For this basic module we simply take the
suitable module of the
Hyperscript "Introduction to Materials Science II" |
|||

This module is the newest and updated version. The module as it existed in Sept. 01 is reprocuced below. |

In this subchapter we will look at the classical treatment of the movement of electrons in an electrical field. It is a direct continuation of subchapter 1.1.3 in backbone II and again more closely matched to the actual lecture. | |||

In this preceding subchapter we obtained the most basic
formulation of Ohms law in material terms.s = q·n·m |
|||

For a homogeneous and isotropic material (e.g. polycrystalline
metals or single crystal of cubic semiconductors), the concentration of
carriers n and their mobility m do not depend
on the coordinates - they have the same value everywhere in the material and
the specific conductivity s is a scalar. |
|||

In general terms, we may have more than one kind of carriers
(this is the common situation in semiconductors) and n and m could still be more or less complicated functions of
the temperature , the local field strength
T resulting from an applied external voltage, the
detailed structure of the material (e.g. the defects in the lattice), and so
on.E_{loc} |
|||

We will see that these complications are the essence of
advanced electronic materials, especially the semiconductors, but in order to
make life easy, we now will restrict ourselves to the special class of ohmic
materials. We have seen
before that this requires and nm to be independent of the local field strength. We
still may have a temperature dependence of s;
even commercial ohmic resistors, after all, do show a more or less pronounced
temperature dependence which increases roughly linearly with T. |
|||

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. |

**Basic Equations and the Nature of
the "Frictional Force"**

We consider the electrons in the metal to be "free", i.e. they can move unhindered in any direction. | |||

The electrical field then exerts a force
E 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 +).F = -e·E |
|||

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. | |||

Basic mechanics yields for a single particle with momentum
p with
F = dp/dt = m·dv/dt = momentum of the electron.pNote that does not have to be zero when the field is switched
on.p |
|||

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 |
|||

In classical mechanics this is done by introducing a
frictional force
with F_{fr} = k_{fr}·v being some friction constant. But this, while
mathematically sufficient, is devoid of any physical meaning with regard to the
moving electrons. So we have to look for another approach.k_{fr} |
|||

The best way to thing about it, is to assume that the
electron, flying along with increasing velocity, will hit something else in its
way every now and then, which will change its momentum (and thus the magnitude
and the direction of v) as well as its kinetic energy
1/2·m·v.^{2} |
|||

In other words, we consider collisions with other particles where the total energy and momentum of the particles is preserved, but the individual particles loses its "memory" of its velocity before the collision and starts with a new momentum after every collision. | |||

What are the "partners" for collisions
of an electron, or put in standard language, what are the
scattering mechanisms? There are
several possibilities: |
|||

Other electrons. While
this happens, it is not the important process in most cases. |
|||

Defects, e.g. foreign
atoms, point defects or dislocations. This is a more important scattering
mechanism and moreover a mechanism where the electron can transfer its surplus
energy (obtained through acceleration in the electrical field) to the lattice
which means that the material heats up |
|||

, i.e. "localized"
lattice vibrations traveling through the crystal. In a quantum mechanical
treatment of lattice vibrations it can be shown that these vibrations, which
contain the thermal energy of the crystal, show typical properties of (quantum)
particles: They have a momentum and an energy given byPhonons h·n (h = Plancks constant, n = frequency of the vibration), and treating the
interaction of an electron with a lattice vibration as a collision with a
phonon gives correct results. This is the most
important scattering mechanism. |
|||

It would be far from the truth to assume that only accelerated electrons scatter; scattering happens all the time. If electrons are accelerated in an electrical field and thus gain energy, scattering is the way to transfer this surplus energy to the lattice which then will heat up. Generally, scattering is the mechanism to achieve thermal equilibrium and equidistribution of the energy of the crystal. |

Lets look at some figures illustrating the scattering processes. | |||||

Shown here is the magnitude of the velocity of an electron in
and x direction without an external field. The
electron moves with constant velocity until it is scattered, then it continues
with a new velocity. -x |
|||||

The scattering processes, though unpredictable as single
events, must lead to the average <v> characteristic for the
material and its conditions. |
|||||

Whereas <, v> = 0<v>
has a finite value and <. This is bit tricky
since the way we are writing formulas here does not allow easily to distinguish
betwen vectors and scalars: Here, to emphasize the point, v_{x}> = -
<v_{-x}> is a
v, and v is a scalar (its magnitude).vector |
|||||

From
classical thermodynamics
we know that the electron gas in thermal equilibrium with the environment
possesses the energy per particle and degree of freedom with
E =
(1/2)k_{kin}Tk = Boltzmanns constant and
= absolute temperature. T(We write energies in magenta to
avoid confuison with electrical fields E). E |
|||||

