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In this subchapter
we will give an outline of how to progress from the simple version of
Ohms "Law", which is a kind of
"electrical" definition for a black box, to a formulation of the same
law from a materials point of view
employing (almost) first principles. |
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In other words: The electrical engineering point of view is: If a
"black box"
exhibits a linear relation
between the (dc) current I flowing through it and the voltage
U applied to it, it is an ohmic
resistor. |
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That is illustrated in the picture: As long as the
voltage-current characteristic you measure between two terminals of the black
box is linear, the black box is called an (ohmic) resistor). |
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Neither the slope of the I-U-characteristics
matters, nor the material content of the box. |
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TheMaterials
Science point of view is quite different. Taken to the extreme, it
is: |
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Tell me what kind of material is in the black box, and I tell
you:
- If it really is an ohmic resistor,
i.e. if the current
relates linearly to the voltage for
reasonable voltages and both polarities.
- What its (specific) resistance will be, including its temperature
dependence.
- And everything else of interest.
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In what follows we will see, what we have to do
for this approach. We will proceed in 3 steps |
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In the first two steps, contained in this
sub-chapter we simply reformulate Ohms law in physical quantities that are
related to material properties. In other words, we look at the properties of
the moving charges that produce an electrical current. But we only define the necessary quantities; we do not calculate
their numerical values. |
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In the third step - which is the content of many
chapters - we will find ways to actually calculate the important quantities, in particular
for semiconductors. As it turns out, this is not just difficult with classical
physics, but simply impossible. We will need a good dose of quantum mechanics
and statistical thermodynamics to get results. |
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First we switch from current I and
voltage U to the current
density j
and the field
strength
E, which are not only independent of the (uninteresting)
size and shape of the body, but, since they are
vectors, carry much more information about
the system. |
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This is easily seen in the schematic drawing
below. |
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Current
density j and field
strength E may depend on the coordinates,
because U and I depend on the coordinates, e.g. in
the way schematically shown in the picture to the left. However, for a
homogeneous material with constant cross section, we may write |
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with F = cross sectional area. The direction of
the vector j would be
parallel to the normal vector f of the reference area
considered: it also may differ locally. So in full splendor we must write |
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| j(x,y,z) = |
I(x,y,z)
F |
· f |
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The "global" field strength is |
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With l = length of the body. If we
want the local field strength
E(x,y,z) as a vector, we have, in
principle, to solve the
Poisson
equation |
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| Ñ ·
E(x,y,z) = |
r(x,y,z)
ee0 |
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With r(x,y,z) =
charge density. For a
homogeneous materials with constant cross section, however,
E is parallel to f and constant everywhere,
again which is clear without calculation. |
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So. to make things easy, for a homogenous material
of length l with constant cross-sectional area F,
the field strength E and the current density
j do not depend on position - they have the same numerical
value everywhere. |
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For this case we can now write down Ohms law with
the new quantities and obtain |
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| j · F =
I = |
1
R |
· U |
= |
1
R |
· E · l |
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The fraction l/ F ·
R obviously (think about it!)
has the same numerical value for any homogeneous cube (or homogeneous whatever) of a
given material; it is, of course, the specific
conductivity s |
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and
r is the specific
resistivity. In words: A 1
cm3 cube of homogeneous material having the specific resistivity
r has the resistance R = (r · l)/F |
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Of course, we will never mix up
the specific resistivity r with the charge density
r or general densities r, because we know from the context what is
meant! |
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The specific
resistivity obtained in this way is necessarily identical to what
you would define as specific resistivity by looking at some rectangular body
with cross-sectional area F and length l. |
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The specific
conductivity has the dimension [s] =
W–1cm–1, the
dimension of the specific resistivity is
[r] = Wcm. The
latter is more prominent and you should at least have a feeling for
representative numbers by remembering |
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| r (metal) |
» |
2 µWcm |
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| r (semicoductor) |
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1 Wcm |
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| r (insulator) |
» |
1 GWcm |
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Restricting ourselves to
isotropic and homogenoeus materials, restricts s and r to being
scalars with the same numerical value everywhere, and Ohms law now
can be formulated for any material with weird shapes and being quite
inhomogeneous; we "simply" have |
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Ohms law in this vector
form is now valid at any point
of a body, since we do not have to make assumptions about the shape of the
body. |
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Take an arbitrarily shaped body with current
flowing through it, cut out a little cube (with your "mathematical"
knife) at the coordinates (x,y,z) without changing the flow of
current, and you must find that the local current density and the local field
strength obey the equation given above
locally. |
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Of course, obtaining the external current
I flowing for the external voltage U now needs
summing up the contributions of all the little cubes, i.e. integration over the
whole volume, which may not be an easy thing to do. |
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Still, we have now a much more powerful version of
Ohms law! But we should now harbor a certain suspicion: |
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There is no good reason why
j must always be parallel to E. This means that
for the most general case s is not a scalar
quantity, but a
tensor; s =
sij.
