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A few words
before you start: |
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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). |
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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. |
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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. |
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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. |
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The essential parameters of interest
for conductors are: |
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1.
Specific resistivity
r or specific conductivity
s = 1/r. |
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The
defining
"master" equation is |
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With q = magnitude of the charge of the current carrying particles;
n = concentration of current
carrying particles (usually electrons in conductors) ; µ =
mobility of the current carrying particles. |
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The units are |
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| [r] |
= |
Wm |
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| [s] |
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(Wm)–1 |
= S/m |
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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
Wcm. |
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A homogeneous material with a constant
cross-sectional area F and a length l thus has a
resistance of R = (r ·
l)/F |
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Or, in other words, a cube with 1 cm
length has a resistance R given in W that is numerically equal to ist specific resistance
r given in Wcm. |
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If electrons are carrying the current, we have
q = – e = elementary charge = 1.602 ·
10–19 C. |
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For units, conversions, and so
on consult
the link! |
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2. Ohm's law.
Ohm's law
(which was not a "law", but an empirical observation) formulated for
the specific quantities writes |
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With j = current density (a
vector); E = electrical field strength (a
vector); s =
specific conductivity, in general a tensor
of 2nd rank and, most important, not
a function of the field strength E 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. |
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Ohm's law thus means that the E - j
characteristics or the easily measured voltage - current characteristics are
always straight lines through the origin!
Within reasonable values of E, or U, of course. |
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If you have any problem with these equations, perhaps because
you feel Ohm's law should read R = U/I, or if you
are not sure about the the meaning of the basic quantities, as e.g., mobility, you have a
problem. Turn to the
required reading
module and other modules accessible from there. |
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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. |
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A remark to the mathematical
notation: 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: |
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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 v is easily mixed
up with the Greek nu n. |
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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. |
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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
E 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 color coding whenever the possibility of negative
symbol interference raises its ugly head. |
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The density and
mobility of mobile charged
carriers thus determines the conductivity. |
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The carrier density is a
function of bonding (metallic, covalent in semiconductor, etc.), defects
(doping in semiconductors) and temperature in general. In metals, however,
ne is nearly constant. |
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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). |
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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". |
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Ohm's "law"
asserting that s is not a function of E but only of some material property that can be
expressed as a number. |
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Matthiesen's rule, stating that |
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| r = rLattice(T) + rdefect(N) |
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With N = some measure of defect density. |
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A "rule of
thumb": r is
proportional to T for T > some
Tcrit |
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| Dr |
= |
ar ·
r · DT |
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0,4%
oC |
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With Temperature
coefficient ar = 1/r · dr /
dT. |
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Then we have the Wiedemann-Franz "law", linking electrical conductivity
to thermal conductivity, and so on. |
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The links give some graphs and numbers for
representative metals. |
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Table of
some metal properties
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r(T) for different defect densities in
Na
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r(T) for different metals |
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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 s and r,
while closely related, are quite different parameters with a numerical value
that depends on the choice of the units! Do not mix up cm and
m! |
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| Metal |
Ag |
Cu |
Au |
Al |
Na |
Zn |
Ni |
Fe |
Sn |
Pb |
Hg |
| r
[mWcm] |
1,6 |
1,7 |
2,2 |
2,7 |
4,2 |
5,9 |
6,8 |
9,7 |
12 |
21 |
97 |
s =
1/r
[106 · W–1cm–1] |
0.625 |
0.588 |
0,455 |
0.37 |
0.238 |
0.169 |
0.147 |
0.103 |
0.083 |
0.046 |
0.01 |
s =
1/r
[106 · W–1m–1] |
62,5 |
58.8 |
45.5 |
37 |
23.8 |
16.9 |
14.7 |
10.3 |
8.3 |
4.6 |
1 |
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The temperature dependence,
expressed e.g. in r(300K)/r(100K) may be a factor of 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. |
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This is something you should be aware of; cf. the
anecdote in the link. |
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The specific resistivity, however, is not the only
property that counts. In selecting a metal, important design parameters might
also be: |
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Weight, mechanical strength, corrosion resistance, prize, compatibility with other materials, ..... |
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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. |
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One very simple example: Lets say, weight is important. Define a
figure of
merit = F = r/
d, with d = density. The bigger F, the
better. |
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You now get the following ranking (normalized to
FNa = 1): |
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| Metal |
Na |
K |
Ca |
Al |
Mg |
Cu |
Ag |
| F |
1 |
0,77 |
0,69 |
0,56 |
0,52 |
0,28 |
0,25 |
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The winner sodium! So you are going to use
Sodium - Na for wiring? |
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Certainly not. Because now you will either include chemical
stability C in your figure of merit (just multiply with
C and assign values C = 1 for great stability (e.g.
Au, Al,), C = 2 for medium stability (Cu,
Mg) and C = 5 for unstable stuff (Na, 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. |
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And while you are at it, divide by some price index
P. Use the price per kg, or just a rough grade scale. |
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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. |
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© H. Föll (Electronic Materials - Script)