(We write energies to avoid confuison
with electrical fields E in magenta). E |
|||||

The three degrees of freedom are the velocities in
-, x- and y-direction, so we must
havez orE_{kin,x} =
1/2m<v_{x}>^{2} = 1/2 kT<v._{x}> = (kT/m)^{1/2} |
|||||

Similarly, for the total energy ,E_{kin} =
1/2m<v_{x}>^{2} + 1/2m<v_{y}>^{2} +
1/2m<v_{z}>^{2} = 1/2m<v>^{2} =
1/2mv_{0}^{2}we have E_{kin} =
1/2m<v>^{2} = 1/2mv_{0}^{2} = 3/2 kTwith v._{0} = <v> |
|||||

Now lets turm on an electrical field
. It will accelerate the electrons between collisions; their
velocity increases linearly.E |
|||||

In our diagram from above this looks like this: | |||||

Here we have an electrical field in -direction.
Between collisions, the electron gains velocity in x-direction
at a constant rate. +x |
|||||

The average velocity in directions,
+x<v, is now larger than in _{+x}>
direction, -x<v. For real electrons, however, the
difference is very small; the drawing is very exaggerated._{-x}> |
|||||

The drift velocity is contained in the difference
<v; it is completely
described by the velocity gain between collisions. We may thus symbolically
neglect the velocity right after a collision because it averages to zero
anyway, and just plot the _{+x}> - <v_{-x}>velocity gain in
a simplified picture; always starting from zero after a collision. |
|||||

The picture now looks quite simple; but remember that it
contains some not so
simple averaging. |
|||||

At this point it is worthwhile to point out that we can define
a new average: The mean time between
collisions, or more conventional, the mean time t for reaching the drift velocity v
in the simplified diagram. _{D} |
|||||

This is most easily seen 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. |
|||||

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 |
|||||

While this diagram is not wrong, it is a highly abstract rendering of the underlying processes 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! |

With the scattering concept, we now can introduce two new (related) material parameters: | |||

The mean (scattering)
time t between two collisions as
defined before, and a directly related quantity: |
|||

The mean free
path between collisions; i.e. the distance travelled by
an electron (on average) before it collides with something else and changes its
momentum. We haveL. L = 2t·(v_{0} +
v_{D})Note that v enters the equation!_{0} |
|||

Using t as a new
parameter, we can rewrite the mechanics equations: |
|||

dv/d can be written as tDv/Dt =
v_{D}/tbecause the velocity change during the time t
is just v. From this we obtain_{D} |
|||

v, or_{D}/t
= -E·e/mv ._{D} = -(E·e·t)/m |
|||

Inserting this equation for v
in the old definition of
the current density, _{D}
yieldsj = -n·e·v_{D} |
|||

,j =
-E·(n·e^{2}t)/m :=
s·Eand thus |
|||

s =
(n·e^{2}t)/m |
|||

This is the classical formula for the conductivity
of a classical "electron gas"
material; i.e. metals. We do not yet know t, but we may turm the equation around
and use it to calculate the order of magnitude of _{class}t, since we_{class}
know the order of magnitude for the conductivity of metals. The
result is: |
|||

t_{class}
=_{ca.} (10^{-13} .... 10^{-15}) sec |
|||

"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? Well, . So let's
look at the mean free path you can't instead.L |
|||

We have t = or L/(2(v_{0} +
v_{D})) and
L = 2t(v_{0} + v_{D})v_{0}^{2} = (3kT/m)^{1/2}. |
|||

This gives us a value v at room temperature! Now we need _{0} =_{ca.}
10^{5} m/sv,
and this we can estimate from the equation _{D}v given above:_{D}/t = -E·e/mv _{D} = -E·t·e/m
=_{ca.} 1 mm/secif we use the value for t dictated by the
measured conductivities. It is much smaller than v!_{0} |
|||

The mean free path between collisions thus
isL = 2t(v_{0} + v_{D})
=_{ca.} 2tv_{0} =_{ca.}
(10^{1}..10^{-1})nmand this is certainly too small!. |
|||

Why is a mean free path in the order of the size
of an atom too small? Well, think about the scattering mechanisms. The distance
between defects is certainly much larger, and a phonon itself is
"larger", too. Moreover, consider what happens at temperatures below
room temperatures: would become even smaller - somehow this
makes no sense.L |
|||

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 - maybe reluctant but unavoidable - must be: | |||

There is no way to describe conductivity (in metals) with classical physics. |

2.2.1 Intrinsic Properties in Equilibrium

2.2.2 Doping and Carrier Concentration

2.2.3 Life Time and Diffusion Length

5.3.2 Isotype Junctions, Modulation Doping, and Quantum Effects

8.1.2 Basic Time Consuming Processes

© H. Föll (Semiconductor - Script)