A few words before you start: |
||

Conductors in general are a bit boring, whereas conductors
in particular applications are often hot topics (take the recent switch from Al
to Cu in chip technology, for example). | ||

There is a large number of highly optimized materials which are used as conductors nowadays. Just enumerating them is tiresome and not very interesting. Still, some knowledge of this issue is a must for materials scientists in the context of electronic materials. | ||

As far as "theory" goes, there is either not much that goes beyond a basic knowledge of solid state physics (which, it is assumed, you already have), or very involved special theory (e.g. for superconductors or conducting polymers) - for which there is no time. | ||

In conclusion, we will only touch the issue, trying to present all major facets of the topic. In particular, the long list of applications for conductors (much longer than you probably would imagine) will be covered. This chapter, however, will be brief and mostly lists topics and key words. | ||

**The Basics**

The essential parameters of interest for conductors are: | |||||||||||||||

1. Specific resistivity
or rspecific conductivity
s
= 1/.r | |||||||||||||||

The defining "master" equation is | |||||||||||||||

| |||||||||||||||

With magnitude of the q =charge of the
current carrying particles; = nconcentration of current carrying particles
(usually electrons in conductors) ; of the current carrying particles.µ = mobility |
|||||||||||||||

The units are | |||||||||||||||

| |||||||||||||||

Note that S
= "Siemens" = 1/W = A/V is a
bit old fashioned, but still in use. Note, too, that while the SI standard units call for the meter
(m), you will find many values given in W.cm |
|||||||||||||||

A homogeneous material with a constant cross-sectional area and a length F
thus has a resistance of lR = (r · l)/F |
|||||||||||||||

| |||||||||||||||

Or, in other words, a cube with 1 cm length has a resistance given
in RW that is numerically equal to ist specific resistance
given in rWcm. | |||||||||||||||

If electrons are carrying the current, we
have = elementary charge q = – e= 1.602 · 10.^{–19} C | |||||||||||||||

For units, conversions, and so on consult the link! | |||||||||||||||

2. Ohm's
law. Ohm's law (which was
not a "law", but an empirical observation) formulated for the specific quantities writes | |||||||||||||||

| |||||||||||||||

With current density (a =jvector);
electrical field strength (a vector);
=Es = specific conductivity, in general a tensor of
2nd rank and, most important, not a function of the field strength if not specifically noted. In other words, if the specific conductivity of a material
is a constant, i.e. a fixed number with respect to E, the material obeys
Ohm's law.E | |||||||||||||||

Ohm's law thus means that the - E characteristics
or the easily measured voltage - current characteristics are always jstraight lines through
the origin! Within reasonable values of , or E, of course.U |
|||||||||||||||

If you have any problem with
these equations, perhaps because you feel Ohm's law should read , or if you are not sure
about the the meaning of the basic quantities, as e.g., R = U/Imobility, you
have a problem. Turn to the required reading module
and other modules accessible from there. | |||||||||||||||

More about Ohm's law and the failure of classical physics in explaining the conductivity of metals can be found in a second required reading module. Add to this the required reading module for averaging vector quantities and you are ready for this chapter and others to come. | |||||||||||||||

A remark to the : HTML possibilities are limited and it is difficult to adhere to all rules of notation. In case
of doubt, clarity and easy reading will have preference to formal correctness. This means:mathematical
notation | |||||||||||||||

Whenever sensible, cursive symbols will
be used for variables. It is not sensible, e.g., to use cursive letters for the velocity
v, because the cursive is easily mixed up with the Greek nu vn. |
|||||||||||||||

All equations and the quantities used in equations are always bold
- this greatly improves readability. However, it leaves little room for symbolizing vectors by bold lettering, and
since underlining is cumbersome and not particularly helpful, we simply will mostly not use special notation for vectors.
If you are able to understand this lecture course at all, you will know the vector (or tensor) quantities anyway. |
|||||||||||||||

There are not enough letters in the alphabet to give every
physical quantity an unambiguous symbol. One and the same symbol thus traditionally has several meanings, usually quite
clear from the context. Occasionally, however, the danger of mix-up occurs. An example in case is the traditional use of
the letter for electrical field strength and for energies (and for Young's modulus in German). While in
conventional texts one must give a different letter to these quantities, we will use the advantage of HTML and use Ecolor coding whenever the possibility of negative symbol interference raises its ugly head. |
|||||||||||||||