(There is no good way to write tensors in html; we use
the ij index to indicate tensor properties. |
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Ohms law then writes |
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jx = sxx · Ex +
sxy ·
Ey + sxz
· Ez
jy = syx
· Ex + syy · Ey +
syz ·
Ez
jz = szx
· Ex + szy · Ey +
szz ·
Ez |
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For anisotropic inhomogeneous materials you have
to take the tensor, and its components will all depend on the coordinates -
that is the most general version of Ohms law. |
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Note that this is not so general as to be meaningless: We still have
the basic property of Ohms law: The local current density is directly
proportional to the local field strength (and not, for example, to exp–
[const. · E] ). |
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We have a new thing. however:
The current density vector j points no longer in the
direction of the electrical field E. In other words: The vector
response of an anisotropic material to some
disturbance or "driving force" still produces a vector but with a
direction and amplitude that is determined by a tensor that describes the
material properties. While this used to be a somewhat exotic material behavior
for practitioners or engineers in the past, it is quickly becoming mainstream
now., So you might as well acquaint yourself with tensor stuff right now.
This link gives a
first overview. |
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Our goal now is to find a relation that allows to
calculate sij for a given material
(or material composite); i.e. we are looking for |
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sij =
sij(material, temperature, pressure,
defects... ) |
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2. Step: Describe sij in
Terms of the Carrier Properties |
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Electrical current needs mobile charged "things" or carriers that are mobile. Note that we do not automatically assume
that the charged "things" are always electrons. Anything charged and mobile will do. |
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What we want to do now is to express sij in terms of the properties of the carriers present in the material
under investigation. |
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To do this, we will express an electrical current as a
"mechanical" stream or current of (charged) particles, and compare
the result we get with Ohms law. |
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First, lets define an electrical current in a wire
in terms of the carriers flowing through that wire. There are three crucial points to consider |
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1. The external electrical current as
measured in an Ampèremeter is the result of the net current flow through any cross section of an
(uniform) wire. |
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In other words, the measured current is proportional to the
difference of the number of carriers of the
same charge sign moving from the left to
right through a given cross sectional area minus the number of carriers moving from the
right to the left. |
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In short: the net current
is the difference of two partial currents
flowing in opposite directions: |
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Do not take this point as something simple! We
will encounter cases where we have to sum up 8 partial currents to
arrive at the externally flowing current, so keep this in mind! |
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2. In summing up the individual
current contributions, make sure the signs are
correct. The rule is simple: |
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The electrical current is (for historical reasons)
defined as flowing from + to –. For a particle current this means: |
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In words: A technical current I flowing from
+ to – may be obtained by negatively charged carriers flowing in the opposite direction (from – to +),
by positively charged carriers flowing in
the same direction, or from both kinds of
carriers flowing at the same time in the proper directions. |
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The particle currents of differently charged particles then must be added! Conversely, if negatively charged carriers
flow in the same directions as positively charged carriers, the value of the
partial current flowing in the "wrong" direction must be subtracted
to obtain the external current. |
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3. The flow of
particles through a reference surface as symbolized by one of arrows above, say
the arrow in the +x -direction, must be seen as an average
over the x -component of the velocity of the individual particles
in the wire. |
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Instead of one arrow, we
must consider as many arrows as there are particles and take their average. A more detailed picture of a wire at
a given instant thus looks like this |
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An instant later it looks entirely different in detail, but exactly the same on average! |
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If we want to obtain the net flow of particles through the wire (which is obviously
proportional to the net current flow), we
could take the average of the velocity components <v+x>
pointing in the +x direction (to the right) on the left hand
side, and subtract from this the average <v–x>
of the velocity components pointing in the –x direction (to
the left) on the right hand side. |
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We call this difference in
velocities the drift
velocity
vDof the ensemble
of carriers. |
