Again, we look at a "perfect" semiconductor,
where doping has been achieved by replacing some lattice atoms
by suitable doping atoms without – in the ideal world – changing anything else. |
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We have now additional allowed energy levels in the band gap, belonging to the doping atoms, i.e. they are localized states. | |||

These levels may or may not be occupied by electrons; the Fermi distribution will give the probability for occupancy as before. In analogy to the intrinsic case, we now have the following highly stylized picture: | |||

| |||

As
in the intrinsic case, in this drawing no distinction was made between carrier
density (in cm^{–3}) and carrier concentration
(in ppm, ppb, or the like). |

**Charge Neutrality Condition **

While the picture may look complicated, it is actually just a means to keep track of the number of charges in the semiconductor. Since there is an over-all charge neutrality, we have | ||||||||||||||

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This equation can be used to calculate the exact position of the Fermi energy (as we will see). Since almost everything else follows from the Fermi energy, let's look at this in some detail. | ||||||||||||||

But wait – why are there positive
charges at all? Aren't we just discussing the occupation of electronic
levels/states? After all, electrons are negatively charged. And by which rule is it that some
states in the above figure count the one way (negative), but some the other way (positive)? |
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This
rule is simple: Everything counts relative to the individual behavior at zero kelvin
(*) where the conduction band is completely empty, the valence band is completely filled,
all donor states are occupied, and all acceptor states are empty. | ||||||||||||||

Of course, if both donor and acceptor states
were present together in a material, then at zero kelvin the electrons would not be distributed
like that. And that's why it is emphasized that one has to look at the individual behavior
of these states at zero kelvin, because charge neutrality is defined relative to the
latter (for which it obviously holds). | ||||||||||||||

(* Yes: The units of measurement are spelled with small letters – because it's only the natural persons' names that are spelled with capital letters.) | ||||||||||||||

First, let's count the negative
charges which, following the above rule, means to look for those places where at zero
kelvin no electrons are to be found. | ||||||||||||||

There
are, .
Their density, as spelled out before, was first, the electrons in the conduction band |
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More generally and more correctly, however, we have | ||||||||||||||

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The Fermi energy is now included as a E_{F}
variable in the Fermi function, because the density of electrons depends on its
precise value which we do not yet know. In this formulation the electron density comes out
always correct, no matter where the Fermi energy is positioned
in the band gap. | ||||||||||||||

Then there are,
. Their density second, the negatively charged acceptor atoms is given by the density of
acceptor states (which is just the N_{A}^{–} doping density ) N_{
A}times the probability that the states are
occupied, and that is given by the value of the Fermi distribution at the energy of the acceptor
state, f(. We have accordingly
E_{A}, E_{F},T) | ||||||||||||||

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Note: The
Fermi distribution in this case should be slightly modified to be totally correct. The difference
to the straight-forward formulation chosen here is slight, however, so we will keep it in
this simple way. More about thisin an
advanced module. | ||||||||||||||

Now
let's count the positive charges which, following the
above rule, means to look for those places that at zero kelvin are completely
filled with electrons. | ||||||||||||||

, we have the
First holes in the valence band. Their number is given by the
number of electrons that do not occupy states in the
valence band; in other words we have to multiply the effective density of states with the
probability that the state is not occupied. | ||||||||||||||

The probability that a state
is not occupied is just 1
minus the probability that it is occupied, or simply 1 – f(. This gives the density of holes to E,
E_{F}, T) | ||||||||||||||

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, we have the Second
positively charged donors, i.e.
the donor atoms that lost an electron. Their density
is equal to the density of states at the donor level (which is again identical to the density
of the donors themselves) times the probability that the level is N_{D}^{+}not
occupied; so, we have | ||||||||||||||

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Charge neutrality thus demands | ||||||||||||||

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If we insert the expression for the Fermi distribution | ||||||||||||||

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where
stands for E_{n}, we have, for a
given material with a given doping, E_{C,V,D,A} one equation for the
one unknown quantity !E_{F} |
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Solving this equation for any given semiconductor
and any density of (ideal) donors and acceptors will not only give us the exact value of the
Fermi energy for any temperature E_{F}, i.e. T, but E_{F}
(T)all the carrier densities as specified
above. | ||||||||||||||

Unfortunately, this is a messy transcendental equation – it has no direct solution that we can write down. | ||||||||||||||

Before the advent of cheap computers, this was a problem – you had to do case studies and use approximations: High or low temperatures, only donors, only acceptors and so on. This is still very useful, because it helps to understand the essentials. | ||||||||||||||

However, here we could use a program (written by J. Carstensen) that solves the transcendental equation and
provides all functions and numbers required. | ||||||||||||||

Unfortunately, the relevant JAVA applet doesn't work anymore, so the link to this illustration module is deactivated. | ||||||||||||||

Therefore, here we show only a screen shot
with the result for the typical case of Si with | ||||||||||||||

1. An acceptor density
of 10 (red line).^{15} cm^{–3} |
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2.
A donor density of 10
(blue line).^{17} cm^{–3} |

| |||

One of many important points to note about carrier densities is the simple, but technologically supremely important fact that the majority carrier density for many semiconductors in a technically useful temperature interval is practically identical to the dopant density. | |||

This is then the temperature
regime were we can hope to have only a weak temperature dependence of electronic properties
and thus of devices made from Si. |

