# Junctions

The contact potential is given as

where,

V_{0} = contact potential

k = Boltzman constant

T = Temperature

N_{a} = Concentration of acceptors

N_{d }= Concentration of donors

n_{i} = Intrinsic carrier.

# Space Charge at Junction

= penetration of space charge region in p- material.

= penetration of space charge region in n- material.

The maximum value of electric field is at *x* = 0 (i.e at junction)

E_{0} = =

w = width of depletion region

V_{0} = contact potential

= width of depletion region.

The width of depletion region can be given as:

w =

where,

N_{a} and N_{d} = concentration of acceptor and donor (in/cm^{3})

âˆˆ = permitivity of material used for formation p â€“ n junction diode.

q = electronic charge (1.6 Ã— 10^{â€“19 }c)

We can also calculate the penetration of transition region into the n and p material

x_{po} = =

x_{no} = =

We can see from the above equations that the depletion region is inversely proportional to the square root of concentration. i.e.

w

# Carrier Injection

Excess holes on the n - side (i.e. Î”p_{n})

Î”p_{n} = p_{n} (e^{qv/kT }â€“ 1)

Excess electron on p - side

Î”n_{p} = n_{p} (e^{qv/kT} â€“ 1)

where,

Î”p_{n} = excess hole concentration on n side

Î”n_{p} = excess electron concentration on p side

p_{n} = equilibrium hole concentration on n side

n_{p} = equilibrium electron concentration on p side

So, the resulting excess electron distribution is obtained using following formula

(*x*_{p}) = n_{p} (e^{qv/kT} â€“ 1)

Similarly excess hole distribution is

(*x*_{n}) = p_{n} (e^{qv/kT} â€“ 1)

Where, x_{p} = Distance in the p-material measured in the â€“x direction with x_{po} as origin.

x_{n} = Distance in the n-material measured in the + x direction with x_{no} as origin.

(*x*_{p}), (*x*_{n}) = Excess carrier at *x*_{n} and *x*_{p}

p_{n} = equilibrium hole concentration on n - side

n_{p} = equilibrium electron concentration on p - side

L_{n}, L_{p} = electron and hole diffusion length.

The total current across the diode is given by,

I = qA

This equation is also called diode equation where,

D_{p}, D_{n} = hole and electron diffusion coefficient respectively.

# Capacitance of p - n Junction

Basically two types of capacitances are associated with a junction

- Junction capacitance
- Charge storage capacitance (Diffusion capacitance)

**Junction capacitance :**_{J}= =_{j}= Voltage variable capacitance_{0}= built - in potential (contact potential)**Diffusion Capacitance :**

Diffusion capacitance is the capacitance due to transport of charge carriers between two terminal of a device.

Let the time to cross the device is the forward transit time t_{f}. So, the amount of charge in transit through device at this particular moment, Q, is given by

Q = I (V)