of a large difference in energy gaps in semiconductors, from a very low value for InSb (0.17 eV) to GaAs in which the energy gap changes slightly as the temperature changes: about five-percent for silicon between absolute zero and room temperature."/>
Figure 2.11 There is a large difference in energy gaps in semiconductors, from a very low value for InSb (0.17 eV) to GaAs (1.43 eV). Notice that the energy gap changes slightly as the temperature changes: about 5% for silicon between absolute zero and room temperature.
There is an interesting website with a huge amount of information on the elements: http://periodictable.com/Elements/050/data.html. Clicking any element in the periodic table will tell you its properties. Furthermore, if you click any property, it will tell you the value of that property for all the elements in the periodic table. It is interesting to see, using this website, that the great majority of elements are conductors. According to this site, of all the elements in the periodic table, 76 are conductors, 5 are insulators, and only 3 are semiconductors: Si, Ge, and Te. Te has an energy gap of 0.45 eV. The rest of the elements are either gases or have unknown values.
Appendix 2.2 Number of Electrons and the Fermi Function
To help you better understand how many electrons and holes are free in a material, let me state, without a proof, the Fermi–Dirac function. Enrico Fermi (1901–1954) was an Italian physicist who emigrated to the United States in 1938 to escape the persecution of Jews (his wife was Jewish). Paul Dirac (1902–1984) was a British scientist. These two physicists did a huge amount of work on quantum theory (Figure 2.12).
For our purposes, we will look at the results of their quantum statistics: the Fermi–Dirac function, or F‐D function for short, which both scientists developed independently in 1926.
Figure 2.12 Enrico Fermi (left) and Paul Dirac (right), who developed the statistics for particles that obey the quantum physics theory.
Source: https://en.wikipedia.org/wiki/Enrico_Fermi#/media/File:Enrico_Fermi_1943‐49.jpg (left); https://en.wikipedia.org/wiki/Paul_Dirac#/media/File:Dirac_4.jpg (right).
Ludwig Boltzmann (1844–1906), an Austrian physicist, developed his Boltzmann distribution function, which related the behavior of gases and their pressure to their temperature. This applied only to classical particles. The F‐D equation applies the quantum mechanical theory to particles that behave under the Pauli exclusion principle. By the way, these statistics apply to all particles that obey the Pauli exclusion postulate, not just electrons, and these particles are called fermions.
Anyway, the F‐D formula is
where F(E) is the probability that an energy level E is occupied by an electron, E is the energy of that specific level, Ef is the Fermi level, k is the Boltzmann constant (k = 1.38 × 10−23 m2 kg/s2 T), and T is the temperature in units Kelvin. Note that the only variable for a given energy is the temperature.
At absolute zero, T = 0, the term e(E–Ef)/kT is equal to infinity if E < Ef and zero if E > Ef (remember that e raised to plus infinity is infinity and e raised to minus infinity is zero). Therefore, the F‐D function tells us that at absolute zero all energy levels above Ef are empty, and its probability of occupancy is zero, and all energy levels below Ef are full, with a 100% probability of being occupied, as I said earlier. As the temperature increases to room temperature (300 K), a slight probability exists that there are some electrons in the higher allowed energy levels, i.e. in the conduction band, and a loss of electrons in the allowed energy levels, the valence band. The F‐D function is symmetrical, as it should be. It confirms mathematically what we saw intuitively and graphically in Figure 2.8. You can see right away that the probability of allowed energy bands greater than 0.1 eV above the Fermi level at 300 K is only 2%, and if we go 0.3 eV above the Fermi level, it goes down to about 10−7 or just one electron in 10 million sites (Figure 2.13).
Let me now show you graphically how the F‐D statistics represent mathematically exactly what I have just explained. Figure 2.14 is the same as Figure 2.13, except that I have exchanged the axes and drawn only one curve: the 300 K (room temperature) curve. Figure 2.14 shows that at the Fermi level, the probability of electrons occupying an energy level is 50%, assuming, of course, that there are allowed energy levels in that range. Allowed energy levels higher than the Fermi level have a lower and lower probability of being occupied (Figure 2.15). As the energy increases, the probability of having an electron occupying a valid energy level decreases, quickly going to zero. If the energy decreases, the opposite is true: at some point, 100% of all the allowed energies will be occupied by electrons. This F‐D curve at 300 K never changes. To change the shape, we need to change the temperature, as shown in Figure 2.13. In Figure 2.15, notice that the F‐D functions on the right for each energy band are identical to each other and also identical to the one in Figure 2.14, except that they are smaller to fit the figure. In the case of the insulator (A), the probability that there are any electrons in the conduction band is as close to nothing as it can get, if you consider that the energy gap is 3 eV, the Fermi level is in the middle, and the difference between the lowest level and the Fermi level is 1.5 eV. If I insert this energy difference in Eq. (2.1), I get the probability that the lowest energy in the conduction band is occupied:
(2.2)
Figure 2.13 The probability that electrons are free as a function of the difference between their energy and the Fermi energy. As the temperature increases, the probability that there are free particle also increases.
