**Quantitative Theory of PN Junction Diode:**

Quantitative Theory of PN Junction Diode – Here an expression for the total current will be derived as a function of the applied voltage with the assumption that depletion layer thickness is negligible (i.e., barrier width is zero).

When a P-N diode is forward biased, holes are injected from P-region into the N-region. The concentration p_{n }of the holes in the N-region is increased above its thermal equilibrium value p_{no}. The hole concentration in N-region is given as

where p_{no} is the hole concentration in thermal-equilibrium condition, L_{h} is the diffusion length for holes in N-type material and x is the distance from the junction where concentration is considered.

The injected, or excess, concentration of holes at x = (0), p′_{n}(0) is given as

These several hole concentration components are shown in Fig. 7.13, which shows that the concentration p_{n}(x) falls exponentially with distance x into the N-type material.

The diffusion hole current in the N-region is given as

Taking the derivative of Eq. (7.4) and substituting the value of dp_{n}/dx in above Eq. (7.6), we have

At junction i.e., x = 0

where A is area of material in m^{2}, D_{h} is diffusion constant for holes in m^{2}/s, e is the magnitude of charge on holes, L_{h} is the diffusion length of holes in N-type material in metre, and p′_{n}(0) is the excess of hole concentration at Quantitative Theory of PN Junction Diode.

Using Boltzmann relationship of kinetic theory of gases, we have

where V is the voltage applied across the P-N diode and V_{T} is the volt-equivalent of temperature and it is given as

The Eq. (7.9) is known as the **law of junction**.

**Diffusion Length:**

Diffusion length is defined as the distance travelled by free charge carriers (electrons or holes) before recombination. It may also be defined as the average distance covered by an excess charge carrier during its lifetime τ.

The diffusion length L_{h} and lifetime τ_{h} are related to each other as :

Similarly in case of P type semiconductor

**Forward Currents:**

The total diode current I at x = 0 is given as

where I_{hn} (0) is the current caused by holes entering the N-region and I_{ep} (0) is the current caused by electrons entering the P-region.

From Eqs. (7.8) and (7.9)

Similarly,

From Eqs. (7.12), (7.13) and (7.14) we have total diode current

where

**Reverse Saturation Current:**

In the foregoing discussion a positive value of V indicates a forward bias. Equation (7.15) is equally valid for reverse bias [i.e., for negative values of applied voltage V]. For a reverse bias voltage exceeding V_{T} (i.e., 26 mV) at room temperature (27 °C or 300 K), current I → – I_{0}. Thus I_{0} is called the reverse saturation current.

The concentration of holes in N-type region,

The concentration of electrons in P-type region,

where N_{D} and N_{A} are the concentrations of donor and acceptor atoms respectively.

Substituting p_{no} = n^{2}_{i}/N_{D} and n_{po} = n^{2}_{i}/N_{A } in Eq. (7.16), we have

where V_{GO} is a voltage numerically equal to the forbidden energy gap, E_{GO} in electron volts, and V_{T} is the volt equivalent of temperature and is equal to T/11,600.

For germanium the diffusion constants D_{h} and D_{e} vary approximately inversely proportional to temperature T. Hence I_{0} may be given as

where K_{1} is a constant, independent of temperature T.

In the discussion made above, the effect of carrier generation and recombination in the space-charge region has been ignored. Such an assumption is valid for a germanium diode (not for a silicon diode). For the silicon diode, a diffusion current is negligible in comparison with the transition-layer charge-generation current, which is given approximately by

where η ≈ 2 for small (rated) currents and η ≈ 1 for large currents.

Also I_{0} is found to be proportional to n_{i }instead of n^{2}_{i} so

where K_{2} is a constant, independent of temperature T.