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In this subchapter we will look at the classical treatment of the movement of electrons
inside a material in an electrical field. |
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In the preceding subchapter we obtained the most basic
formulation of Ohms law, linking the
specific conductivity to two fundamental material parameters: |
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For a homogeneous and isotropic material (e.g.
polycrystalline metals or single crystal of cubic semiconductors), the concentration of carriers n and their
mobility µ have the same value
everywhere in the material, and the specific conductivity s is a scalar. |
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This is boring, however. So let's look at useful
complications: |
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In general terms, we may have more than one kind
of carrier (this is the common situation in semiconductors) and n
and µ could be functions of the temperature T, the
local field strength
Eloc resulting from an applied external voltage, the detailed structure of the
material (e.g. the defects in the lattice), and so on. |
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We will see that these complications are the essence of
advanced electronic materials (especially semiconductors), but in order to make
life easy we first will restrict ourselves to the special class of ohmic materials. |
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We have
seen before
that this requires n and µ to be independent of the
local field strength. However, we still may have a temperature dependence of
s; even commercial ohmic resistors, after
all, do show a more or less pronounced temperature dependence - their
resistance increases roughly linearly with T. |
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In short, we are treating metals, characterized by a constant density of one kind of carriers (= electrons) in the
order of 1 ...3 electrons per atom in the metal. |
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We consider the electrons in the metal to be
"free", i.e. they can move freely in any direction - the atoms of the
lattice thus by definition do not impede
their movement. |
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The (local) electrical field
Eloc then exerts a force
F = – e · Eloc on any
given electron and thus accelerates the electrons in the field direction (more
precisely, opposite to the field direction because the field vector points from
+ to – whereas the electron moves from – to
+). |
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In the fly
swarm analogy, the electrical field would correspond to a steady airflow -
some wind - that moves the swarm about with constant drift velocity. |
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Now, if a single electron with
the (constant) mass m and momentum p is subjected
to a force F, the equation of motion from basic mechanics
is |
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Note that p does not have to be zero
when the field is switched on. |
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If this would be all, the velocity of a given
electron would acquire an ever increasing component in field direction and
eventually approach infinity. This is obviously not possible, so we have to
bring in a mechanism that destroys an unlimited increase in
v. |
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In classical mechanics this is done by introducing a
frictional force
Ffr that is proportional to the velocity. |
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with kfr being some friction
constant. But this, while mathematically sufficient, is devoid of any physical meaning with regard to the
moving electrons. |
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There is no "friction" on an
atomic
scale! Think about it! Where should a friction force come from? An electron
feels only forces from two kinds of fields
- electromagnetic and gravitational (neglecting strange stuff from particle
physics). |
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It thus makes no sense to complement the
differential equation above with a friction term - we have to look for a better
approach. |
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All that friction does to big classical bodies is
to
dissipate ordered kinetic
energy of the moving body to the environment. Any ordered movement gets slowed down to zero (surplus)
speed, and the environment gets somewhat hotter instead, i.e. unordered movement has increased. |
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This is called energy
dissipation, and that is what we
need: Mechanisms that take kinetic energy away from an electron and
"give" it to the crystal at large. The science behind that is called
(Statistical) Thermodynamics - we have
encountered it before. |
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The best way to think about this, is to assume
that the electron, flying along with increasing velocity, will hit something else along its way every now and then;
it has a collision with something else, or,
as we will say from now on, it will be scattered at something else. |
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This collision or scattering event will change its momentum, i.e. the magnitude and the direction of
v, and thus also its kinetic energy
Ekin, which is always given by |
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| Ekin =
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m · v2
2 |
= |
p · v
2 |
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In other words, we consider
collisions with something else, i.e. other particles (including
"pseudo" particles), where the total energy and momentum of all the
particles is preserved, but the individual particle looses its
"memory" with respect to its velocity before the collision, and
