Scientific Notes on Power Electronics: doping of semiconductors

In a previous tutorial, we considered the ideal case of a monatomic semiconductor without foreign atoms in the crystalline matrix. Physically, the presence of impurities is inevitable. Technologically, an appropriate modulation of impurities favors greater conductivity.

N-type semiconductors

The following considerations refer to a monatomic semiconductor (for example made of silicon or germanium) but are easily generalized. Let us consider silicon in particular, imagining replacing any atom with an atom of phosphorus (P). Since it is a pentavalent element, the first four valence electrons bond to the first four neighboring silicon atoms (exactly as happens for an intrinsic semiconductor, i.e., free of impurities). The fifth electron appears to be in excess (Figure 1) and is subject to the attractive (Coulomb) force field of its P⁺ ion, but also to the force field exerted by the ions of the crystal which as established in the previous number, it has a periodic potential energy V (x, y, z) with a period equal to the lattice pitch.

As we know, the effects of V(x,y,z) translate mathematically into the substitution m_e → m^∗ (effective mass of the electron) which allows us to treat the electron as a free particle; electrostatically, the crystal behaves as a homogeneous medium with dielectric constant ∈_r (for silicon ∈_r= 11.9). In turn, the phosphorus ion is a hydrogenoid system, i.e., an atom which, despite having more electrons, behaves like a hydrogen atom, since the potential energy of the Coulomb attraction in which the valence electron moves is −Ze²/r where: Z is the atomic number; e the absolute value of the electron charge; r is the radial coordinate. The energy levels are:

In the above expression, applied to our case we must perform the following substitutions:

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Whereby the effective energy levels, i.e., modulated by the grating, are given by:

Now, we recall that the Bohr radius can be obtained as follows:

Which gives us an idea of the “size” of the hydrogen atom in the ground state. We can define the Bohr radius of the hydrogenoid atom we are considering, performing the substitutions (2), obtaining a value significantly higher than the Bohr radius of hydrogen. Physically, it means that the electron explores a non-negligible number of cells of the lattice, for which the introduction of ∈_r is justified. Let’s see how the ionization energy changes; in the case of the hydrogen atom, we have:

In our case, we obtain an ionization energy of approximately 0.045 eV ≪ ε_g = 1.12 eV (silicon bandgap). We denote the ionization energy by ε_d where the subscript d refers to the fact that the phosphorus atom behaves as a donor atom since it provides an electron. Other pentavalent atoms that behave similarly with semiconductor lattice structures are arsenic (As) and antimony (Sb).  The ionization energy ε_d is therefore much lower than the energy of covalent bonds, so even at temperatures lower than 300 K K, the phosphorus atom ionizes and the electron energetically transitions to the conduction band.

However, to achieve an experimentally detectable effect we must remove not a single silicon atom, but a macroscopic number of atoms, replacing them with an equal number of pentavalent impurities. The final result is the generation of an extremely dense and −ε_d centered level spectrum, as illustrated in Figure 2.

The extreme proximity of −ε_d to the levels of the conduction band favors the transition of the corresponding electrons to the conduction band. We have thus created a semiconductor where the dominant contribution to conductivity comes from electrons. Hence the name n-type semiconductors.

P-type semiconductors

A dual scenario to the previous one consists of replacing a silicon atom with an atom of a tetravalent element such as boron (B). Missing an electron in the formation of the corresponding covalent bond, a hole is automatically generated and if the thermal energy is sufficient, one of the electrons of a neighboring atom can fill the hole but generate another one (Figure 3).

With the new electronic configuration, the boron atom becomes a negative ion (B⁻), so this atom acts as an acceptor. Since the “added” electron comes from a covalent bond, its initial energy belongs to the valence band and the final energy will be slightly higher. The replacement of a macroscopic number of silicon atoms with trivalent atoms generates a very dense spectrum of levels centered in a given value slightly higher than the energy of the top of the valence band, as illustrated in Figure 4. These levels are populated by a macroscopic number of electrons which corresponds to an equal number of holes in the valence band. We have therefore created an extrinsic p-type semiconductor since the dominant contribution to conductivity comes from holes.

Conclusion

We have presented a heuristic model to explain the doping of monatomic semiconductors. Technologically, doping was the first step for the creation of pn junctions. Phenomenologically, it would have been more interesting to study the behavior not of disjoined and metallurgically interfaced p-type and n-type semiconductors, but a single semiconductor with both dopings.

References

¹ Millman J., Grabel A., Microelectronics.
² Shokley W., Electrons and Holes in Semiconductors. Van Nostrand, Princeton. 1950.
³ Kittel C., Introduction to Solid State Physics.
⁴ Scientific notes on power electronics.

Graduated in Physics, since 2008 he has been involved in online teaching of Physics and Mathematics at university level through the extrabyte.info platform. In recent years he has published various articles on mathematical physics in the journal scientiajournal.org, then making “more technological” contributions published in “Elettronica Open Source” and “Electronic Design Master”. An avid reader of visionary/cyberpunk fiction, he has tried to make a transition to the status of “cyber writer”, publishing various anthologies of short stories.