The equations used for charge neutrality also make possible
to deduce an extremely important relation, the mass action law (for electrons and holes), as follows: | |||||||||||||||||||||||

First we consider the product | |||||||||||||||||||||||

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Then we insert the formula for the Fermi distribution and note that | |||||||||||||||||||||||

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Using the Boltzmann approximation for the Fermi distribution, we get the famous and very important mass action law (try it yourself!): | |||||||||||||||||||||||

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We are now in a position to calculate the density of majority and minority carriers with very good precision if we use the complete formula, and with a sufficient precision for the appropriate temperature range where we can use the following very simple relations: | |||||||||||||||||||||||

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We know that the conductivity s
of the semiconductor is given by | |||||||||||||||||||||||

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With µ = mobility of the carriers. | |||||||||||||||||||||||

We now have the densities; obviously we now
have to consider the mobility of the carriers. | |||||||||||||||||||||||

Mobility |
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Finding simple relations for the mobility of the
carriers is just not possible. Calculating mobilities
from basic material properties is a far-fetched task, much more complicated and involved than
the carrier density business. | ||||||

However, the carrier densities (and their redistribution in contacts and electrical fields) is far more important for a basic understanding of semiconductors and devices than the carrier mobility. | ||||||

At this point we will therefore only give a cursory view of the essentials relating to the mobility of carriers. | ||||||

The mobility µ
of a carrier in an operational sense is defined as the proportionality constant between the
average drift velocity v of an ensemble
of carriers in the presence of an electrical field _{D}: E | ||||||

| ||||||

The average (absolute) velocity of a carrier
and its drift velocity v must not be confused; for a detailed discussion
consult the links to two basic modules: v_{D} | ||||||

The simple linear relationship between the drift velocity and
the electrical field as a driving force is pretty
universal – it is the requirement for ohmic behavior (look it up in the link) –
but not always obeyed. In particular, the drift velocity may saturate
at high field strengths, i.e. increasing the field strength does not increase anymore. We come back to that later.v_{D}
| ||||||

Here we only want to get a feeling for the order of magnitudes of the mobilities and the major factors determining these numbers. | ||||||

As we (should) know, the prime factor influencing
mobility is the average time between scattering processes. In fact, the mobility µ
may be written as | ||||||

| ||||||

With
tmean scattering time. We thus
have to look at the major scattering processes in semiconductors._{s} = | ||||||

There are three important mechanisms: | ||||||

The first
(and least important one) is scattering at crystal defects
like dislocations or (unwanted) impurity atoms. | ||||||

Since we consider only "perfect" semiconductors at this point, and since most economically important semiconductors are pretty perfect in this respect, we do not have to look into this mechanism here. | ||||||

However, we have to keep an open mind because semiconductors
with a high density of lattice defects are coming into their own (e.g. GaN or CuInSe) and we should be aware that the mobilities in these semiconductors might be impaired
by these defects._{2}
| ||||||

Second, we have the scattering
at wanted impurity atoms, in other word at the (charged) dopant
atoms. | ||||||

This is a major scattering process which leads to decreasing mobilities with increasing doping
density. The relation, however, is non-linear and the influence is most pronounced for larger
doping densities, say beyond 10 for ^{17} cm^{–3}Si. |
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Examples for the relation between doping and mobilities can be found in the illustration. | ||||||

As a rule of thumb for Si, increasing the doping level
by 3 orders of magnitude starting at about about 10
will decrease the mobility by one order of magnitude, so the change in conductivity will be
about only two orders of magnitude instead of three if only the carrier density would change.^{15} cm^{–3} |
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The
scattering at charged dopant atoms decreases with increasing
temperature. (This results mainly from the scattering being Rutherford scattering via Coulomb
interaction, for which the scattering cross section reduces with increasing temperature.)
| ||||||

Third, we have
scattering at phonons – the other important process.
Phonons are the (quantum-mechanical) "particles" corresponding
to the thermally stimulated lattice vibrations and thus strongly depend on temperature. |
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This
part scales with the density of phonons, i.e. it increases
with increasing temperature (with about ). It is thus not surprising
that it dominates at high temperatures (while scattering at dopant atoms may dominate at low
temperatures).T ^{3/2} | ||||||

Scattering at phonons and dopant atoms together essentially dominate the mobilities. | ||||||

The different and opposing temperature dependencies almost cancel each other to a certain extent for medium to high doping levels (see the illustration), again a very beneficial feature for technical applications where one doesn't want strongly temperature dependent device properties. | ||||||

Conductivity | ||

With a relatively simple law for the carrier densities and no direct equations for the mobility, but relatively simple behavior of the mobility, conductivity data can now be compiled. | ||

Some data giving the relationship between doping and resisitivity
(=1/conductivity) or the temperature dependence of the conductivity are shown in the illustration. | ||

The illustrations show different behavior for n-
and p-type Si, which could not be predicted in our present simplified treatment;
it is due to different mobilities of the different carriers. | ||

Our present assumptions of equal parameters (mass, mobility,
life time, and so on) for electrons and holes, while justifiable in a 1st degree approximation,
are too simple. Specific differences between holes and electrons exist and will be dealt with
in subsequent chapters. They are responsible for the differences in mobility, conductivity,
and so on between holes and electrons. |

© H. Föll (Semiconductors - Script)