Читать книгу Perovskite Materials for Energy and Environmental Applications - Группа авторов - Страница 15

First, we consider the intrinsic condition applied to the semiconductor without any impurities. The current in a semiconductor is contributed by two types of charge carrier, the electron and the hole. To evaluate the carrier concentration the function of the density of the allowed electron energy states and the occupation function of the allowed energy states must be established. The function of energy density states, N(E), defines the number of allowed electronic states per unit volume and energy. The function describing occupation is the well-established distribution function given by Fermi-Dirac, f(E), which defines the ratio of filled states having an electron to maximum states allowed with the given energy E.

The density of states at an energy E in the conduction band, which is given as NC(E) is close to EC, and in the valance band, NV(E) is close to EV and are given by [2]:

(1.1a)

(1.1b)

with me and mh being electron and hole effective mass respectively. EC denotes the conduction band bottom, whereas EV represents the valance band top.

The Fermi-Dirac distribution function is expressed as follows:

(1.2)

with kB representing the Boltzmann constant, the absolute temperature is represented by T and EF the Fermi energy level.

For energy much greater than kBT (i.e. E – EF ≫ kBT), the Fermi function can be approximated by the Boltzmann function:

(1.3)

The density of free electrons (n) (conduction-band levels which have been occupied) is found after multiplying the total number of states NC(E) with occupation function f(E) and then integrating over the conduction band:

(1.4)

By replacing equations 1.1(a) and 1.3 into 1.4 the resultant expression for n is found after resolving the equation:

(1.5)

The parameter NC is termed the conduction band effective density of states which is:

(1.6)

We can derive an expression for the concentration of holes in a similar manner. The probability that a state of energy is captured by a hole is the same as its not being held by an electron, i.e., 1-f(E). Thus,

(1.7)

Applying the Boltzmann approximation, we can get:

(1.8)

Substituting Equations 1.1(b) and 1.8 into Eq. 1.7 yields

(1.9)

(1.10)

NV shows the valance band effective density of states. The difference in the values of NC and NV is only because of the difference in me and mh. The variation in NC and NV from one semiconductor to another is also due to the dissimilarity in the effective masses, also.