Diffusion Current and Diffusion Current Density

Diffusion Current and Diffusion Current Density

• This is the current which is due to the transport of charges occurring because of nonuniform concentration of charged particles in a semiconductor.

• Consider a piece of semiconductor which is nonuniformly doped. Due to such nonuniform doping, one type of charge carriers occur at one end of a piece of semiconductor.

• The charge carriers are either electrons or holes, of one type depending upon the impurity used. They have the same polarity and hence experience a force of repulsion between them.

• The result is that there is a tendency of the charge carriers to move gradually i.e. to diffuse from the region of high carrier density to the low carrier density. This process is called diffusion.

• This movement of charge carriers under the process of diffusion constitutes a current called diffusion current. This is shown in Fig. 1.16.1.

• The diffusion current continues till all the carriers are evenly distributed throughout the material.

• A diffusion current is possible only in case of nonuniformly doped semiconductors while drift current is possible in semiconductors as well as conductors.

Key Point: The diffusion current exists without external voltage applied while drift current can not exist without an external voltage applied.

1. Concentration Gradient

• Consider a p‒type semiconductor bar which is nonuniformly doped along its length, in the direction of x as shown in Fig. 1.16.2 (a). As x increases, the doping concentration decreases.

• To form p‒type of semiconductor, acceptor impurity is added which creates holes as the majority charged particles.

• Let P be the concentration of holes, but due to nonuniform doping it is not constant but is changing with respect to x.

• Let concentration of holes at x = 0 is p = p(0) and is maximum as bar is heavily doped at x = 0. As x increases, the concentration of holes decreases. The nature of the variation in p against distance x is shown in Fig. 1.16.2 (b).

• The slope of the graph can be observed from Fig. 1.16.2 (b) is the ratio of change in concentration to change in distance. It is called rate of change of concentration or concentration gradient.

  Slope of graph = Concentration gradient = dp/dx

  ….. (1.16.1)

• Hence nonuniform doping produces a concentration gradient in a semiconductor.

• Due to such concentration gradient, holes move from the higher concentration area to the lower concentration area to adjust the concentration. Such a movement of holes, due to the concentration gradient in a semiconductor is called diffusion.

• Due to the movement of holes, current is constituted in a bar which is called diffusion current.

• There exists such a diffusion current in n‒type semiconductor if it is nonuniformly doped, due to movement of electrons which are majority carriers.

Key Point: Nonuniform doping creats concentration gradient, due to which diffusion of charge carriers exists.

2. Diffusion Current Density

• Consider a nonuniformly doped p‒type semiconductor bar as shown in Fig. 1.16.2 (a).

• The diffusion current density is proportional to the concentration gradient, which is responsible for the diffusion and hence the diffusion current.

 J_p ∝ dp/dx

 where Jp Diffusion current density due to holes

• Hence the diffusion current density is expressed by,

 J_p = q D_p dp/dx

 where D_p = Diffusion constant for holes expressed in square metres per second. (m²/sec).

Key Point: The current due to holes is in the direction of the conventional current and hence treated as positive. But slope of the graph i.e. dp/dx is negative giving the negative diffusion current density for holes. But to get positive sign for holes, an additional negative sign is used to compensate for negative sign of dp/dx. Hence diffusion current density for holes is mathematically expressed as,

 J_p = ‒ q D_p dp/dx

• In case of n‒type bar, such diffusion current is due to the electrons. Current due to the electrons is in opposite direction to the conventional current and mathematically treated to be negative. The concentration gradient dn/dx is negative where n is concentration of electrons.

• Hence diffusion current density to electrons is expressed by,

 J_p = + q D_n dn/dx

where D_n = Diffusion constant for electrons in square metres per second (m²/sec).

• Fig. 1.16.3 (a) shows the direction of diffusion of holes and corresponding diffusion current density. Fig. 1.16.3 (b) shows the direction of diffusion of electrons and corresponding diffusion current density.

• Observe that the charge carriers whether it is hole or electron, always move from high concentration area towards low concentration area. Hence direction of diffusion is same in both the cases. But resulting current densities have opposite directions.

3. Total Current Density Due to Drift and Diffusion

• We have seen that the drift current is due to the applied voltage while the diffusion current is due to the concentration gradient.

• But in semiconductor it is very much possible that both the types of currents may exist simultaneously.

• In practice in such situation there exists four components of current as the drift current due to electrons and due to holes, while the diffusion current due to electrons and due to holes.