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If there is no driving force, e.g. an electrical field, the
velocity vectors are randomly distributed and <v+x> =
<v–x>; the drift velocity and thus net current is zero
as it should be. |
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Average properties of ensembles
can be a bit tricky. Lets look at some properties by considering the analogy to
a localized swarm of summer flies
"circling" around like crazy, so that the ensemble looks like a small
cloud of smoke. This link
provides for a more detailed treatment about averaging vectors. |
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First we notice that while the individual fly moves around quite fast, its vector velocity vi averaged
over time t, <vi>t, must be zero as long as the swarm as
an ensemble doesn't move. |
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In other words, the flies, on
average, move just as often to the left as to the right, etc. The
net current produced by all flies at any
given instance
or by
one individual fly after sufficient time is
obviously zero for any reference
surface. |
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In real life, however, the fly swarm
"cloud" often moves slowly around
- it has a finite drift velocity which must
be just the difference between the average movement in drift direction minus
the average movement in the opposite direction. |
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The drift velocity thus
can be identified as the proper average that gives the net current through a
reference plane perpendicular to the direction of the drift velocity. |
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This drift velocity is usually much smaller than the average
magnitude of the velocity <v> of the individual flies. Its value is the
difference of two large numbers - the average velocity of the individual flies in the drift direction minus the
average velocity of the individual flies in
the direction opposite to the drift direction. |
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Since we are only interested in the drift velocity
of the ensemble of flies (or in our case, carriers) we may now simplify our
picture as follows: |
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We now equate the current density with the particle flux density by the basic law of current
flow: |
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Current density j = Number
N of particles carrying the charge q flowing
through the cross sectional area F (with the normal vector
f and |f| = 1) during the time
interval t, or |
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In scalar notation, because the direction of the current flow
is clear, we have |
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The problem with this formula is N,
the number of carriers flowing through the
cross section F every second. |
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N is not a basic property of the material; we
certainly would much prefer the carrier density n = N/V of
carriers. The problem now is that we have to chose the volume V =
F · l in such a way that it contains just the right
number N of carriers. |
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Since the cross section F is given, this means
that we have to pick the length l in such a way, that all
carriers contained in that length of material will have moved across the
internal interface after 1 second. |
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This is easy! The trick is to give l just that
particular length that allows every carrier
in the defined portion of the wire to reach the reference plane, i.e. |
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This makes sure that all
carriers contained in this length, will have reached F after the
time t has passed, and thus all carriers contained in the volume V =
F· vD · t will contribute to the current
density. We can now write the current equation as follows: |
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| j |
= |
q · N
F · t |
= |
q · n · V
F · t |
= |
q · n · F · l
F · t |
= |
q · n · F · vD ·
t
F · t |
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This was shown in excessive
detail because now we have the fundamental law of
electrical conductivity (in obvious vector form) |
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This is a very general equation relating a particle current (density) via its drift velocity to an electrical current (density) via the charge
q carried by the particles. |
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Note that it does not matter at all, why an ensemble of charged particles moves on
average. You do not need an electrical field as driving force anymore. If a
concentration gradient induces a particle flow via diffusion, you have an
electrical current too, if the particles are charged. |
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Note also that electrical current flow
without an electrical field as primary
driving force as outlined above is not some
odd special case, but at the root of most electronic devices that are more
sophisticated than a simple resistor. |
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Of course, if you have different particles, with different
density drift velocity and charge, you simply sum up the individual
contributions as pointed out above. |
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All we have to do now is to compare our equation
from above to Ohms law: |
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We then obtain |
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| s |
= |
q · n · vD
E |
:= |
constant |
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If Ohms law holds, s must be a
constant, and this implies by necessity
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And this is a simple, but far reaching equation saying
something about the driving force of electrical currents (= electrical field