starts with a new momentum after every collision. |
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What are the "partners"
for collisions of an electron, or put in standard language, what are the
scattering mechanisms? There are
several possibilities: |
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Other electrons. While this
happens, it is not the important process in most cases. It also does not
decrease the total energy contained in the electron movement - the losses of
some electrons are the gains of others. |
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Defects, e.g. foreign atoms, point
defects or dislocations. This is a more important scattering mechanism,
moreover, it is a mechanism where the electron can transfer its surplus energy
(obtained through acceleration in the electrical field) to the atoms of the
lattice, which means that the material heats up |
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Phonons, i.e. "quantized" lattice vibrations traveling
through the crystal. This is the most important
scattering mechanism. |
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Now that is a bit strange. While we (hopefully)
have no problem imagining a crystal lattice with all atoms vibrating merrily,
there is no immediate reason to consider these vibrations as being localized (whatever this means) and particle-like. |
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You are right - but nevertheless: The lattice vibrations
indeed are best described by a bunch of particle-like phonons careening through the crystal. |
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This follows from a quantum mechanical treatment of lattice
vibrations. Then it can be shown that these vibrations, which contain the
thermal energy of the crystal, are quantized and show typical properties of
(quantum) particles: They have a momentum,
and an energy given by hn (h = Plancks constant, n = frequency of the vibration). |
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Phonons are a
first example of "pseudo" particles; but there is no more
"pseudo" to phonons than there is to photons. |
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We will not go into more details here. All we need to know is
that a hot crystal has more phonons and
more energetic phonons than a cold crystal, and treating the interaction of an
electron with the lattice vibration as a collision with a phonon gives not only
correct results, it is the only way to get results at all. |
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It this point comes a crucial insight: It would be
far from the truth to assume that only accelerated electrons scatter; scattering happens
all the time to all the electrons moving randomly about because they all have
some thermal energy. Generally, scattering is the mechanism to achieve thermal
equilibrium and equidistribution of the energy of the crystal. |
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If electrons are accelerated in an electrical field and thus
gain energy in excess of thermal equilibrium, scattering is the way to transfer
this surplus energy to the lattice which then will heat up. If the crystal is
heated up from the outside, scattering is the mechanism to turn heat energy
contained in lattice vibrations to kinetic energy of the electrons. |
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Again: Even without an electrical field, scattering is the
mechanism to transfer thermal energy from the lattice to the electrons (and
back). Generally, scattering is the mechanism to achieve thermal equilibrium and equidistribution of the
energy of the crystal. |
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Our free electrons in metals behave very much like a gas in a
closed container. They
careen around with some average
velocity that depends on the energy contained in the electron gas, which is - in classical terms- a
direct function of the
temperature. |
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Let's look at some figures illustrating the
scattering processes. |
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Shown here is the magnitude of the velocity v
±x of an electron in +x and
–x direction without an
external field. The electron moves with constant velocity until it is
scattered, then it continues with some new velocity. |
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The scattering processes, though unpredictable at single
events, must lead to the averages of the velocity, which is characteristic for
the material and its conditions. |
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The plural in "averages" is intentional: there are different averages of the velocity |
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Whereas <v> = 0, <v> has a finite value; this is also true for
<vx> = – <v
–x> . Consult the
"fly swarm
modul" if you are unsure about this. |
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From
classical
thermodynamics we know that the (classical) electron gas in thermal
equilibrium with the environment contains the energy Ekin
= (1/2)kT per particle and degree of freedom, with k = Boltzmanns constant and
T = absolute temperature. If
you forgot all about this, check this Link, too. The three degrees of freedom
are the velocities in x-, y- and
z-direction, so we must have |
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| Ekin,x |
= |
½ · m · <vx>2 = ½ · kT
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| <vx> |
= |
æ
ç
è |
kT
m |
ö
÷
ø |
1/2 |
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For the other directions we have
exactly the same relations, of course. For the total energy we obtain |
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| Ekin = |
m · <vx2>
2 |
+ |
m · <vy2>
2 |
+ |
m · <vz2>
2 |
= |
m · <v2>
2 |
= |
m · (v0)2
2 |
= |
3kT
2 |
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with v0 =
<|v|> =
<v>.
