Defect electron

As defect electron , electron vacancy , electron hole or hole is the (imaginary) positive movable charge carriers in semiconductors , respectively. It represents the equivalent description of the absence of a (real) valence electron , which serves the simplified mathematical treatment of the processes in the semiconductor. The real charge transport continues to take place through electrons .

Furthermore, the concept of holes is important for understanding the conduction mechanisms in doped semiconductors.

The defect electron is a quasiparticle , its counterpart is the quasiparticle “ crystal electron ”.

The defective electron was discovered by Karl Baedeker (1911), the name comes from Fritz Stöckmann and Heinz Pick .

In the undoped semiconductor

left: pure silicon crystal; Generation of an electron-hole pair as well as movement and recombination of an electron with a hole
on the right: Representation of the processes in the band diagram

Defect electrons are generally created by the excitation of lattice atoms in a crystal . In pure semiconductor single crystals (made of silicon , germanium , gallium arsenide etc. ) all valence electrons are involved in the bonds (at absolute zero ) , i.e. i.e. all valence electrons are in the valence band , the conduction band is unoccupied. Therefore, there are no holes.

Therefore, some lattice atoms have to be excited in order to generate defect electrons. This can happen , for example, at higher temperatures (thermal excitation) or by absorption of a photon (optical excitation). Valence electrons are excited into the conduction band and leave an unoccupied valence electron site (a defect electron) on the associated lattice atom.

If an electrical voltage is applied to the semiconductor , both the free electron in the conduction band and the defect electron in the valence band contribute to the charge transport. One speaks (in the case of pure semiconductors) of self-conduction .

In contrast to the conduction band electron, however, the defect electron cannot move freely. Rather, it moves through a kind of "advancement" of valence electrons. An adjacent valence electron takes up the vacant position (the defect electron) and in turn leaves an unoccupied position at its original location. Viewed from the outside, this process can be interpreted in such a way that a positively charged particle (the defect electron) moves in the opposite direction (comparable to an air bubble in a liquid).

Mathematical description

The physics of the semiconductor (conductivity, optical transitions) takes place

• in a maximum of the valence band (negative curvature = negative effective mass of the electrons) and ${\displaystyle m^{*}}$
• in a minimum of the conduction band (curvature positive = effective mass of the electrons positive).

while other configurations occur in a metal.

In the fully occupied valence band there is an equally large negative for every positive impulse . If an electron with charge and momentum (index each for English missing electron ) changes from the valence band to an acceptor level or into the conduction band (due to thermal or optical excitation), then an unoccupied state remains in the previously neutral valence band with the resulting momentum and the resulting positive charge. ${\displaystyle q_{\mathrm {me} }=-e<0}$ ${\displaystyle \hbar {\vec {k}}_{\mathrm {me} }}$ ${\displaystyle _{\mathrm {me} }}$ ${\displaystyle -\hbar {\vec {k}}_{\mathrm {me} }}$

This can be equivalent to describe (Engl. As a perforated hole ) with:

• positive charge (with the elementary charge ) ${\displaystyle q_{h}=-q_{\mathrm {me} }=+e}$ ${\displaystyle e}$
• positive momentum (with the reduced quantum of action ) ${\displaystyle {\vec {k}}_{\mathrm {h} }=-{\vec {k}}_{\mathrm {me} }=\hbar {\vec {k}}}$ ${\displaystyle \hbar }$
• positive effective mass . ${\displaystyle m_{\mathrm {h} }^{*}=-m_{\mathrm {me} }^{*}>0}$

The electron removed from the valence band (in contrast to electrons in the conduction band or in metals) had exactly the same speed and direction of movement as the hole left after the excitation: with an external electric field it moves to lower electric potentials , i.e. H. to the minus pole : ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {v}}_{\mathrm {me} }=\hbar {\frac {{\vec {k}}_{\mathrm {me} }}{m_{\mathrm {me} }^{*}}}=\hbar {\frac {{\vec {k}}_{h}}{m_{\mathrm {h} }^{*}}}={\vec {v}}_{\mathrm {h} }}$

The acceleration due to an external electric field is just as great for the missing electron, if it were in its original state, as for the hole:

${\displaystyle {\frac {\mathrm {d} {\vec {v}}_{\mathrm {me} }}{\mathrm {d} t}}={\frac {q_{\mathrm {me} }}{m_{\mathrm {me} }^{*}}}{\vec {E}}={\frac {q_{\mathrm {h} }}{m_{\mathrm {h} }^{*}}}{\vec {E}}={\frac {\mathrm {d} {\vec {v}}_{\mathrm {h} }}{\mathrm {d} t}}}$

Other important characteristic quantities of semiconductors are the charge carrier mobility and their effective mass. Both, however, are not automatically the same for electrons and holes and hang example, on the material, doping, mechanical stress condition , temperature, direction of movement, etc.

In the doped semiconductor

Another way of generating defect electrons is to excite foreign atoms in semiconductor crystals. In a semiconductor single crystal, foreign atoms create energy levels within the band gap . Therefore, less energy is required for an excitation than in a pure semiconductor crystal. For this reason, a significant increase in conductivity can be observed even at low temperatures ; in this case one speaks of fault line .

In semiconductor technology, foreign atoms (usually boron or phosphorus for silicon ) are introduced into the semiconductor crystal (doping) in order to specifically change the conductivity of the starting material. Depending on the valence of the foreign atom, various impurities can arise.

In the case of p-doping, an atom with one less valence electron ( acceptor ) replaces a lattice atom, so that the defect acts like a positive charge carrier

The p-doping should be emphasized in particular for the generation of defect electrons. In this case, a semiconductor is doped with a foreign atom of lesser valence , i. H. this foreign atom has one or more valence electrons less than would be necessary to replace the substituted semiconductor atom; in the case of a tetravalent semiconductor such as silicon, for example boron. In terms of energy, these unoccupied sites are only slightly above the valence band, so that an electron from the valence band needs little energy to change to the (stationary) level of impurities. The associated lattice atom is again ionized and a defect electron is generated in the valence band.

literature

• Karl Nitzsche, Hans-Jürgen Ullrich: Functional materials in electrical engineering and electronics . Hüthig, 1986, ISBN 3-7785-1264-1 .
• Konrad Kopitzki, Peter Herzog: Introduction to Solid State Physics. 6th, revised edition. Vieweg + Teubner, 2007 ISBN 978-3-8351-0144-9 .
• Charles Kittel: Introduction to Solid State Physics . Oldenbourg Publishing House.
• Neil W. Ashcroft, David N. Mermin: Solid State Physics . 3rd, improved edition. Oldenbourg, 2007, ISBN 978-3-486-58273-4 .

Individual evidence

1. ↑ Rudolf Müller: Fundamentals of semiconductor electronics. 5th edition. Springer-Verlag, Berlin 1987. ISBN 3-540-18041-9 , pp. 25 and 30.
2. ↑ Horst Hänsel, Werner Neumann: Physics. Volume 4 - Molecules and Solids. Spektrum-Akademischer Verlag, 2000, ISBN 3-8274-1037-1 , p. 381 ff. And p. 377 ff.
3. ^ Biographical notes by Robert Wichard Pohl, pdf, Universität Göttingen 2013 , p. 20. With an interview by Pohl and Pick.