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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, so in full splendor we
must write |
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The field strength is |
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With l = length of the body. If we want
E as a vector, we have, in principle, to solve the
Poisson
equation |
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With r = charge density. For
the homogeneous case E is parallel to f
again which is clear without calculation. |
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We now write down Ohms law with the new quantities
and obtain |
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| j · F = I =
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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 ....) of a given material; it is, of course, the
specific
conductivity s |
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and
r is the specific
resistivity. |
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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 of course 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) |
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2 µWcm |
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| r (semicoductor) |
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1 Wcm |
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| r (insulator) |
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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, however, 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
equation given above locally. |
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This is a much more powerful version of Ohms law!
Especially, because we now harbor a suspicion: There is no good reason why
j must always be parallel to E. This
means that for the most general case 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 basis property of Ohms law: The
local current density is directly proprotional to the local field strength (and
not, for example, to exp– (const. · E). |
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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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Electrical current needs mobile charged carriers. We therefore want 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 look at 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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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 historic 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. 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 thus looks
like this |
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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 <vx>
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
the drift velocityof the ensemble of carriers. If there is no driving force,
e.g. an electrical field, the velocity vectors are randomly distributed and
<vx>
= <v-x>; the 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 of
a localized swarm of summer flies
"circling" around like crazy, so that the ensemble looks like a small
cloud of smoke. A more detailed
treatment can be found in the advanced section. |
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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 theindividual 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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With n = N / v = density of carriers in a
volume V, and V = F · l with l
being the required certain length of the wire needed to obtain V,
we now must consider how many carriers contained in the volume V
will flow through the reference plane F. |
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The trick is to take |
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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 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 fundamtental 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 carries 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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Of course, if you have different particles, with different
density drift velocity and charge, you simple sum up the individual
contributions as pointed out above. |
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All we have to do now is to compare this equation
to Ohms law: |
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We then obtain |
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| s |
= |
q · n · vD
E |
:= |
constant |
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This implies by necessity |
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A simple, but far reaching equation! |
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What im 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
requires a constant average velocity in field direction for the particle which
is directly proportional to E. This leads to a simple
conclusion: |
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Since a constant electrical field exerts a constant force on a
charged particle, its velocity would grow to infinity without some kind of
friction. 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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It is called the
mobility
m of the carriers, with the unit
[m] = (m/s)/(V/m) = m2/V ·
s. |
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The
mobility m (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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Thinking ahead a little
bit, we realize that m is a material constant
even in the absence of electrical fields -
it simple expresses how fast carriers give up surplus energy to the lattice;
and it does not matter how the got the surplus energy. It is therefore no
suprise if m 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
m = const. everywhere. It is expressed
completely in particle (= material) properties. |
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Sinc 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 ·
me 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 tast 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 m 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 m 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 (Semiconductor - Script)
2.2.1 Intrinsic Properties in Equilibrium 
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