v0 is thus the
average velocity of a
carrier careening around in a crystal. We can easily calculate it from
the formula given above; we have |
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At this point you should stop a moment and think
about just how fast those electrons will be careening around at room
temperature (300K) without plugging numbers in the equation. Got a
feeling for it? Probably not. So look at the exercise question (and the
solution) further down!. |
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Now you should stop another
moment and become very aware of the fact that this equation is from purely
classical physics. It is absolutely true
for classical particles - which electrons
are actually not. Electrons obey the
Pauli
principle, i.e. they behave about as non-classical as possible. This should
make you feel a bit uncomfortable. Maybe the equation from above is not correct
for electrons then? Indeed - it isn't. Why, we will see later; also how we can
"repair" the situation! |
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Now lets turn on an
electrical field. It will accelerate the electrons between the collisions. Their velocity in field
direction then increases linearly from whatever value it had right after a
collision to some larger value right before the next collision. |
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In our diagram from above this looks like
this: |
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Here we have an electrical
field that accelerates electrons in x-direction (and
"brakes" in –x direction). Between collisions, the
electron gains velocity in +x-direction at a constant rate (=
identical slope). |
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The average velocity in +x directions,
<v+x>, is now larger than in –x
direction, <v–x>. |
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However, beware of the pitfalls of schematic drawings: For
real electrons the difference is very small as we shall see shortly; the slope
in the drawing is very exaggerated. |
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The
drift velocity is contained in the difference
<v+x>
– <v–x>; it is completely described by the velocity gain
between collisions. For obtaining a value, we may neglect the instantaneous
velocity right after a scattering event because they average to zero anyway and
just plot the velocity gain in a simplified
picture; always starting from zero after a collision. |
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The picture now looks quite simple; but remember
that it contains some not so
simple averaging. |
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At this point it is time to define a very meaningful
new average quantity: |
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The mean time between
collisions, or more conventional, the mean time t for reaching the drift velocity v in the
simplified diagram. We also call t the mean scattering
time or just scattering time for
short. |
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This is most easily illustrated by simplifying the
scattering diagram once more: We simply use just one time - the average - for the time that elapses
between scattering events and obtain: |
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This is the standard
diagram illustrating the scattering of electrons in a crystal
usually found in text books; the definition of the scattering time t is included |
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It is highly idealized, if not to say just wrong if you
compare it to the correct picture above. Of
course, the average velocity of both pictures will give the same value, but
that's like saying that the average speed va of all real cars
driving around in a city is the same as the average speed of ideal model cars,
which are going at va all the time. |
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Note that t is only half of the average time between
collisions. |
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So, while this diagram is not wrong, it is a
highly abstract rendering of the underlying processes obtained after several
averaging procedures. From this diagram only, no conclusion whatsoever can be
drawn as to the average velocities of the electrons without the electrical
field! |
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With the scattering concept, we now have two new
(closely related) material parameters: |
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The mean (scattering)
time t between two collisions as
defined before. |
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The mean free
path l between collisions; i.e. the distance travelled by
an electron (on average) before it collides with something else and changes its
momentum. We have |
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Note that v0 enters the defining equation
for l, and that we have to take twice the scattering time
t because it only refers to half the time
between collisions! |
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After we have come to this point, we now can go
on: Using t as a new parameter, we can
rewrite Newtons equation from above for an
electron (q = -e) as follows: |
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| m · |
dv
dt |
= m · |
Dv
Dt |
= m · |
vD
t |
= F = q · E =
– e · E |
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We now only consider what happens to the electron as long as
it doesn't hit anything. Then it is possible to equate the differential quotient with the difference quotient, because the velocity change is
constant. After a scattering event has taken place, the process is completely
interrupted and starts under "virgin" conditions again. |
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We obtain immediately the relation between the drift velocity
vD and the applied field E: |
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vD
t |
= – |
E · e
m |
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| Þ |
vD |
= – |
E · e · t
m |
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Inserting this equation for vD
in the old definition of the current
density j = – n · e ·
vD and invoking the general version of
Ohms law, j =
s · E, yields |
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| j = |
n · e2 · t
m |
· E |
: = |
s · E |
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This gives us the final result |
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This is the classical formula for the conductivity of a
classical "electron gas" material; i.e. metals. The conductivity
contains the density n of the free electrons and their mean
classical scattering time t as material parameters. |
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We have a good
idea about n, but we do not yet know tclass, the mean classical scattering time for classical electrons.
However, since we know the order of
magnitude for the conductivity of metals, we may turn the equation around
and use it to calculate the order of magnitude of tclass. If you do the exercise farther
down, you will see that the result is: |
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| tclass = |
s · m
n · e2 |
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(10– 13 .... 10– 15) sec |
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"Obviously" (as stated in many
text books), this is a value that is far too
small and thus the classical approach must be wrong. But is it really too small? How can you tell without knowing a lot more about electrons
in metals? |
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Let's face it: you
can't !!. So let's look at the mean free path l
instead. We have |
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The last equation gives us a value
v0 » 104 m/s at
room temperature! Now we need vD, and this we can estimate
from the equation given above to
vD = – E · t
· e/m » 1 mm/sec,
if we use the value for t dictated by the measured conductivities. It is much
smaller than v0 and can be safely neglected in calculating
l. |
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We thus can rewrite the equation
for the conductivity and obtain |
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| s = |
n · e2 · l
2 · m · (v0 + vD) |
» |
n · e2 · l
2 · m · v0 |
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Knowing s from experiments, but not l, allows to
determine l. The smallest possible mean free path
lmin between collisions (for vD = 0)
for typical metals thus is |
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| lmin = |
2 · m · v0 · s
n · e2 |
= 2 · v0 · t |
» (10–1 –
101) nm |
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And this is
certainly too small!. |
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But before we discuss these results, let's see if
they are actually true by doing an exercise: |
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Now to the important question: Why is a mean free path in the order of the size of
an atom too small? |
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Well, think about the scattering
mechanisms. The distance between lattice defects is certainly much larger,
and a phonon itself is "larger", too. |
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Moreover, consider what happens at temperatures below room
temperatures: l would become even smaller since
v0 decreases - somehow this makes no
sense. |
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It does not pay to spend more
time on this. Whichever way you look at it, whatever tricky devices you
introduce to make the approximations better (and physicists have tried very
hard!), you will not be able to solve the
problem: The mean free paths are never even coming close to what they need to
be, and the conclusion which we will reach - maybe reluctantly, but unavoidably
- must be: |
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There is no way to describe conductivity (in metals)
with classical physics!
