrons that reach it are free to move through the whole crystal.

1:09The forbidden gap Eg

Got it? Now look at the middle picture first. Energy goes up the arrow on the left; the upper box is the conduction band, the lower box the valence band. The double arrow marks the energy gap, capital E with a small g below, E g. It equals E C minus E V, with small C and V below for conduction and valence: the bottom of the conduction band minus the top of the valence band. Okay? No electron can have an energy inside this gap, so we also call it the forbidden gap. With me? Now, at zero kelvin the valence band is completely full of electrons. So many electrons, so it must conduct very well, right? No, it does not, so be careful here: a completely filled band carries no current at all. Clear? For conduction we need electrons in a partly filled band, where they can gain energy from an electric field and move. Got it?

2:05Gap and resistivity of three kinds of solid

Now look at all three pictures. On the left, a metal: its conduction band is partly filled, or overlaps the valence band as marked, so E g is zero. Its resistivity is tiny, between ten to the power minus eight and ten to the power minus six ohm metre; the Greek capital omega is the ohm. Okay? See, the middle and the right pictures both have a gap, right? So what decides whether it is a semiconductor or an insulator? Yes, exactly, it is the size of the gap. Now, a semiconductor has a small gap, E g less than three electron volts, written e V; below the middle picture, silicon has one point one, and germanium zero point seven. At room temperature a few electrons cross, the black dots, leaving empty places, the open circles. Its resistivity lies between ten to the power minus five and ten to the power six ohm metre. With me? An insulator has a large gap, E g more than three electron volts, and diamond has five point four. Its resistivity is huge, between ten to the power eleven and ten to the power nineteen ohm metre. So our puzzle is solved: room temperature lifts electrons across the small gaps of silicon and germanium, but not across the large gap of diamond. Clear?

3:26Light across the gap

Light can also lift an electron across the gap. Each photon carries energy h nu, where h is Planck's constant, and nu, the Greek letter written like a small v, is the frequency. It works only if h nu is greater than or equal to E g. Okay? In terms of the wavelength, the Greek letter lambda, this becomes lambda less than or equal to lambda max, which equals h c over E g, with c the speed of light. Lambda max, with max as a subscript, is the longest wavelength that works. With me? Now here is the topper's trick, so remember it. Lambda max in nanometres equals one thousand two hundred and forty, over E g in electron volts, where n m is the nanometre. Got it? So, back to diamond, with its gap of five point four electron volts. Can diamond absorb visible light? One thousand two hundred and forty over five point four gives about two hundred and thirty nanometres; the wavy equals sign means approximately. Only shorter wavelengths, in the ultraviolet, are absorbed. So visible light passes straight through, and that is why diamond is transparent. Clear?

4:42Heat across the gap

Now let us look at heat. At temperature T, the thermal energy available is about k T, Boltzmann's constant k times the temperature. At three hundred kelvin, k T is only about zero point zero two six electron volts. Okay? The number of electrons excited varies as e to the power minus E g over two k T; the symbol like an open alpha means proportional to. See, the gap sits in the exponent, so a slightly bigger gap means far fewer electrons. Right?

5:14Example 1: longest wavelength for Si and Ge band gaps

Time for our first example. Find the longest wavelength of light that can create an electron hole pair in silicon, and in germanium. The pair is the lifted electron and the hole, the empty place it leaves behind. Pause the video and try it first. Done? Okay, let's check. For silicon, S i on screen, lambda max is one thousand two hundred and forty over one point one, which gives about one thousand one hundred and twenty seven nanometres. Clear? Now germanium, G e on screen, has a smaller gap of zero point seven electron volts. So will its lambda max be longer or shorter than for silicon? Yes, exactly, it is longer, because a smaller gap needs less energy. One thousand two hundred and forty over zero point seven gives about one thousand seven hundred and seventy one nanometres. Right?

6:12Example 2: Ge vs Si carriers at 300 K, the exponent trap

Now let us try a harder one, with the same e to the power minus E g over two k T, and k T equal to zero point zero two five nine electron volts. Estimate how many times more carriers germanium has than silicon at three hundred kelvin, other factors being equal. Pause the video and try it first. Done? Okay, let's think. One point one over zero point seven is about one point six, so germanium has about one point six times more carriers, right? No, and that is exactly the trap, so be careful! The gap sits in the exponent, so we divide the two exponentials, germanium on top, as on screen. Subtracting the exponents, the ratio is e to the power, one point one minus zero point seven, over two k T, okay? Let us put in the numbers. One point one minus zero point seven is zero point four, and two k T is two times zero point zero two five nine. So the exponent comes to about seven point seven two. With me? And e to the power seven point seven two is a little over two thousand. So germanium has about two times ten to the power three times as many carriers. Not one point six, but about two thousand times. Clear?

7:34Quick revision: bands, gap values, 1240/Eg and heat

So, quick revision. First, the valence band, full at zero kelvin, then the gap E g, then the conduction band. Second, a metal has no gap, a semiconductor below three electron volts, and an insulator above three. And third, light lifts electrons across only for wavelengths up to lambda max, one thousand two hundred and forty over E g, and heat lifts electrons as e to the power minus E g over two k T, okay? Now go and try the practice problems on this page. All the best!