Conduction of electricity in materials is by means of ‘charge carriers’, of which there are three types:
The conductivity of a material depends on three factors:
It is clear from Figure 1 that, as regards conductivity, different types of material fall into three radically different categories:
These three groups of materials, with their vastly different properties, are very important in electronics. Whilst detailed consideration of the physics2 behind their conductivity behaviour is beyond the scope and needs of this module, a simple summary is that:
Without an applied electric field, the charge carriers move randomly, at a rate dictated by the temperature. However, when an electric field is applied to a conductor, a gradual drift is superimposed on this random movement.
Figure 2 is a simplification of the situation in a conductor – to start with, there is no indication of any random movement. The current flowing is the rate of movement of charge, so, if there are n free charge carriers in unit volume, the amount of charge moved through the cross section in the conductor during time t is given by the equation
q = charge of the charge carrier
v = drift velocity2
This equation is often converted into the form
where J is called the ‘current density’.
This current density concept is very important for the designer, because there are limitations on the achievable current density – what may seem a small current in a cable is an extremely high current density in a fine track. You know what happens to fuses!
2 Metals have very high numbers of free electrons, so drift velocities are actually very small, of the order of microns (µm) per second.
You have designed a board with a 0.5 mm wide track in 35 µm thick copper. When this is carrying a current of 10mA, what is the current density in the track?
Pure silicon contains equal densities of holes and electrons. This ‘intrinsic carrier density’ (ni) is approximately 1.5×10−16m−3 at 25°C and increases by 8% by degree.
However, the intrinsic conductivity of semiconductor materials is altered greatly by the presence of very small amounts (a little as parts per thousand million) of foreign atoms. During the semiconductor ‘doping’ process, impurities are added intentionally, either to make more electrons available for conduction (creating an ‘n-type’ semiconductor) or to create holes into which electrons can move (a ‘p-type’ semiconductor). The impurities used for silicon are ‘acceptor’ elements from Group III of the Periodic Table, such as boron, gallium, indium or aluminium, which create p-type silicon, and Group V ‘donor’ atoms, such as arsenic, antimony or phosphorus, which form n-type silicon.
The flow of mobile carriers constitutes an electric current; in p-type silicon, mobile holes drift towards the more negative terminal, in the conventional direction of current flow; in n-type material, electrons drift towards the more positive terminal, in the opposite direction to the conventional current flow. The conductivity of the material is the conductivity due to the holes, plus the conductivity from the electrons. [Note that holes drifting negative and electrons drifting positive both constitute an electric current in same direction]
As with conduction in metal, the current that flows results from the drift of carriers in the direction of the electric field, the average velocity with which carriers move, the drift velocity, being proportional to the applied field. We define the ‘mobility’ (µ) of the carrier as the drift velocity per unit electric field – its units are (velocity/electric field) or m2V−1s−1. Carrier mobility varies with temperature, and the mobility of holes is only 30% of that of electrons (approximately 0.15 and 0.045 respectively), though the exact values depend on doping densities.
The current that flows is a function of the number of carriers, the mobility of carriers, and the charge per carrier. For both electrons and holes, the unit of charge is the ‘electronic charge’, q, where q = 1.6×10−19C. For a sample of n-type silicon doped to a density of Nd donor atoms per cubic metre, the conductivity σn is given by:
Note that the conductivity is directly proportional to the doping density, which is why selected areas within an integrated circuit are heavily doped in order to reduce their series resistance. The equation also explains why the conductivity of p-type silicon will be lower than n-type for a given level of doping.
The volume resistivity of a material, symbol ρ (Greek letter rho), is the resistance between opposite faces of a ‘unit cube’. For a test piece, the relationship between resistance and resistivity is given by the equation:
ρ = volume resistivity in ohm.cm
R = resistance in ohms between faces
A = area of the faces
l = distance between faces
Note that this is not resistance per unit volume, which would be ohm/cm3, although this term is sometimes incorrectly used.
The surface resistivity, symbol σ (Greek letter sigma), is the resistance between two opposite edges of a square of film. Using the equation above, where l is the length of each side, and t the thickness of the film:
For a constant film thickness, the resistance is independent of the length of the path. The units of surface resistivity are actually ohms, but are more frequently quoted in ‘ohms per square’ (Ω/sq.) to avoid confusion with usual resistance values.
The electrical resistance of an insulating material, like that of a conductor, is the resistance offered by the conducting path to the passage of current. Insulating materials are very poor conductors when dry, so that resistance values tend to be in Megohms, rather than ohms. However, the concept of surface resistivity is equally applicable, and you will find this concept when you read about ESD. In the case of insulators, the thickness of the film may be constant, but it is poorly defined – typically conduction takes place in the top layers of the surface and in any contamination or moisture on top of it.
You will also come across the term insulation resistance, which is a measurement of ohmic resistance for a given configuration, rather than a specific resistivity test. For example, insulation resistance is often measured at high voltage, in order to check for electrical safety. In that case, the term ‘proof test’ is also used.
Whether we are referring to volume resistivity, surface resistivity or insulation resistance, the values we obtain will depend on a number of factors, including temperature, humidity, moisture content, applied voltage, and the duration of voltage application. Comparing or interpreting data is difficult unless the test is controlled and defined, especially when a specimen is drying out after being subjected to moist or humid conditions. Results can be particularly affected by surface wetting or contamination, which greatly reduce surface resistivity.
Some general points can, however, be made about the likely changes in resistivity and insulation resistance as temperature rises:
3 A particular case of an insulator which becomes a conductor of electricity when heated is soda-silica glass, which contains enough sodium ions for the resistivity at 300°C to be six orders of magnitude lower than at room temperature.4 The rate of change of temperature gives useful information about the basic physics of the device.
A protective bag for boards has an internal static-dissipative layer of polyethylene whose surface resistivity is quoted as being <1012 ohms. What does this measurement mean, and how would you expect it to change with the conditions in the room?