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Somewhere on the
way, we have also indirectly found that the
mobility
µ as defined before is just another way
to look at scattering mechanisms. Let's see why. |
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All we have to do is to compare the
equation s = (n · e2
· t)/m for the conductivity
from above with the
master equation s = q · n · µ. |
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This gives us
immediately |
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| µ |
= |
e · t
m |
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| µ |
» |
e · l
2 · m · v0 |
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In other words: |
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| The decisive
material property determining the mobility µ is the average time between
scattering events or the mean free path between those events. |
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The mobility µ thus is a basic
material property, well-defined even without electrical fields, and just another way to characterize the scattering processes taken
place by a single number. |
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We even can go one stage further with
this: If we envision the movement of an electron again, as described above in
many words, analogies ("fly swarm"), graphs and equations, we
"see" exactly the same thing we envisioned when we looked a diffusing
particle or vancany when we learned about
diffusion and
random
walk. |
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"Something" bounced round in a random
matter, and everything important about the "something" was captured
in its diffusion coefficient D. This diffusion coefficient was
either defined via Fick's laws (e.g.
Fick's
first law jx = –D ·
dn/dx) or by looking at the atomic mechanisms that got
us something like D »
a2 · r (a = lattice
constant, r = jump rate). From the
random walk
consideration we had for the "diffusion length" L
= (D · t)½, a relation that also
could be used to define D. |
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You should now have a certain feeling that all
this old stuff from diffusion and what we just learned about the random
bouncing around of electrons, must be somehow connected. After all, we always
have the element of something moving around (mostly) at random. |
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Right you are! Again, it was
Einstein
(and independently Smoluchowski) who found the proper relation, the
Einstein-Smoluchowski relation hinted
at a chapter ago: |
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| D |
= |
µ · kT
e |
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| µ |
= |
D · e
kT |
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The mobility µ thus is
"almost" the same as the diffusion coefficient D; for a
given temperature T they are proportional to each other, |
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How do we obtain this simple relation? Well -
we won't at this point. It's not all that
difficult to derive, but it is no accident either that it's called after
Einstein (that's actually part of what he got the Nobel prize for). |
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If you are not satisfied with that, check
this
link for a derivation, or
this
one for an alternative way. More to the relation between diffusion
coefficient and mobility in this (German) link. |
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If you are a bit exhausted and confused by now - that's
OK!
This German link might help, where things are summed
up once more.
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In the equations above slumbers an
extremely important aspect of semicoductor technology: |
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In all electronic devices carriers have to travel
some distance before a signal can be produced. A
MOS
transistor, for example, switches currents on or off between its
"Source" and "Drain" terminals depending on what voltage is
applied to its "Gate". Source and drain are separated by some
distance lSD, and the "Drain" only
"feels" the "on" state after the time it takes the carriers
to run the distance lSD. |
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How long does that take if the
voltage between Source and Drain is USD? |
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Easy. If we know the mobility µ of
the carriers, we also know their (average) velocity vSD in
the source-drain region, which by
definition is vSD = µ ·
USD/lSD. |
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The traveling time
tSD between source and drain for obvious reasons
defines roughly the maximum frequency fmax the
transistor can handle, we have tSD =
lSD / vSD or |
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| tSD = |
lSD2
µ · USD |
» |
1
fmax |
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The maximum frequency of a MOS transistor
thus is directly proportional to the mobility of the carriers in the material
it is made from (always provided there are no other limiting factors). And
since we used a rather general argument, we should not be surprised that pretty
much the same relation is also true for most electronic devices, not just
MOS transistors. |
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This is a momentous statement: We linked a prime
material parameter, the material constant
µ, to one of the most important parameters of electronic circuits.
We would like µ to be as large as possible, of course, and now we
know what to do about it! |
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Actually, we
do not really know what to do, but other people do - and act on it. See the
link to find out how it is
done. |
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A simple exercise is in order to see
the power of this knowlegde: |
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© H. Föll (MaWi 2 Skript)
