
In this chapter we deal with basic
semiconductor properties and simple devices like pnjunctions on a
somewhat simplified, but easy to understand base. The chapter serves to give a
good understanding, if not "gut feeling" to what happens in
semiconductors, leaving more involved formal theory for later. 


However: Intrinsic semiconductors are theoretical
concepts, requiring an absolutely perfect infinite crystal. Finite crystals
with some imperfections may have properties that are widely different from
their intrinsic properties. 


As a general rule of thumb: If you cannot come up
with a material that is at least remotely similar to what it should be in its
"intrinsic" state, it is mostly useless because then you cannot
manipulate its properties by doping. 


That is the major reason, why we utilize so few
semiconductors  essentially Si, GaAs, GaP, InP,
GaN, SiC and their relatives  and tend to forget that there is a
large number of "intrinsically" semiconducting materials out there.
For a short
list activate the (German) link. 

Silicon crystals
are pretty good and thus are closest to truly intrinsic properties. But even
with the best Si, we are not really close to intrinsic properties, see
exercise
3.11 for that. Nevertheless: This chapter
always refers to Silicon, if not otherwise stated! 


A few very basic topics about semiconductors,
including some specific expressions and graphical representations will be taken
for granted; in case of doubt refer to the link with an
alphabetical list of essential
topics. 

In this first subchapter we review
the properties of
intrinsic
semiconductors. We make two simplifying assumption at the beginning
(explaining later in more detail what they imply): 


The semiconductors is "perfect",
i.e. it contains no crystal defects whatsoever. 


The effective density of states in the conduction
and valence band, the mass, mobility, lifetime, and so on of electrons and
holes are identical. 

All we need to know for a start then
is the magnitude of the band gap
E_{G}.
The Fermi energy then is
exactly in the middle of the forbidden band, we can deduce that in two
ways: 


Simply by looking at a drawing
schematically showing the concentration of electrons in the valence band. 





For ease of drawing, the Fermi distribution is
shown with straight lines instead of the actual curved shape. 

The concentration of
electrons, n_{e}, in the conduction band is given exactly by 


n^{e} 
= 
E '
ó
õ
E_{c} 
D(E) · f(E,T) · dE 




With E ' = energy of the upper band
edge. 


With the usual approximations:



we obtain 


n^{e} 
= 
N^{e}_{eff } · exp – 
E_{C} – E_{F}
kT_{ } 



N^{e}_{eff} (with the factor
two for spin up/spin down included) can be
estimated from the
free electron gas model in a fair approximation to 


N^{e}_{eff} =
2 
æ
ç
è 
2 pm
kT
h^{2}

ö
÷
ø 
^{3/2} 




How this is done and
how some numbers can be generated from this formula (look at the dimensions
in the formula above and start wondering) can be found in the link. 


The light blue triangle in the
picture symbolizes this concentration! 

All electrons in the conduction band
in thermal equilibrium must come from the valence band. The concentration of
holes in the valence band, n_{h} thus must be exactly
equal to the concentration of electrons in the conduction band, or 


n^{e} 
=^{ } 
n^{h} 
=^{ } 
n^{i} 
=^{ } 
intrinsic
concentration 




The dark blue triangle in the
picture then symbolizes the hole concentration! 


Given the assumptions made above and the symmetry
of the Fermi distribution, the unavoidable conclusion is that the Fermi energy
must be exactly in the middle of the band gap. 

The carrier densities are decisive
for the conductivity (or resistivity) of the material. If you are not familiar
(or forgot) about conductivity, mobility, resistivity, and so on and how they
connect to the average properties of an electron gas in thermal equilibrium, go
through the basic modules 


Ohms Law and Materials Properties 


Ohms Law and Classical Physics 

We thus have the concentration of
mobile carriers in both bands and from that we can calculate the conductivity
s via the standard formula 


s = e ·
(µ^{e} · n^{e} + µ^{h} ·
n^{h}) 




provided we know the
mobilities
µ of the electrons
and holes, µ^{e} and µ^{h},
respectively. 

Again, simplifying
as much as sensibly possible, with µ^{e} = µ^{h} =
µ we obtain 


s = 2eµ ·
N^{e} _{eff} · exp – 
E_{C} –
E_{F}
kT_{ } 
= 
2eµ ·
N^{e}_{eff} · exp – 
E_{g}
2kT_{ } 




because we have E_{C} –
E_{F} = E_{g}/2 for the intrinsic case as
discussed so far. 


This gives us already a good idea about the
comparable magnitudes and especially the temperature dependences of
semiconductors, because the exponential term overrides the preexponential
factor which, moreover, we may expect not to be too different for perfect intrinsic semiconductors of various
kinds. 