Mobility and Conductivity

Transcript

0:00Introduction

Hello everyone! Pure silicon conducts electricity very poorly. Now, in every million silicon atoms, swap just one for an arsenic atom, a pentavalent impurity. You might expect a small change. In fact, the conductivity goes up about two and a half million times! How can one atom in a million do so much? By the end of this video, you will calculate it yourself. This idea is useful for JEE, so let's begin.

0:28Mobility: drift speed per unit field

So, in an electric field, free electrons and holes drift with a small average speed, the drift speed. Now tell me, if I double the field, what happens to the drift speed? Yes, exactly, it doubles too. So we use the drift speed per unit field, the mobility, mu, the Greek letter mu. Mu equals v d over E, where v d, the v with a small d below it, is the drift speed, and capital E is the field, okay? Its unit is metres per second, divided by volts per metre, the division sign on screen, which gives metre squared per volt per second. The small minus ones mean per, right? Now, which carrier do you think moves more easily, the free electrons or the holes? That is correct, the electrons are more mobile. So mu e, the mu with a small e below it, is greater than mu h, the mu with a small h below it, okay? For silicon, mu e is zero point one three five, and mu h is zero point zero four eight, almost three times smaller. For germanium, mu e is zero point three nine, and mu h is zero point one nine. With me?

1:46Current and conductivity

Now let us look at the current. Electrons drift against the field, and holes drift along it, in opposite directions. So tell me, do their currents cancel? No, they add up, because negative charge moving one way is a current in the opposite direction. So through a cross-section of area capital A, the current capital I equals e times capital A, times the bracket, n e times v e, plus n h times v h. Here e is the electronic charge, n e and n h are the numbers of free electrons and holes per unit volume, and v e and v h are their drift speeds, okay? And drift speed is mobility times field. So v e equals mu e times E, and v h equals mu h times E, right? Put these in, and divide by the area. Current per unit area is the current density, capital J, so J equals capital I over capital A, which is e times the bracket, n e times mu e, plus n h times mu h, all times E, with me? Everything in front of the field depends only on the material and its carriers. This is the conductivity, sigma, the Greek letter sigma. So J equals sigma times E, with sigma equal to e times the bracket, n e times mu e, plus n h times mu h. This is the key formula. Clear? And the resistivity, rho, the Greek letter rho, is one over sigma. Sigma is in siemens per metre, the capital S on screen, and rho is in ohm metre, with the Greek capital omega for ohm. Got it?

3:30Pure, n-type and p-type: which terms count

Now, in a pure, or intrinsic, semiconductor, n e equals n h equals n i, the n with a small i below it, the intrinsic carrier concentration. The double arrow means implies, so sigma equals e times n i, times the bracket, mu e plus mu h, okay? Next, n-type. Electrons are the majority carriers, and holes are very few. So which term can we drop? Yes, exactly, we drop the hole term. Since n e is much greater than n h, the double greater-than sign, sigma is approximately e times n e times mu e, with the wavy equals sign for approximately, right? And p-type is the mirror image. Holes are the majority, so sigma is approximately e times n h times mu h. Now careful, this is a common trap. In p-type, use mu h, the smaller hole mobility, not mu e. Clear? So here is a topper's habit, on screen now. Before putting in any number, first ask whether the sample is pure, n-type or p-type. That tells you which terms to keep. Okay?

4:42Example 1: conductivity and resistivity of pure silicon

Now, example one. Find the conductivity and resistivity of pure silicon at three hundred kelvin, the capital K on screen. Here n i is one point five times ten to the power sixteen per cubic metre, the m to the power minus three. Pause the video and try it first. Done? Okay, let's check. Many students write sigma equals e times n i times mu e, since electrons are more mobile. Is that enough? No, and that is exactly the trap. Pure silicon has as many holes as free electrons, so both terms count. Sigma equals e times n i, times the bracket, mu e plus mu h, okay? Now put in the numbers: one point six times ten to the power minus nineteen for e, times one point five times ten to the power sixteen, times the bracket, zero point one three five plus zero point zero four eight, right? One point six times one point five gives two point four, with ten to the power minus three. The bracket adds up to zero point one eight three. With me? Two point four times zero point one eight three is about zero point four four, so sigma is about four point four times ten to the power minus four siemens per metre, okay? And the resistivity, one over sigma, is about two point three times ten to the power three ohm metre, so pure silicon is a poor conductor. Keep this sigma for example two. Got it?

6:19Example 2: n-type silicon resistivity and doping factor

Now example two, which solves our opening puzzle. For n-type silicon with n e equal to five times ten to the power twenty two per cubic metre, which is about what one arsenic atom per million silicon atoms gives, find the resistivity, and the factor by which doping has increased the conductivity. Pause the video and try it first. Done? Okay, let's check. It is n-type, so which term of the formula do we keep? Only the electron term, that is correct, since the holes are the minority carriers. So sigma is approximately e times n e times mu e, okay? Now put in the numbers: one point six times ten to the power minus nineteen, times five times ten to the power twenty two, times zero point one three five, right? One point six times five gives eight, and with ten to the power three, that is eight thousand. Eight thousand times zero point one three five gives one thousand and eighty siemens per metre. With me? So the resistivity is one over one thousand and eighty, about nine point three times ten to the power minus four ohm metre. Clear? Now the factor. Doping increased the conductivity, but compared with what? Yes, exactly, compared with pure silicon from example one. So the factor is one thousand and eighty, over four point four times ten to the power minus four, about two point five times ten to the power six. That is two and a half million, and our puzzle is solved. Got it?

8:02Quick revision: mobility, conductivity, n-type and p-type

So, quick revision. Mobility is drift speed per unit field, and electrons are more mobile than holes. Sigma equals e times the bracket, n e times mu e, plus n h times mu h, rho is one over sigma, and J equals sigma times E, okay? Keep both terms for a pure sample, and only the majority-carrier term for n-type or p-type. Now try the practice problems on this page. All the best!