Continuity Equation

Continuity Equation

• The carrier concentration in the body of a semiconductor is a function of time and distance.

• Mathematically, a partial differential equation governs this functional relationship between carrier concentration, time and distance. Such an equation is called continuity equation.

• The equation is based on the fact that charge can neither be created nor destroyed.

• Consider the infinitesimal n type element of volume of area A and length dx as shown in Fig. 1.17.1.

• The average hole concentration is p/m³.

• The current entering the volume at x is I and leaving at x + dx is I + dI. This change in current is because of diffusion.

• Now due to diffusion the concentration of charge carriers decreases exponentially with the distance.

Hence,

 dI = Number of coulombs per second decreased within the volume

  ……... (1.17.1)

• Now if τ_p is the mean life time of the holes then, Р/τ_p = Holes per second lost by recombination per unit volume.

• Due to recombination, number of coulombs per second decreased within the given volume is,

 = (Charge on hole) × (Holes/sec per unit volume) × (Volume)

 = q × P/τ_P × (Adx) = qA dx × P/τ_P

         ……... (1.17.2)

• While let g is the rate at which electron hole pairs are generated by thermal generation per unit volume. Due to this, number of coulombs per second increases with the volume

 = (Charges on hole) × (Rate of a generation) × (Volume) = q g A dx

     ……… (1.17.3)

• Thus the total change in number of coulombs per second is because of three factors as indicated by the equations (1.17.1), (1.17.2) and (1.17.3).

• Total change in holes per unit volume per second is dp/dt. Hence the total change in coulombs per second within the given volume

 = q dp/dt (Volume) = q A dx dp/dt

        ….. (1.17.4)

• According to law of conservation of charges,

 q A dx dp/dt = ‒ q A dx P/τ_P + qgAdx ‒ dI

           ……... (1.17.5)

Key Point: The negative sign indicates decrease while positive indicates increase in number of coulombs per second.

 J = I/A = Current density

 I = JA

i.e. dI = AdJ as A is constant.

       ………..(1.17.7)

• The total current density J is due to drift and diffusion current.

 J = −qDp dp/dx + pqμ_PE = Diffusion + Drift

           ……… (1.17.8)

where

E = Electric field intensity within the volume

• If the semiconductor is in thermal equilibrium and subjected to no external electric field then hole density will attain a constant value P₀. Under this condition I = 0 i.e. J = 0 and dp/dt = 0 due to equilibrium. Using in (1.17.7),

 0 = ‒P₀/τ_P + g ‒ 0

i.e.  g = Р₀ / τ_р

         ………. (1.17.9)

• The equation (1.17.9) indicates the thermal equilibrium i.e. the rate at which holes are thermally generated just equal to the rate at which holes are lost due to the recombination.

• Using (1.17.8) and (1.17.9) in (1.17.7),

        ………. (1.17.10)

• This is called equation of conservation of charge or the continuity equation.

• As holes in n type material are considered, let us use the suffix n. And as concentration is a function of both time t and distance x, let us use partial differentiation, Hence the final continuity equation takes the form as,

• Similarly the continuity equation for the electrons in p type material can be written as,

Let us see the special cases for continuity equation under the condition of zero electric field.

1. Concentration Independent of Distance and E = 0

• As concentration is not dependent on x and E = 0, the last two terms on right hand side of the continuity equation become zero. Hence equation reduces as,

• This is a differential equation interms of p_n and the solution of this equation is,

 P_n‒P_n0 = Kq^‒t/τp

where K = Constant

• τ_p is mean life time but also time constant of the above equation.

Key Point: Thus τ_p can be defined as the time taken by the injected concentration to fall to 36.78 % of its initial value.

2. Concentration Independent of Time and E = 0

• As concentration is not dependent on t and E = 0, the left hand side of continuity equation is zero and last term on right hand side is zero.

       ………. (1.17.14)

 But

 √[D_Pτ_P] = L_P = Diffusion length of holes

• Hence the solution of the equation (1.17.14) takes the form,

  P_n ‒ P_n0 = K₁q^‒x/LP + K₂q^‒x/LP

where K₁, K₂ = Constantsis

3. Concentration Varies Sinusoidally with Time and E = 0

• The injected concentration varies sinusoidally with an angular frequency of ω. The sinusoidal variation is indicated in phasor form as,

  P_n(x, t) = P_n(x) q^jwt

• Substituting in continuity equation with E = 0,

• For zero frequency (ω = 0), the equation takes the form as obtained for special case discussed in 1.17.2.

• Hence a.c. solution for w ≠ 0 can be obtained from d.c. solution by replacing L_p by L_p/√[1+jωτ_p]⋅

• Thus continuity equation is the fundamental law governing the flow of charge.

Review Question

1. State and explain the continuity equation.