Introduction:

Let consider a piece of doped n-type semi-conductor and a piece of intrinsic semi-conductor. Both pieces of semiconductor are electrically neutral that is, negative and positive charges are balanced. The doped, n-type material is, though, much richer in free electron carriers than the intrinsic material in the conduction band, that is, the concentration of electrons in conduction band is higher in n-type material. (The figure is as shown below).

Figure: Joining n-type and Intrinsic Material

When the two pieces of material are joined altogether perfectly and thus the crystalline structures of each meet exactly, the higher concentration of free electrons in n-type causes diffusion across the boundary among the two materials. Electrons migrate from n-type material to the intrinsic material in an effort to balance the electron concentration all through the material. Note, though, that as electrons cross the boundary from n-type to intrinsic material, positively charged, ionized atoms that donated such electrons remain behind in n-type material.

The p-n Junction:

Consider the separate n-type and p-type materials shown in figure below. In isolation, both materials possess charge neutrality and conduction and the valence band energies are at similar level in each type. Note, though, that due to the doping included, the Fermi levels of two materials are different, being near to the donor level in n-type and acceptor level in the p-type.

Whenever the two pieces of material are joined altogether, there is an imbalance of free carrier concentrations in both valence and conduction bands on each side of the junction. This gives mount to carrier concentration gradients across the junction. As a result, electrons diffuse across the junction from n-type to p-type, whereas holes diffuse from p-type to n-type material, setting up related diffusion currents. Though, electrons entering the p-type material readily re-join with the abundant holes present here and similarly, holes entering the n-type material readily re-join with the abundant electrons here. However, as this recombination occurs, the concentration of free carriers in the vicinity of junction drops dramatically. Therefore, the junction region is termed to as depletion area.

Moreover, as the diffusion of carriers and resultant recombination progresses, there is a build-up of ionized dopant atoms in the area of junction, negatively charged acceptor atoms in p-type and positively charged donor atoms in n-type material. This outcome in build-up of electric field acting from n-type to p-type material. This field, due to ionization, tends to resist the diffusion of electrons from n-type to p-type and as well the diffusion of holes from p-type to n-type. However, the electric field sets up drift currents, internally in semiconductor, in the opposite direction to the diffusion currents for both kinds of carrier. This electric field builds up until equilibrium is reached where the drift and diffusion currents are equivalent and opposite in each case and hence there is no longer any total transfer of either kind of carrier from one material to the other. The internal built-up electric field then remains at this level giving mount to a "barrier-potential" across the junction where n-type material is positive with respect to p-type material.

 Figure: Changes in Energy Bands in the Formation of p-n Junction

The built-in field caused by ionization of junction gives mount to an electrostatic potential difference between the two sides of junction. This is the voltage or potential measured with respect to absolute ground or zero potential. This comprises a shift in energy levels of both valence and conduction bands on each side of the junction. Energy levels readjust and hence they are higher in the p-type material relative to n-type material (Note that this is compatible with electric field). However, equilibrium is reached if the Fermi level in the p-type material becomes equivalent to the Fermi level in n-type material. In another words, the Fermi level is constant across the whole p-n semiconductor and has a gradient of zero all through. If this is not the case, then the transfer of electrons on one hand and holes on the other, between the two kinds of material would not be equivalent in both directions. Note that the electrostatic potential is higher or much positive in n-type material than the p-type whereas the valence and conduction band energy levels are lower in n-type material than the p-type.

Barrier Potential:

Beneath equilibrium conditions, the barrier potential exists wholly across the junction as it is due to the ionization of fixed dopant atoms as can be seen in figure below. The carrier concentrations in neutral areas away from the junction can be closely approximated as such in purely n-type and p-type materials, respectively.

Figure: Barrier Potential Developed across Ionized Depletion Region

Beneath equilibrium conditions, there is no total transfer of either kind of carrier from one material to the other that is, the combined diffusion and drift currents are zero for both electrons and holes. Considering electrons then:

J_{n drift} + J_{n diff} = 0

n q μ_n E + q D_n (dn/dx) = 0

μ_n E = - D_n (dn/dx)

- (μ_n/ D_n) = (1/n) (dn/dx)

From Einstein relation we have, (μ_n/ D_n) = (q/kT) and hence after substituting:

- (q/kT) E = (1/n) (dn/dx)

The electric field E exists across the junction and gives mount to the potential barrier, V_O. The field can be considered as the gradient of electrostatic potential and hence E = - dv/dx;  where V can be taken as electrostatic potential as the function of x.

Note that the negative sign is comprised to signify that the electric field is specified in the direction of reducing potential. Then:

(q/kT) [dv(x)/dx] = (1/n) [dn(x)/dx]

Integrating across the junction with respect to x from p-type side to the n-type side and applying the suitable limits gives:

(q/KT) _Vp∫^Vn dV = _ⁿpo∫^n_no (1/n) dn

Hence,

(q/kT) (V_n - V_p) = ln (n_no) - ln (n_po)

The potential difference across the junction with voltages stated from p-type to n-type material is V_p - V_n = V₀, the barrier potential. Additionally, the electron concentrations in the n-type and p-type materials are as follows:

 n-type: n_no ≈ Nd

p-type: n_po ≈ ni²/Na

 This gives:

 - (q/kT) V_o = ln N_d - ln (ni²/Na)

And hence,

V_o = - (kT/q) ln (N_a N_d/ ni²)

 Note that the barrier potential is negative as evaluated in a positive x-direction from p-type to n-type material.

Junction Capacitance:

The depletion area is greatly decreased in its concentration of free charge carriers and is oppositely charged on either side due to the presence of ionized dopant atoms in each kind of doped material. This is analogous to two charged plates of a capacitor and gives mount to the property of junction capacitance. The capacitance is dependent on the width of junction and therefore on the doping concentrations of both n-type and p-type materials. It is as well dependent, though, on any electric field that might be applied externally and is therefore a bias voltage dependent property in the semiconductor devices.

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