Question

For silicon at room temperature verify that N_c=1.8x10¹⁹ cm^-3. Show both the numerical and units compution.

For silicon at room temperature verify that n_i~1x10¹⁰ cm^-3.

Answer 1

The probability that an electronic state with energy E is occupied by an electron is given by the Fermi-Dirac distribution function:
f(E) = E E kT Fe ( )/1 1 −+ (Equation 1.2)
where EF is the Fermi level, the energy at which the probability of occupation by an electron is exactly one-half. At room temperature, the intrinsic Fermi level lies very close to the middle of the bandgap.
The effective density of states in the conduction band NC is equal to 2[2πmnkT/h2]3/2. Similarly, the effective density of states in the valence band NV is 2[2πmpkT/h2]3/2. At room temperature, NC for silicon is 2.8 x 1019 atoms/cm3. For an intrinsic semiconductor, the number of electrons per unit volume in the conduction band is equal to the number of holes per unit volume in the valence band. That is, n = p = ni where ni is the intrinsic carrier density. Since n = NCexp{-(EC-EF)/kT} and p = NVexp{-(EF-EV)/kT}, where n is the electron density and p is the hole density, np = ni2 = NCNVexp{(EV-EC)/kT} = NCNVexp{-Eg/kT}. Therefore,
n N N
E kTi C V g= − ( ) exp{ / 12 2 }
For a doped, or extrinsic, semiconductor, the increase of one type of carriers reduces the number of the other type. Thus, the product of the two types of carriers remains constant at a given temperature. For Si, ni = 1.45 x 1010 cm-3 . .