Fundamentals of Semiconductor physics

An intrinsic semiconductor is an undoped semiconductor. This means that holes in the valence band are vacancies created by electrons that have been thermally excited to the conduction band, as opposed to doped semiconductors where holes or electrons are supplied by a “foreign” atom acting as an impurity.

In order to better understand the behavior of semiconductors, we need to investigate more deeply what are the electron and hole densities in the conduction and valence bands, respectively. We first need to introduce the notion of energy density of states N(E). This parameter gives the number of states (per unit volume and per unit energy) between E and E+dE: Nc(E) (respectively, Nv(E)) physically represents the “room” available for electrons (resp. holes) in the conduction band (resp. valence band). For energies that are close to the extrema of these two bands, the density of states has a quadratic dependence with E:

Where is the normalized Planck constant (h=6.626.10-34Js) and mc (resp. mv) is the average effective mass of the conduction band (resp. of the valence band). For a direct gap semiconductor, mc (resp. mv) is the effective mass of an electron me (resp. a hole mh) in the crystal.

The above-mentioned concept of effective mass allows considering electrons (and holes) inside the crystal as almost free particles, like free quasi-particles. The semiconductor then becomes an electron (and hole) gas, but in which electrons and holes have an effective mass which may be very different of the mass of the particle moving in free space. For example, for GaAs mc/m0=0,066 avec m0=0,911.10-30kg is the free electron mass.

In order to know what is the number of electrons and holes present in each band, the density of states is not the only information that we need. We also have to know the probability for an electron to occupy a level with a given energy E. This probability is given by the Fermi-Dirac distribution function :

Where k=1,38.10-23 JK-1 is the Boltzmann constant, T the temperature and EF the Fermi energy, which is the chemical potential for semiconductors.

The probability for a hole to occupy a level of energy E is simply given by 1-f(E) since a hole is by definition the absence of an electron.

The electron density n [cm-3] in the conduction band is obtained by integrating, over the range of energies accessible by electrons in the band, the number of states that may be occupied by electrons of energy E, weighted by the probability to “find” an electron having this given energy :

Similarly, the hole density p [cm-3] in the valence band writes :

For a semiconductor whose Fermi level EF is located more than 3 kT away from the extrema, the Fermi distribution function can be written under the form of a simple exponential, so that the expressions for the charge carriers densities become :

Where Nc et Nv are called the effective densities of states. They represent the density of “useful” states, at a given temperature T, in their respective energy band.

Let's note that the product of the two densities turns out to be independent on the position of the Fermi level. This is still true even for extrinsic semiconductors (see below). It is known as the law of mass action since it resembles the law of mass action encountered in chemistry (for instance the auto-ionization equilibrium of water is governed by [H+][OH-]=Ke).

Where ni is the density of intrinsic carriers (for Si at 300K, ni 1010cm-3).

Figure 3 : Diagram showing the electronic bonds in an intrinsic semiconductor (Si)
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Figure 3 shows that for an intrinsic semiconductor (without impurities), each electron in the conduction band is associated with a hole in the valence band. We conclude that the electron and hole densities are equal :

By substituting the carrier densities by their respective expressions, this relationship allows defining the Fermi level for an intrinsic semiconductor EFi. Since at room temperature kT is significantly lower than the energy gap, this level is located near the middle of the forbidden band :

In figure 4 are graphically summarized the important points raised above for intrinsic semiconductors.

Figure 4 : Intrinsic semiconductor
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