strength E) and the drift velocity of the particles in the
material. |
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What this means is that if
vD/E = const. holds for any (reasonable) field E, the material
will show ohmic behavior. We have a first condition
for ohmic behavior expressed in terms of material properties. |
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If, however, vD/E is constant (in
time) for a given field, but with a value
that depends on E, we have s =
s(E); the behavior will not be ohmic! |
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The requirement vD/E =
const. for any electrical field thus
requires a drift velocity in field direction for the particle, which is
directly proportional to E. This leads to a simple
conclusion: |
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This is actually a rather strange result! A charged particle
in an electrical field experiences a constant force, and Newtons first law
tells us that this will induce a constant accelerations, i.e. its velocity
should increase all the time! Its velocity therefore would grow to infinity -
if there wouldn't be some kind of friction. |
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We thus conclude that there must exist some mechanism that acts like a
frictional force on all accelerated particles, and that this frictional force
in the case of ohmic behavior must be in a form where the average drift velocity obtained is proportional to the
driving force. |
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Since vD/E =
constant must obtain for all (ohmic) materials under investigation, we may
give it a name: |
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vD
E |
= |
µ |
= Mobility = |
Material constant |
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The mobility
µ of the carriers
has the unit
[µ] = (m/s)/(V/m) = m2/V · s. |
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The
mobility µ
(Deutsch: Beweglichkeit) then is
a material constant; it is determined by
the "friction", i.e. the processes that determine the average
velocity for carriers in different materials subjected to the same force
q · E. |
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Friction,
as we (should) know, is a rather unspecified term, but always describing energy
transfer from some moving body to the environment. |
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Thinking ahead a little
bit, we might realize that µ is a basic material constant even in the absence of electrical fields. Since it
is tied to the "friction" a moving carrier experiences in its
environment - the material under consideration - it simply expresses how fast
carriers give up surplus energy to the lattice; and it must not matter how they
got the surplus energy. It is therefore no suprise if µ pops up in
all kinds of relations, e.g. in the famous
Einstein -
Smoluchowski equation linking diffusion
coefficients and mobility of
particles. |
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We now can write down the
most general form of Ohms law applying to
all materials meeting the two requirements: n = const. and
µ = const. everywhere. It is expressed completely in particle (=
material) properties. |
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The task is now to calculate n and
µ from first priciples, i.e. from only knowing what atoms we are
dealing with in what kind of structure (e.g. crystal + crystal defects) |
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This is a rather formidable task since s varies over a extremely wide range, cf. a
short
table with some relevant numbers. |
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In order to get acquainted with the new entity
"mobility", we do a little exercise: |
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Since we like to give s as a positive number, we always take only the
magnitude of the charge q carried by a particle. |
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However, if we keep the sign, e.g. write
s = – e · n ·
µe for electrons carrying the charge q = –
e; e = elementary charge, we now have an indication if the particle
current and the electrical current have the same direction (s >
0) or opposite directions s < 0)
as in the case of electrons. |
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But it is entirely a matter of taste if you like to
schlepp along the signs all the
time, or if you like to fill 'em in at the end. |
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Everything more detailed then this is no longer
universal but specific for certain materials. The remaining task is to
calculate n and µ for given materials (or groups of
materials). |
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This is not too difficult for simple
materials like metals, where we know that there is one (or a few) free
electrons per atom in the sample - so we know n to a sufficient
approximation. Only µ needs to be determined. |
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This is fairly easily done with classical physics; the
results, however, are flawed beyond repair: They just do not match the
observations and the unavoidable conclusion is that classical physics must not
be applied when looking at the behavior of electrons in simple metal crystals
or in any other structure - we will show this in the immediately following
subchapter 2.1.3. |
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We obviously need to resort to
quantum theory and solve the Schrödinger
equation for the
problem. |
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This, surprisingly, is also fairly easy in a simple
approximation. The math is not too complicated; the really difficult part is to
figure out what the (mathematical) solutions actually mean. This will occupy us for quite some
time. |
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© H. Föll (Advanced Materials B, part 1 - script)