The density
and mobility of mobile
charged carriers thus determines the conductivity. |
|||||||||||||||

The carrier density is a function of bonding (metallic, covalent in
semiconductor, etc.), defects (doping in semiconductors) and temperature in general. In metals, however,
is nearly constant.n_{e} | |||||||||||||||

The mobility is a function of collisions between carriers (e.g. electrons
and holes) and/or between carriers and obstacles (e.g. phonons
and crystal lattice defects). | |||||||||||||||

Carrier concentration and mobility are, in general, hard to calculate from first principles. In semiconductors, the carrier density is easy to obtain, mobility is somewhat harder. In metals, the carrier density is rather fixed, but mobility is quite difficult to calculate, especially for "real" i.e. rather imperfect crystals. There are however, empirical rules or "laws". | |||||||||||||||

Ohm's "law" asserting that
is snot a function of but only of some material
property that can be expressed as a number.E | |||||||||||||||

Matthiesen's
rule, stating that | |||||||||||||||

| |||||||||||||||

With some measure of defect density.N = | |||||||||||||||

A "rule of thumb":
is proportional to r for TT > some T_{crit} | |||||||||||||||

| |||||||||||||||

With Temperature coefficient
a._{r} = 1/r
· dr / dT | |||||||||||||||

Then we have the Wiedemann-Franz
"law", linking electrical conductivity to thermal conductivity, and so on. |
|||||||||||||||

The links give some graphs and numbers for representative metals. | |||||||||||||||

Table of some metal properties |
|||||||||||||||

r(T) for different
defect densities in Na | |||||||||||||||

r(T) for different
metals |

**Some Values and Comments**

The range of resistivity values (at room temperature) for metals
is rather limited; here are some values as well as a first and last reminder that
and s, while closely related, are quite different parameters with a numerical value
that depends on the choice of the units! Do not mix up rcm and m! | ||||||||||||||||||||||||||||||||||||||||||||||||||

| ||||||||||||||||||||||||||||||||||||||||||||||||||

The temperature dependence, expressed e.g. in may be a factor of r(300K)/r(100K)5 ...10, so it is not
a small factor. It may be used and is used, for measuring temperatures, e.g. with well-known Pt resistivity thermometers.
| ||||||||||||||||||||||||||||||||||||||||||||||||||

This is something you should be aware of; cf. the anecdote in the link. | ||||||||||||||||||||||||||||||||||||||||||||||||||

The specific resistivity, however, is not the only property that counts. In selecting a metal, important design parameters might also be: | ||||||||||||||||||||||||||||||||||||||||||||||||||

Weight, mechanical strength, corrosion resistance, prize, compatibility
with other materials, ..... | ||||||||||||||||||||||||||||||||||||||||||||||||||

Sometimes it is advisable to look at "figures
of merit", i.e. the numerical value coming out of a self-made formula that contains your important criteria
in a suitable way. | ||||||||||||||||||||||||||||||||||||||||||||||||||

One very simple example: Lets say, weight is important. Define a figure of merit = , with F = r/
d density. The bigger d =, the better.F | ||||||||||||||||||||||||||||||||||||||||||||||||||

You now get the following ranking (normalized to ): F_{Na} = 1 |
||||||||||||||||||||||||||||||||||||||||||||||||||

| ||||||||||||||||||||||||||||||||||||||||||||||||||

The winner sodium! So you are going to use Sodium - Na
for wiring? | ||||||||||||||||||||||||||||||||||||||||||||||||||

Certainly not. Because now you will either include chemical stability in your figure of
merit (just multiply with C and assign values C for great stability (e.g. C = 1Au, Al,),
for medium stability (C = 2Cu, Mg) and for unstable stuff (C = 5Na, K,
Ca). Or any other number reflecting the importance you put on this parameter. There is no ready made recipe - if
precise numbers are not existing, you take your personal grading scale. | ||||||||||||||||||||||||||||||||||||||||||||||||||

And while you are at it, divide by some price index . Use the price per Pkg, or just
a rough grade scale. | ||||||||||||||||||||||||||||||||||||||||||||||||||

In this simple example, you will get a surprising result: No matter what you do, the winner will be Al.
It is the (base) material of choice for heavy duty applications when weight matters. In not so simple questions, you really
may benefit from using the figure of merit approach. | ||||||||||||||||||||||||||||||||||||||||||||||||||

| ||||||||||||||||||||||||||||||||||||||||||||||||||

© H. Föll (Electronic Materials - Script)