Understanding Density of States, Bloch Function, and Fermi Energy

Density of States

Density of states: Each value of K determines an orbital state of e-, double degeneracy due to spin. It is agreed that:

1. The electronic states always include spin degeneracy.
2. When an electron current is assumed, the electron charge of each spin is always included.
3. Electron density N is defined as the total number of electrons per unit volume.

Now the following is discussed:

1. E is a quantized continuous distribution of values, as the quantized E is increased by small integer steps of K, leading to small ΔE of the order of h².
2. Given the continuous distribution of E, the probability of a particular value of E must be zero.
3. We define the density of states n(E), which is the density of energy electrons in the range around energy E per unit energy interval.
4. The density of states of the electrons dE is divided by defining a separate amount dE. [E = N(E) dE].

Bloch Function

Bloch function: The general solution of the equation is: u(x)

Fermi Energy and Fermi Surface

Fermi energy: If we have N electrons in the crystal, we should start filling the states from lower energy levels up until we find the N electrons. The level of the highest occupied energy state is equal to the Fermi energy, E_F.

Fermi surface: The surface in K-space, outside which all the states are empty (isoenergetic surfaces).

Effective Mass

The effective mass: At the bottom of a band, the effective mass is defined by setting an electron-free formula changed to real curvature E(k). Thus: E(k) = V_o + [(ħk)² / 2m], m = ħ² / 2∂²a, this formula, the electron has a small effective mass in a broad band, where [∂] is large, and a large effective mass in narrow bands. The justification for using the effective mass as a kind of "trap-factor" in the WEF is to allow many of the simple predictions of the model to be applied more generally than would otherwise be possible. The effective mass can be defined more generally as d²E / dk² = ħ² / m, m = ħ² / (d²E / dk²). The effective mass for electrons at any point of the band depends on the curvature of E(k).

Electronic Conductivity

Electronic conductivity: By raising the temperature, the number of thermally excited vibrations is increased, so the electrons are scattered efficiently. In a metal, the conductivity decreases with increasing temperature due to the decrease in mobility. Changing to a semiconductor, however, can have a much greater effect on carrier concentration than on mobility, and conductivity usually increases with temperature.