### 5.1.3 Doping of Compound Semiconductors

Some Basic Considerations

| Essentially, the semiconducting properties of Silicon stem from the sp3 hybrid bonds formed between electrically neutral atoms.
|  | Two Si atoms donate one electron each to all four sp3 hybrid bonds, forming the familiar diamond type lattice.
|  | 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.
| 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):
|  | 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.
|  | This allows both partners to form the sp3 hybrid orbitals needed for forming a diamond type lattice.
| 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
|  | 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.
|  | To give some examples: For Si we have p = 0, for GaAs we find p = 0,08.
| 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.
II | III | IV | V | VI
| B | C (2,5) | N (3,1) | O (3,5)
| 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
|  | The numbers in brackets give the electronegativity, the elements in yellow cells are never used for doping of compound semiconductors.
| 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.
|  | 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.
| We now can look a the basic possibilities open for doping:
|  | 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.
|  | 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.
|  | 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.
|  | We could even replace an atom of the lattice by a small molecule - isoelectronic or not - and achieve a doping effect.
| 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.
|  | 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.
| The same situation occurs with compound semiconductors.
|  | 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.
|  | There are therefore no general rules for optimal doping and here we will only discuss amphoteric doping and isoelectronic doping in somewhat more detail.

Amphoteric Doping

| 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?
|  | 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
|  | 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.
|  | 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.
| 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.
|  | 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?
| This question cannot be easily answered from first principles. Two guidelines are:
|  | 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.
|  | 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.
|  | If Si is incorporated into a GaAs melt, large Si concentrations tend to produce donors, small concentrations acceptors.
| More about amphoteric doping and its practical aspects in the various chapters about specific compound semiconductors.

Isoelectronic "Doping"

| 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.
|  | 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.
|  | 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.
|  | 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.
| How does this work? There are several crucial ingredients, all from rather involved solid state physics:
|  | First, we need excitons.
|  | Second, we need to have the excitons bound to isoelectronic dopants.
|  | Third, we need to have a radiative recombination of the electron and hole constituting the exciton - despite the nominal violation of the crystal momentum.
| 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.

| What is an exciton? Imagine the generation of an electron - hole pair, e.g. by irradiating a semiconductor with light.
|  | 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.
|  | 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
| 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)
|  | 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:
Eex (K) = Egap - | mred q4 8(n e0 er h)2 | + | h2 K2 8p2 (me + mh) |
q = elementary charge, and n = quantum number = 1,2,3,...
|  | 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
| In total we have a system of energy levels right below the conduction band with the "deepest" level defined by an energy difference
D Eex = | mred q4 2e0 er h |
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.
|  | The calculated value for GaAs: D Eex(N = 1) = 4,4 meV which is close to the experimental one. More values can be found in the advanced module.
|  | In k-space for GaP this looks like this:
|  | It simply takes D Eex less energy to kick an electron from the top of the valence band to an exciton state - because you safe the binding energy.
|  | Since D Eex 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.
|  | 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!
| 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.
|  | 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.
| The net effect of a positive ("binding") interaction of an isoelectronic dopant and an exciton is twofold:
|  | The exciton energy levels move "down" from the free exciton level by an amount equal to the binding energy; i.e. D Eex 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.
|  | 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 D h/2p k·D r < h with h/2p k = momentum. Since D r is in the order of the length scale of the attractive potential, i.e. a lattice constant, D k 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
| 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.
|  | 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
|  | This is essentially the mechanism to extract light out of the indirect semiconductor GaP!

| You should now have a lot of questions:
|  | Why GaP? How about other III-V compound semiconductors?
|  | How about more exotic semiconductors? The II-VI system, organic semiconductors?
|  | Anything similar for elemental semiconductors? After all, putting Ge into Si also changes the potential locally.
|  | How about other defects, not necessarily isoelectronic? For example, ionized donors and acceptors also attract and possibly "bind" free electrons or holes, respectively?
| 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).

| Of course, there is another question looming large by now: How is doping it actually done; at least for the more common non-Si semiconductors out there? What are particular problems and limits?
|  | Again, answering these questions in any detail would far exceed what is possible to do during a lectures course.
|  | We will, however, touch upon the subject in various and any places in the course of the following chapters.