Bucknell Physics 212 Supplement

Subsection 7.5.1 Temperature Dependence of Conductivity for a Semiconductor

If the temperature of the material is not zero, then there is thermal energy that can cause electrons in the valence band to become excited into the conduction band. The larger the temperature, the larger the interatomic collision energies that can excite electrons. The temperature dependence is captured well by the Fermi-Dirac probability distribution function:

This relation is plotted in Figure 7.5a for three different temperatures. For a material at absolute zero (\(T = 0\)), all of the energy levels for \(E \lt E_F\) are filled (with probability \(p(E) = 1\)), whereas all energy levels for \(E > E_F\) are empty (\(p(E) = 0\)). If the temperature is not zero, though, thermal excitations result in a decrease in \(p\) for \(E \lt E_F\) and an increase for \(E > E_F\text{.}\)

If the temperature is large enough, then the probability \(p(E)\) for electrons to be found for energies in the conduction band may become non-negligible, and there may be sufficient electrons excited into the conduction band for the material to become a good conductor. An important question: how large is “large enough?” The answer to this question can be seen by noticing the factor of \(k_BT\) in (7.15). This factor should look familiar to you from our thermodynamics unit from PHYS 211 — this is the well-known Boltzmann factor that gives a characteristic energy for thermal collisions. If the temperatue in a semiconductor is such that \(k_BT\) is comparable to or greater than the band gap energy \(E_g\text{,}\) then there will be sufficient thermal energy for the semiconductor to be a good conductor. This is shown conceptually in Figure 7.5b: for a sufficiently large temperature, some of the higher energy levels in the valence band become empty and some of the lower energy bands in the conduction band contain electrons. This enables conduction not only for electrons in the conduction band but also for electrons in the valence band who now have open energy levels if an electric field is applied.

In the previous example, it is tempting to say that if \(p(E) = 0.0047\) for an electron to be found in the lowest conduction-band energy state, then the material will be an insulator. But remember how many electrons are in a typical solid (a lot!), so even if the probabilities are low, there will still be some electrons that can conduct electricity. But with higher and higher temperatures there would be more and more conduction band electrons.

So, semiconductors conduct electricity better at higher temperatures. That can be both a good thing and a bad thing. On the good side, this principle can be used to make very sensitive and precise temperature measurement devices called thermistors. On the bad side, the electronic devices that we'll discuss in the next section can fail if the semiconductor gets too hot. This is why it is so critical for electronic devices to be cooled.

Subsection 7.5.2 Doping and n- and p-Type Semiconductors

There is another way that you can make a semiconductor conduct: doping, i.e., add impurities to a semiconductor with a different number of valence electrons than the primary semiconductor material. For example, silicon is a common “Group IV” semiconductor element with 4 valence electrons; i.e., 4 electrons in the highest energy (unfilled) shell. If a silicon semiconductor is doped with trace amounts of an impurity “Group V” element such as arsenic with 5 valence electrons, 4 of the valence electrons in the impurity will fill the valence band along with the silicon atoms, but there will be one additional outer-shell electron for each impurity atom. Arsenic is considered to be a “donor impurity” for silicon semiconductors since it donates an additional electron. The additional donated electrons are mobile charge carriers in the conduction band, as shown in Figure 7.7a.¹ Semiconductors doped with donor atoms are called “n-type” semiconductors, because they have mobile charge carriers that have negative charge (electrons).

Doping a semiconductor with “acceptor impurities” results in “p-type” semiconductors, as shown in Figure 7.7b. An example of a p-type semiconductor would be silicon (Group IV) doped with impurities from Group III (with 3 valence electrons), such as boron. Since the impurity atoms have only 3 valence electrons (unlike the 4 for the bulk silicon atoms), each impurity atom would fill only 3 out of the 4 available energy levels, leaving an unfilled “hole” in the valence band. This hole provides other electrons in the valence band with an opportunity to change their energies in response to an applied electric field. Consequently, the presence of holes in the valence band allows for conductivity. An applied electric field results in electrons in the valence band moving opposite the field into available holes; the result is that the holes move in the direction of the electric field. Conceptually, the holes themselves act like positive, mobile charge carriers; hence the term p-type to describe these semiconductors.

Things become very interesting if a p-type semiconductor is joined with an n-type semiconductor. Figure 7.8a shows an n-type semiconductor (on the left) with mobile electrons (dark circles) connected to a p-type semiconductor (on the right) with mobile holes (white circles). But in the absence of an applied electric field, the mobile electrons and holes diffuse around the material, moving randomly in the sample, including moving across the p-n junction. Invariably, mobile electrons and holes near the junction come into contact with each other and annihilate, with the mobile electron dropping into the open energy level left by the hole. This annihilation process removes both the electron and the hole from the conduction process. The result is that in the vicinity of the junction between the n- and p-type materials, there is a depletion zone (DZ) with no mobile charge carriers, as shown in Figure 7.8b. The annihilation process at the junction produces a bias electric field that eventually stops the diffusion of the electrons and holes across the junction, stabilizing the width of the DZ.