• Drift current density for electrons and holes can be expressed as,

 J_n = n q μ_n E and J_p = P q μ_p E

• Diffusion current density due to the electrons and holes can be expressed from equations (1.16.3) and (1.16.4) as,

 J_n = + q D_n dn/dx and J_p = ‒ q D_p dp/dx

  Total current density due to the electrons can be expressed as,

J_n = n q μ_n E  +  q D_n dn/dx

Total current density due to the holes can be expressed as,

J_p = P q μ_p E  ‒  q D_p dp/dx

And hence the total current density due to the electrons and holes (Drift + Diffusion) is,

 J = J_n +J_p

4. Einstein's Relationship

• It is now known that drift current density is proportional to the mobility (μ) while diffusion current density is proportional to the diffusion constant (D). There exists a fixed relation between these two constants which is called Einstein's relation.

• It states that, at a fixed temperature, the ratio of diffusion constant to the mobility is constant. This is Einstein's relation. Mathematically it is expressed as,

 D_P/μ_P = D_n / μ_n = kT = Constant at fixed temperature.

     ……….. (1.16.8)

where T is the temperature in °K and k is the Boltzmann's constant = 8.62 × 10^‒5 eV/°K

5. Voltage Equivalent of Temperature

• In the equation (1.16.8), the product kT is called voltage equivalent of temperature.

• The voltage equivalent of temperature is denoted by V_T.

  V_T = KT = Voltage equivalent of temperature

     ……….. (1.16.9)

At room temperature i.e. at 27 °C,

T = 273 + 27 = 300 °K

 V_T = kT = 8.62 × 10^‒5 × 300 = 0.02586 V ≈ 26 mV at 300 °K

Key Point: The value of V_T = 26 mV at 27° C i.e. 300 °K is very commonly used while solving the examples.

• Substituting this in equation (1.16.8) we get,

 D_n / μ_n = D_p / μ_p = V_T = 0.02586 at room temperature

 μ_n=39 D_n and μ_p=39 D_P at room temperature.

……….. (1.16.10)

• In general we can express the relation between mobility and diffusion constant as,

 μ = 39 D at room temperature

       ……. (1.16.11)

6. 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 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_P 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 Lenth L

Let

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 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 = √(Dτ)

Thus,

L_n = √(D_nτ_n)

L_p = √(D_pτ_p)

τ_n = L_n² / D_n

τ_p = L_P² / D_P

Ex. 1.16.1 Find out the diffusion constant of holes if their mobility is given as 0.039 m²/v‒sec.

Solution:

 μ_p = 0.039 m²/V‒s

According to Einstein's relation, D_p / μ_р = KT

 At 300 °K,

D_p / μ_р = D_p / 0.039 = 26 × 10^‒3

i.e. Dp = 1.01 × 10^‒3

Ex. 1.16.2 The phosphrous (donor) concentration in a region of a silicon crystal varies linearly from a concentration of n₀ = 10¹⁴ cm^‒3 at x = 0 mm to a concentration of n₁ = 10¹⁷ cm^-3 at x = 1 mm. The diffusion constant for electrons is  D_n = 22.5 cm²/s, the diffusion constant for holes is D_p = 5.2 cm²/s and the temperature is 300 °K. What is the diffusion  current density in the positive x‒direction?

Solution:

 n₀ = 10¹⁴ cm^-3

 n₁ = 10¹⁷ cm^-3,

x₁ = 1 mm,

x₀ = 0 mm,

 D_n = 22.5 × 10^-4 m²/s,

D_p = 5.2 × 10^-4 m²/s.

For silicon, n_i = 1.5 × 10¹⁰ cm^‒3

 n_i² = np

        ... Law of mass action

 P₀ = n_i² / n₀ = (1.5×10¹⁰)² / 10¹⁴ = 2.15 × 10⁶ cm^-3

P₁ = n_i² / n₁ = (1.5×10¹⁰)² / 10¹⁷ = 2.15 × 10³ cm^-3

dp/dx = [P₁‒P₀] / [x₁‒x₀] = [ 2.25×10³ ‒ 2.25×10⁶] / 10^‒1 = ‒2.248 × 10⁷ cm^‒4

 dn/dx = [n₁‒n₀] / [x₁‒x₀] = [10¹⁷ ‒ 10¹⁴] / 10^‒1 = 9.99 × 10¹⁷ cm^‒4

 J = J_n + J_p = [ qD_n dn/dx ] ‒ [ qD_p dp/dx ]

= 1.602×10^‒19 [ 22.5×9.99×10¹⁷ ‒ 5.2×(‒2.248×10⁷)]

= 3.6 A/cm²

= 3.6×10⁴ A/m2

Review Questions

1. What is diffusion current? Derive the expressions for the diffusion current densities.

2. Explain Einstein's relationship and voltage equivalent of temperature.

3. Define mean life time and diffusion length for the charge carriers.