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Essentially, the semiconducting
properties of Silicon stem from the sp3 hybrid bonds formed between
electrically neutral atoms. |
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Two Si atoms donate one electron each to all four
sp3 hybrid bonds, forming the familiar diamond type lattice. |
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Substituting a Si atom by a group III or group V
element produces a mobile hole or electron and an immobile ion in the familiar way. |
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All III-V compounds essentially keep
this structure. They form sp3 hybrid orbitals, but there is now a
big difference to Si (or Ge, or diamond-C): |
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The atoms must become ionized, at least to some
extent. The group V elements N, P, As, or Sb donate an electron to the group
III elements Ga or In. |
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This allows both partners to form the
sp3 hybrid orbitals needed for forming a diamond type lattice. |
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The binding, which was totally
covalent for the elemental semiconductors, now has an ionic component. The
percentage p of the ionic binding energy varies for the various
compounds |
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The percentage p of the ionic binding
energy is closely related to the so-called electronegativity of the elements
and varies for the various compounds. The electronegativity describes the
affinity to electrons of the element; in a binding situation the more
electronegative atoms will more strongly bind the electrons of its partner an
therefore carry a net negative charge. The difference in electronegativity of
the atoms in a compound semiconductor therefore give a first measure for
p. |
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To give some examples: For Si we have p =
0, for GaAs we find p = 0,08. |
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Doping is still achieved by
introducing specific atoms as substitutional impurities. But in contrast to
elemental semiconductors, we now have more possibilities as can be seen by
looking at the relevant part of the periodic table. |
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II |
III |
IV |
V |
VI |
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B |
C (2,5) |
N (3,1) |
O (3,5) |
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Al (1,5) |
Si (1,7) |
P (2,1) |
S (2,4) |
Zn (1,7) |
Ga (1,8) |
Ge (2,0) |
As (2,2) |
Se |
Cd (1,5) |
In (1,5) |
Sn (1,7) |
Sb (1,8) |
Te (2,0) |
Hg |
Tl |
Pb (1,6) |
Bi (1,7) |
Po |
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The numbers in brackets give the
electronegativity, the elements in yellow cells are never used for doping of
compound semiconductors. |
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A more electronegative element
replacing a certain lattice atom will attract the electrons from the partner
more strongly, become more negatively charged, and thus increase the ionic part
of the binding. |
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That has nothing to do with its ability to donate
electrons to the conduction band or to accept electrons from the valence band,
however. If a foreign atom acts as a donor or acceptor depends only on the
energy levels it introduces in the band gap. |
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We now can look a the basic
possibilities open for doping: |
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We can replace the group III elements by group II
elements to produce acceptors and the group V elements by group VI elements to
produce donors - in principle. |
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But we can also replace both the group III and
group V elements by group IV elements - which may generate donors or acceptor
levels in the band gap of some compounds - we have amphoteric doping. |
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We could also replace the atoms of the compound
by an isoelectronic atom - group III
elements by some other group III elements and the same thing for the group V
partner. In Si, this would mean replacing a Si atom by e.g. a Ge or C atom -
which is not very exciting. In compounds, however, doping with isoeletronic
atoms produces differences in the ionic part of the binding and therefore local
potential differences with noteworthy effects as we shall see below. |
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We could even replace an atom of the lattice by a
small molecule - isoelectronic or not - and achieve a doping effect. |
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Well, in Si we could also use
molecules and all group III or group V elements - but in reality was only use
B, As, P, and sometimes Sb. |
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We do not use the group III elements Ga, Al, or
In; neither the group IV elements N and Bi. The reasons are
"technical": Them ay be difficult to incorporate in a crystal, their
solubility may be too small, their diffusivity too high (or too low?), or their
energy levels in the band gap not suitable. |
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The same situation occurs with
compound semiconductors. |
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There are optimum solutions to doping depending
on the type of semiconductor, the technology available or mandated by other
criteria, and so on and so forth. |
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There are therefore no general rules for optimal
doping and here we will only discuss amphoteric doping and isoelectronic doping
in somewhat more detail. |
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A prime case of amphoteric doping is
the incorporation of Si into GaAs. If the Si atoms replace Ga atoms, they act
as donors, and as acceptors if they occupy As lattice sites. How does this
work? |
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A Si atom has four electrons disposable for
