Derivation of V‒I Characteristics of PN Junction Diode (Diode Current Equation)

Derivation of V‒I Characteristics of P‒N Junction Diode (Diode Current Equation)

• Let us study the derivation of the mathematical expression for the current through a diode, which gives its V‒I characteristics.

Let

P_p = Hole concentration in p‒type at the edge of depletion region

n_n = Electron concentration in n‒type at the edge of depletion region

Pn = Hole concentration in n‒type at the edge of depletion region

n_p = Electron concentration in p‒type at the edge of depletion region

Key Point: Note that in the symbol, basic letter indicates type of charge carrier concentration hole (p) or electron (n). The base indicates type of material in which it exists.

• Under unbiased condition, when holes move from p‒side to n‒side due to diffusion, their concentration behaves exponentially. This is mathematically expressed as,

P_p = P_n e^Vj/VT

              ...(2.10.1)

where

 V_J =  Barrier potential or junction potential

• Now consider forward biased diode as shown in x = 0.

Fig. 2.10.1 p‒n junction diode

• Though the proportion of holes and electrons in constituting a current through the p‒region is changing, the hole concentration throughout the entire p‒region is constant and denoted as,

 P_p0 = Hole concentration in p‒region

• As holes cross the junction, this, concentration becomes P_n(0) which is concentration of holes on n‒side just near the junction. This further behaves exponentially as given in the equation (2.10.1).

• From equation (2.10.1) we can write,

 P_p0 = P_n(0) e^(VJ‒V)/.VT

              ………..(2.10.2)

Key Point: The term V_J becomes V_J ‒ V as the forward biased voltage V opposes the barrier potential. So net voltage across the junction becomes V_J‒V.

• The equation (2.10.2) can be written for open circuited unbiased p‒n junction diode by putting V = 0 as,

P_p0 = P_n0 e^VJ/VT

           ….....(2.10.3)

where P_no is the concentration of holes on n‒side just near the junction when diode is open circuited i.e. at thermal equilibrium and hence different than p_n(0).

• As the concentration of holes in entire p‒region is constant, equating equations (2.10.2) and (2.10.3) we get,

P_n(0) e^VJ‒V/VT = P_n0 e^VJ/VT

P_n(0) = P_n0 e^VJ/VT

            ...(2.10.4)

• This equation represents boundary condition and called law of junction. This indicates that the hole concentration P_n(0) at the junction under forward biased condition is greater than its thermal equilibrium value P_n0 For large forward biasing P_n(0) becomes much larger compared to P_n0.

Key Point: The discussion is equally applicable for the electron concentration on the p‒side.

• Thus, n_p(0) = n_p0 e^V/VT

           ...(2.10.5)

• Now the difference between two concentrations at the junction under unbiased and biased condition is called injected or excess concentration denoted as P_n(0).

 P_n(0) = P_n(0) ‒ P_n0

           ...(2.10.6)

• Using equation (2.10.4) in equation (2.10.6),

 P_n(0) = P_n0 e^V/VT‒ P_n0

 P_n(0) = P_n0 (e^V/VT‒ 1)           ...(2.10.7)

Similarly, N_p(0) = P_p0 (e^V/VT‒ 1)           ...(2.10.8)

• The hole current crossing the junction from p‒side to n‒side is given by,

 I_pn (0) = [ Aq D_p P_n (0) ] / L_p           

...(2.10.9)

While an electron current crossing the junction from n‒side to p‒side is given by,

I_np(0) = [ Aq D_n N_p(0) ] / Ln

   ...(2.10.10)

where

A = Area of cross‒section of junction

D_P = Diffusion constant for holes,

D_n = Diffusion constant for electrons

L_p = Diffusion length for holes,

L_n = Diffusion length for electrons

• Using equations (2.10.7) and (2.10.8) in equations (2.10.9) and (2.10.10), the total current I at the junction is given by,

I = I_pn(0) + I_np(0)

I₀ = Reverse saturation current

• The equation (2.10.11) is the required expression for diode current.

Key Point: In the derivation, the generation and recombination in the depletion region is neglected. To consider its effect, which is dominant in Si diodes, the factor η is introduced in the equation.

I₀ = I₀ (e^V/ηVT ‒1)

 The value of η = 1 for Ge diodes and η = 2 for Si diodes.

Carrier Lifetimes

• In a pure semiconductor the number of free electrons is always equal to number of holes. The thermal agitation continues to produce new hole‒electron pairs while previous pairs disappear. This disappearing of pairs is due to the process called recombination. The merging of of a free electron and hole is called recombination. The amount of time between the creation and disappearence of a free electron and hole pair is called its lifetime.

• Thus a hole exists for τ_р seconds before it recombines while a free electron exists for τ_n seconds before it recombines. τ_p and τ_n are called carrier lifetimes of hole and electron respectively. Carrier lifetime is also called mean lifetime of the hole and electron. Carrier lifetimes range from nanoseconds to hundreds of microseconds.

Diffusion Length L

• Due to the recombination, the concentration of charge carriers decrease exponentially with the distance at the time of diffusion. The charge carriers have mean life time denoted by τ for which they exist before recombination.

τ_n = Mean life time of free electron

τ_p = Mean life time of free hole.

• After recombination, these charge carriers vanish and concentration decreases exponentially with distance.

Key Point: The average distance covered by an excess charge carrier while diffusion during its life time is called diffusion length of that charge length of that carrier. It is denoted by L.

L_n = Diffusion length of free electron

L_p = Diffusion length of free hole.

• The diffusion length L is related to main life time τ through the diffusion constant D of the charge carrier. Mathematically this relationship is given by,

L = √D_τ

Thus,

 L_n = √(D_nτ_n)

i.e. τ_n = L_n² / D_n

L_p = √(D_pτ_p)

i.e. τ_p = L_p² / D_p

Ex. 2.10.1: Determine the ideal reverse saturation current density in a silicon pn junction at T = 300 °K. Consider the following parameters in the silicon pn junction: N_A = N_D = 10¹⁶ cm³, n_i = 1.5×10¹⁰ cm^-3, D_n =25 cm²/s. τ_p0 = τ_n0 =5×10^‒7 s, D_p = 10 cm² / s, ε_r = 117. Comment on the result.

Solution:

The reverse current density is given by,

The diffusion length for the holes and electrons is given by,

L_p = √(D_pτ_p)

L_n = √(D_nτ_n)

n_p0 = n_i² / N_A

P_no = n_i² / N_D

  ….. by law of mass action

L_p = √(D_pτ_p) = √[10×10^‒4×5×10^‒7] = 2.236×10^‒5

L_n = √(D_nτ_n) = √[25×10^‒4×5×10^‒7] = 3.535×10^‒5

n_p0 = n_i² / N_A = (1.5×10¹⁶)² / 10²² =2.25×10¹⁰

P_no = n_i² / N_D= (1.5×10¹⁶)² / 10²² =2.25×10¹⁰

q = 16×10^‒19 C and using in (1),

J₀ = 1.6×10^‒19 [ (10×10^‒4×2.25×10¹⁰ / 2.236×10^‒5 ) + (25×10^-4×2.25×10¹⁰ /3.535×10^‒5) ]

 = 4.15×10^‒7A/m²

i.e.4.15×10^‒11 A/cm²

Comment: Ideally reverse saturation current density is very small. If the area of cross‒section A is given as 2×10^‒8 m² then I₀ becomes J₀ × A i.e., 4.15×10^‒7×2×10^‒8 = 8.3119×10^‒15 A. Thus ideally reverse saturation current is also very small.

Review Question

1. Derive the p‒n diode current equation.