Tutorial 6 - Electrical Properties

What are models?

We usually refer to models as:

• A model aeroplane or a model train, i.e., a miniature reproduction of an original;

• Model girls or model boys in the fashion industry.

Models in physics are ways of explaining difficult concepts in easier terms.  Often models may not describe a system completely accurately, but most people who use the model have an idea of what is happening, even though the reality is more complicated.  Mathematical modelling is often done in physics.  The simplest way to do this is to put a formula into a spreadsheet.  The computer does lots of number-crunching, and draws graphs of the results.  You can change the different parameters (values) that the formula uses to see what happens.

How is electricity conducted?

Electricity moves due to the movement of charge carriers.  If we think about an ionic solution, the positive ions are attracted to the negative terminal (the cathode), while the negative ion are attracted to the positive terminal (the anode).

In a metallic conductor (wire), the simplest model of conduction is to consider the metal as a lattice of metal ions in a sea of free electrons.  The electrons move about randomly.

When a voltage is applied across the ends of the wire, the electrons  continue to move randomly, but there is an overall drift to the positive end of the wire.  So you will (rightly) think that electrons go from negative to positive.  The protons don't move.  So this idea is opposite to what you have been told.  The explanation is that the earliest physicists got it wrong.  They didn't know about electrons in the Eighteenth Century.  So instead of rewriting all the rules of electricity, people talked about conventional current going from positive to negative.

We can write an equation for the conduction of a current in a wire.  The current depends on:

• The speed of the charge carriers (v (m/s));

• The area of the wire (A (m2));

• The charge on the electrons (e = 1.6 × 10-19 C);

• The number of charge carriers per unit volume (n (m-3)).

The symbols in italics are the physics codes for the various quantities, and the units are given as well.

The formula is:

I = nAve

The number of charge carriers per unit volume is probably the hardest quantity to get your head round.  It means the number of electrons in a cubic metre of the material.  For example, copper has 8 × 1028 electrons in each cubic metre of material.  You can look up the number of electrons per cubic metre in a data book.

 Worked example A wire is carrying a current of 200 A.  Its cross sectional area is 7.85 × 10-5 m2.  If copper has  8 × 1028 electrons per cubic metre, what is the speed of the electron drift? Answer Rearrange the equation: v = 200 ÷ (8 × 1028 × 7.85 × 10-5 × 1.6 × 10-19) = 2.0 × 10-4 m s-1

The result from this calculation shows that the speed of electron drift is slow, 1 mm every 5 seconds.  The propagation of the message "that the current is flowing" is very fast, not far off the speed of light.  But the drift speed of electrons is not at all fast.  We can show this by putting a small crystal of potassium permanganate (a dark purple ionic compound) onto a filter paper which has been soaked in sodium chloride solution.  When a current flows, the purple permanganate ions move towards the positive terminal, but it takes some time.

In questions on this, you may be given a diameter of a wire.  To work out the area, you must use the formula:

You must convert millimetres to metres when using the equation.

 Question 1 A copper wire of diameter 1.4 mm connects to the tungsten filament of a light bulb of which the diameter is 0.020 mm.  The current in both materials is 0.52 A. Find the speed of an electron in each of the two materials. Copper has 8 × 1028 electrons per cubic metre.  Tungsten has 3.4 × 1028 m-3.

The tungsten filament has a higher resistance than the copper wire.  For the same current to pass through, the electrons have to go faster, because:

• The number of charge carriers per cubic metre is smaller;

• The area is smaller.

This is opposite to the generally perceived idea that electrons are slowed down by high resistances.

Resistivity

The resistance of a wire depends on three factors:

• the length; double the length, the resistance doubles.

• the area; double the area, the resistance halves.

• the material that the wire is made of.

Resistivity is a property of the material.  It is defined as the resistance of a wire of the material of unit area and unit length.

The formula for resistivity is:

In physics code we write this as r = AR/l

There are three bear traps

• The unit for resistivity is ohm metre (Wm), NOT ohms per metre.

• Notice too that the physics code r (rho, a Greek letter 'r') is the same as that for density.  Resistivity has NOTHING to do with density.

• The area is in square metres.  Real wires have areas in square millimetres; 1 mm2 = 1 × 10-6 m2

 Constantan has a resistivity of 47 ´ 10-8 Wm.  How much of this wire is needed to make a 10 ohm resistor, if the diameter is 0.5 mm?

The reciprocal or inverse of resistivity is conductivity.  It has the physics code s, (“sigma”, a Greek letter ‘s’), and units Siemens per metre (S m-1).

Conductivity is given by the relationship:

Super-conduction

A super-conductor is a material that has zero resistance.  A current flows when there is no potential difference.  The piece of metal floating above the magnet shows that there must be a current flowing.

Picture from Wikimedia Commons.

Authors: Julien Bobroff and Frederic Bouquet

For all metals the resistivity (hence resistance) decreases as they get colder.

For some metals like copper and silver, there is still a tiny bit of resistance left at very low temperatures.

Very low temperatures have to be maintained, which is expensive.  Room temperature superconductivity has not been seen.