binding. If it replaces a Ga atom that had only three atoms, the As partner
does not have to supply an electron any more to make up for its
"deficient" partner Ga, and the surplus electron will only be weakly
bound - it will easily escape into the conduction band |
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Si on a Ga site thus causes the release of an
electron from the As - it acts as donor even so the electron is actually
supplied by the As. |
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Contrariwise, if an As atom is replaced by a Si
atom, the new twosome is now short one electron. It therefore will fill its
hole by an electron from the valence band - Si now acts as an acceptor. |
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While this seems to offer an elegant
way for doping, the tough question now is: How do we control which lattice
sites are occupied by Si. |
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In other words: On what kind of lattice sites
will we find the Si atoms after it was ion- implanted, diffused into the
crystal from the outside world, or incorporated directly during crystal growth
procedures or thin film growth? |
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This question cannot be easily
answered from first principles. Two guidelines are: |
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At low temperatures (T <ca
700 oC), Si will prefer to sit on As sites - it acts as acceptor
with an energy level about 0,35 eV above the valence band edge. |
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At high temperatures (T >ca
900 oC), it tends to sit at Ga sites and acts as donor with an
energy level about 0,006 eV below the conduction band edge. |
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If Si is incorporated into a GaAs melt, large Si
concentrations tend to produce donors, small concentrations acceptors. |
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More about amphoteric doping and its
practical aspects in the various chapters about specific compound
semiconductors. |
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If we introduce isoelectronic
elements in the lattice - e.g. a group V atom like N or B substituting a P
atom, or even a molecule with 5 electrons to share like ZnO substituting a GaP
pair in the GaP lattice - we do not "dope" in the conventional sense
of the word. We rather change the ionic component of the local binding. |
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Since the introduction of isoelectronic elements
is deliberate with a specific purpose in mind, we deal with it under the
general heading "doping", keeping in mind that we do not change the
carrier concentration in this way, but their recombination behavior. |
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Doping with isoelectronic elements may not do
much in most compound semiconductors, but it can have pronounced effects in
others and provide new radiative recombination channels by an interaction of
the isoelectronic dopant and electron-hole pairs called excitons. |
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The paradigmatic semiconductor for isoelectronic
doping is GaP, an indirect semiconductor. It is used as a strong emitter of
green light, however, by doping it with isoelectronic elements and using
the radiative decay of excitons bound to the doping
elements. |
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How does this work? There are several
crucial ingredients, all from rather involved solid state physics: |
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First, we need excitons. |
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Second, we need to have the excitons bound to isoelectronic dopants. |
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Third, we need to have a
radiative recombination of the electron and hole constituting the
exciton - despite the nominal violation of the crystal
momentum. |
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A detailed treatment of these points
(explaining, among other things, why this "works" in GaP, but not in
many other compound semiconductors) is not possible in the context of this
lectures course. We will only superficially look at the basics. |
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What is an exciton? Imagine the generation of an electron - hole
pair, e.g. by irradiating a semiconductor with light. |
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If the photon energy is large enough to lift an
electron all the way from the valence band to the conduction band in a direct
transition, you now have a free electron an
a free hole which move about the crystal in
a random way. There coordinates in real space (r-space) are
arbitrary or undetermined, while their coordinates in k-space are
well-defined, identical and do not change. |
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Now imagine that the hole and the electron stay
so close to each other in real space that
the feel the Coulomb attraction. They are then bound to each other with a
certain binding energy Eex and their coordinates in
r-space are (nearly) identical, but still undetermined |
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We already know a system where one
negative elementary charge is bound to one positive one - it is called
"Hydrogen atom". The only difference between an exciton and an
Hydrogen atom is that the mass of the hole is much smaller that the mass of the
proton (and, of course, that our exciton can only exist in a solid) |
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From a more advanced treatment of the Hydrogen
atom where the electron mass is not neglected with respect to the proton mass,
we immediately can carry over the solution of the relevant Schrödinger
equation to an exciton. Of interest are especially the allowed energy states,
we obtain for Eex: |
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Eex
(K) = Egap -
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mred
