For an electron which behaves as a free particle with effective mass, the density of m*, states was derived in section , yielding: (2.6.3) is the bottom of the conduction band below which the density of states is zero. Ec The density of states for holes in the valence band is given by: (2.6.4) 2.6.2. Calculation of the Fermi integral The carrier density in a semiconductor, is obtained by integrating the product of the density of states and the probability density function over all possible states. For electrons in the conduction band the integral is taken from the bottom of the conduction band, labeled, to the top of the conduction band: Ec, (2.6.5) Where is the density of states in the conduction band and is the Fermi gc(E) f(E) function. This general expression is illustrated with Figure for a parabolic density of states function with = 0. The figure shows the density of states function, the Fermi Ec gc(E), function, as well as the product of both, which is the density of electrons per unit f(E), volume and per unit energy, The integral corresponds to the crosshatched area. n(E). Figure The carrier density integral. Shown are the density of states, the gc(E), 2.6.1 : density per unit energy, and the probability of occupancy, The n(E), f(E). carrier density, equals the crosshatched area. no, The actual location of the top of the conduction band does not need to be known as the Fermi function goes to zero at higher energies. The upper limit can therefore be replaced by infinity. We also relabeled the carrier density as to indicate that the no carrier density is the carrier density in thermal equilibrium. (2.6.6) Using equations () and () this integral becomes:
For an electron which behaves as a free particle with effective mass, m*, the density of states was derived in section , yielding: (2.6.3) Ec is the bottom of the conduction band below which the density of states is zero. The density of states for holes in the valence band is given by: (2.6.4) 2.6.2. Calculation of the Fermi integral The carrier density in a semiconductor, is obtained by integrating the product of the density of states and the probability density function over all possible states. For electrons in the conduction band the integral is taken from the bottom of the conduction band, labeled, Ec, to the top of the conduction band: (2.6.5) Where gc(E) is the density of states in the conduction band and f(E) is the Fermi function. This general expression is illustrated with Figure for a parabolic density of states function with Ec = 0. The figure shows the density of states function, gc(E), the Fermi function, f(E), as well as the product of both, which is the density of electrons per unit volume and per unit energy, n(E). The integral corresponds to the crosshatched area. Figure 2.6.1 : The carrier density integral. Shown are the density of states, gc(E), the density per unit energy, n(E), and the probability of occupancy, f(E). The carrier density, no, equals the crosshatched area. The actual location of the top of the conduction band does not need to be known as the Fermi function goes to zero at higher energies. The upper limit can therefore be replaced by infinity. We also relabeled the carrier density as no to indicate that the carrier density is the carrier density in thermal equilibrium. (2.6.6) Using equations () and () this integral becomes:
topic5countingeandh.docx