Super-conductivity is seen in:

• Aluminium

• Tin;

• Some alloys;

• Some heavily-doped semi-conductors.

All superconductors have a critical temperature above which the phenomenon stops.  The graph below shows the idea:

Above the boiling point of liquid nitrogen, 77 K (-196 oC), superconductivity can be observed in a few materials.  These are called high temperature superconductors.

Very large magnets such as those found in the large hadron collider have coils made of superconducting materials.  It is believed that the superconductivity will last 100 000 years, as long as the coils don’t go above their critical temperature.

The mechanism for super-conduction is complex, and cannot be explained in terms of electrons colliding with ions.

 Explain what happens when a super-conducting metal reaches its critical temperature.

Semiconductors

Semiconductors have a resistivity (or conductivity) that is intermediate between a conductor (like a metal) and a insulator (like glass).  They are usually based on silicon, which is in the same group as carbon.  Group IV elements have 4 valence electrons and form a lattice like this:

The valence electrons form covalent bonds.  There are very few free electrons so pure silicon is a very poor conductor.

However, if we dope the pure silicon crystal by adding an impurity, for example, phosphorus, we can change things.

We have a spare electron left over, because phosphorus is a Group V element, which has 5 valence electrons.  4 are used to make covalent bonds, while the one left over is free.  The free electron acts as a negative charge carrier.  We call this kind of material a n-type semiconductor.

If we use aluminium, or other Group III element as a dopant, we only have 3 valence electrons. Therefore there is one of the silicon electrons that is unpaired.

This missing electron is called a hole and it acts as a positive charge carrier.  Since the silicon atoms can borrow electrons from their neighbours, the hole can move about, and move away from the original aluminium atom.  The hole will move towards the negative terminal of the semiconductor.

In the picture above, we see holes being attracted towards the negative, and electrons being attracted towards the positive.

If we place an n-type semiconductor next to a p-type, we find that some of the holes will diffuse into the n-type material at the interface, or junction, between the materials.  We also find that some of the electrons will diffuse into the p-type material.  The electrons and the holes combine and act as neutral.  So the n-material there has been depleted of electrons, and the p-material has been depleted of holes.  We call this region the depletion layer and it acts as an insulator

If we apply a positive voltage to the n-type material (and a negative voltage to the p-type), the depletion layer gets wider.  This is because more holes are attracted from the p-material towards the negative (and electrons to the positive).  The increased combination of electrons and holes increases the width of the depletion layer.

If we swop the polarity about, so that the n-type material is negative, the holes get attracted to the negative and the electrons get attracted to the positive.  The depletion layer is lost, and conduction starts.

This could be thought of merely as an interesting physics curiosity, but is actually an important concept at the heart of all solid state electronic devices.

Conduction bands

The models that have been explained above are not perfect.  They do not cover every aspect of electrical conduction.  In the early part of the Twentieth Century, quantum physics grew to allow physicists to explain a lot of observations that cannot be explained by normal (or classical) physics.  Quantum physics is not at all easy to understand, but there are two things that we can use:

• Electrons perch on the rungs of an energy ladder;

• Electrons exist in probability clouds.

The key point is that electrons are quantum beings.  If you try to catch an electron, you will never do so.  The closer you are to getting hold of the little brute, the less likely it is that you will catch it.

Let's try to illustrate these ideas with another model:

"Fingers" is a criminal.  He is in prison, but he doesn't think he should be inside.  He wants to be outside to commit more crimes.

In the real world, Fingers can jump, but not that high.  He could not jump from the Inside to the Outside.  So he remains in the nick.   But in the quantum world, Fingers is in a probability cloud.

Fingers' probability cloud extends to just over the prison wall.  So there is a tiny, but real, probability that he could happen to be just on top the prison wall, so he could make good his escape.  Also, in the quantum world, the closer the coppers are to nicking him, the more likely he is to get away.

Now electrons perch on different rungs of the energy ladder.  They have to be on a particular rung; they cannot perch between rungs.  Because they live in probability clouds, there is a chance that they can fly up to the next rung of the energy ladder, as long as another electron comes down from a higher perch (which it will).  For most of their time, electrons are in the valence band, perching on the normal rungs of their ladder.

Above the normal rungs (the valence band) there is a forbidden gap, a space where there are no rungs for them to perch.  Above forbidden gap is the conduction band in which the electrons are free to move about.  It's a bit like aeroplanes flying about an aerodrome.  They can be on the ground, but once in the air, they must not fly below 500 m in the region of the aerodrome.  They can fly at any height above 500 m, but if they fall below, the pilots could be in trouble.

The electrons can get sufficient energy to jump the forbidden gap to go into the conduction band.  This is because of the probability cloud.  If the forbidden gap is small, as in metals, the probability that the electrons will make it to the conduction band is high.  In a pure semiconductor, the forbidden gap is bigger, so the probability of electrons jumping to the gap is smaller, but not impossible.

If we put impurities into the crystal lattice of the semi-conductor, we put extra levels in the forbidden gap, a bit like wires for the electrons to perch on.  This allows for a greater probability of the electrons reaching the conduction band.