q4
8(n e0 er h)2
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+ |
h2 K2
8p2 (me +
mh)
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q = elementary charge, and
n = quantum number = 1,2,3,... |
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The first term simply accounts for the crystal
energy, the second one is straight from the Hydrogen atom, and the third term
is a correction if the two particles are not at the same place in
k-space (it is zero for ke =
-kh or ke = kh = 0
as it will be for most direct semiconductors |
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In total we have a system of energy
levels right below the conduction band with the "deepest" level
defined by an energy difference
which would have a constant value of about 10 meV for all semiconductors -
except that they all have different masses because we have to take the
effective masses, of course, and
different er. |
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The calculated value for GaAs:
DEex(N = 1) = 4,4 meV which
is close to the experimental one. More
values can be found in the advanced module. |
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In k-space for GaP this looks like
this: |
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It simply takes DEex less energy to kick an electron
from the top of the valence band to an exciton state - because you safe the
binding energy. |
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Since DEex for a free exciton - that moves
about the crystal, transporting energy, but not charge - is just a few meV,
comparable to typical donor levels, it will not live very long at room
temperature. The thermal energy then is enough to ionize the exciton, i.e. to
remove the electron (or the hole; your choice; everything is rather
symmetrical), and we are left with a free hole and electron. |
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Even so the electron and the hole are almost at
the same place, recombination is not possible without the help of phonons, so
it is rather unlikely - as stated before.
Excitons in most semiconductors therefore only make their presence known at low
temperatures - and in the absorption of light, because you will already find
some absorption for light with an energy slightly below the band gap
energy! |
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Now imagine a isoelectronic dopant in
GaP, e.g. N instead of P. It distorts the potential for electrons a little bit
and strictly locally; and in GaP this will
lower the energy of the electrons
locally - inside a radius comparable to the lattice
constant. |
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An electron thus may become
bound to the isoelectronic dopant - i.e. it "revolves" around the
isoelectronic atom. It may now attract a hole by Coulomb interaction over
comparatively large distances and thus forms a bound
exciton. This is an easy, albeit oversimplified way to conceive bound
excitons. |
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The net effect of a positive
("binding") interaction of an isoelectronic dopant and an exciton is
twofold: |
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The exciton energy levels move "down"
from the free exciton level by an amount equal to the binding energy; i.e.
DEex increases. In some cases -
naturally for GaP - the binding energy may be in the order of 10 meV, and this
pushes the exciton levels so far below the conduction band that the bound
exciton is now relatively stable at room temperature. |
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The exciton is now localized in space. This
demands that its coordinates in k-space must now be somewhat
undetermined thanks to the uncertainty principle which demands that
Dh/2p
k·Dr < h
with h/2p k =
momentum. Since
Dr is in the order of the length scale
of the attractive potential, i.e. a lattice constant, Dk will be in the order of a/2p, i.e. a Brillouin zone. In other words, the bound
exciton can have any wave vector in the 1st Brillouin zone with a certain, not
too small probability |
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In sum total we conclude:
Recombination for bound excitons is easy! it is still an indirect recombination
that needs a phonon as a third partner. But in contrast to indirect
recombination between free electrons and holes, which needs a photon with a
precisely matched k-vector, any phonon will do in this case
because it always matches one of the k-vectors from the spectrum
accessible to the bound exciton. |
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If bound excitons exist (at room temperature),
their recombination provides a very efficient channel for establishing
equilibrium and thus a possibility to generate light with an energy given by
the |
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This is essentially the mechanism to extract
light out of the indirect semiconductor GaP! |
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You should now have
a lot of questions: |
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Why GaP? How about other III-V compound
semiconductors? |
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How about more exotic semiconductors? The II-VI
system, organic semiconductors? |
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Anything similar for elemental semiconductors?
After all, putting Ge into Si also changes the potential locally. |
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How about other defects, not necessarily
isoelectronic? For example, ionized donors and acceptors also attract and
possibly "bind" free electrons or holes, respectively? |
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Well, this is not an advanced solid
state lecture course. And even there, you may not find all the answers easily.
Some answers I would have to look up, too, some, however, you can work out for
yourself - at least sort of. Otherwise, turn to the advanced module where some
answers will be given (in due time). |