In a semiconductor, electrical current flows due to the motion of charge carriers, namely electrons and holes. Carrier motion can occur due to two different physical causes:
- Applied electric field → causes drift of carriers
- Non-uniform carrier concentration → causes diffusion of carriers
In practical semiconductor devices, both drift and diffusion occur simultaneously. Therefore, the total current in a semiconductor is the sum of the drift current and the diffusion current.
Understanding carrier transport is essential for:
- Semiconductor conductivity
- p–n junction behaviour
- Diodes, transistors, and integrated circuits
Drift of Charge Carriers
Concept of Drift
Drift: Transportation of charge carriers under the effect of an external electric field.
When an electric field is applied across a semiconductor:
- Free electrons experience a force opposite to the field direction
- Holes experience a force in the direction of the field
As a result, carriers acquire a net average velocity called drift velocity.
Drift Velocity
The drift velocity is defined as:
The average velocity acquired by charge carriers under the influence of an externally applied electric field is called drift velocity. It is denoted by vd.
If a semiconductor of length L having a cross-sectional area A is kept in an electric field E, then the average velocity (drift velocity) of a charge particle is given by
for electrons: $$v_{d_{n}}=\mu_nE$$
for holes: $$v_{d_{p}}=\mu_pE$$
where
μn = electron mobility
μp = hole mobility
When a charge carrier q takes time t to travel a distance L inside the semiconductor, then the total charge carriers passing through the cross-section A of the semiconductor in time t will be qN.
Where N is the total number of charge carriers inside the semiconductor. Therefore, the drift current flowing through the semiconductor is
$$I_d=\frac{qN}{t}=\frac{qNv}{L} \qquad \left[ ∵ v=\frac{L}{t} \right]$$
Hence, the drift current density $$J_d= \frac{I_d}{A}=\frac{qvN}{AL}$$
But $$\frac{N}{AL}= n = charge\; carrier\; concentration$$
$$∴\;J_d=nqv_d$$
$$=> J_d= nq\mu E$$
Hence, electron drift current density $$J_{d_n}= nq\mu_n E$$
and hole drift current density $$J_{d_p}= nq\mu_p E$$
Total drift current density $$J_{drift}=J_{d_n}+J_{d_p}$$
$$=>J_{drift}=q\left( n\mu_n+p\mu_p \right)E$$
Physical Interpretation
- Drift current exists only when an electric field is applied
- Higher mobility and higher carrier concentration increase drift current
- Metals conduct mainly by drift mechanism
Diffusion of Charge Carriers
Concept of Diffusion
In a semiconductor, electric current can flow even without applying an external electric field. This happens when there exists a concentration gradient of charge carriers.
This type of current is called a diffusion current.
Diffusion current is the current produced due to the movement of charge carriers from a region of higher concentration to a region of lower concentration.
This phenomenon is based on the natural tendency of particles to spread uniformly due to random thermal motion.
Diffusion is especially important in:
- p–n junctions
- Semiconductor devices without external bias
Diffusion Current Density:
Even without an electric field, diffusion causes a current known as a diffusion current.
Electron Diffusion Current Density:
The electron diffusion current density is proportional to the gradient of electron concentration and to the electronic charge, so we may write $$J_{n_{diff}}=qD_n\frac{dn}{dx}$$
Where Dn is the electron diffusion coefficient, which has units of cm²/s and is a positive quantity. If the electron density gradient becomes negative, the electron diffusion current density will be in the negative x direction.
Hole Diffusion Current Density:
The hole diffusion current density is proportional to the hole density gradient and to the electronic charge, so we may write $$J_{p_{diff}}=-qD_p\frac{dp}{dx}$$
The negative sign appears because:
-
Holes carry a positive charge.
-
The direction of the hole motion directly determines the current direction.
-
Mathematical convention introduces the minus sign.
Where Dp is the hole diffusion coefficient, which has units of cm²/s and is a positive quantity.
If the hole density gradient becomes negative, the hole diffusion current density will be in the positive x direction.
Total Diffusion Current Density:
The total diffusion current density is: $$J_{diff}=J_{n_{diff}}+J_{p_{diff}}$$
$$J_{diff}=qD_n\frac{dn}{dx}-qD_p\frac{dp}{dx}$$
Total Current Density:
The total current density is the sum of these four components, or, for the one-dimensional case,
$$J=J_{drift}+J_{diff}$$
$$J=q\left( n\mu_n+p\mu_p \right)E+qD_n\frac{dn}{dx}-qD_p\frac{dp}{dx}$$
This equation may be generalized to three dimensions as $$J=q\left( n\mu_n+p\mu_p \right)E+qD_n\nabla n-qD_p\nabla p$$
Electron mobility indicates the efficiency of electron movement in a semiconductor due to the influence of an electric field. The electron diffusion coefficient indicates the efficiency of electron mobility in a semiconductor due to a density gradient.
Einstein Relation :
The diffusion coefficient and electron mobility are not independent variables. Likewise, the diffusion coefficient and hole mobility are not independent variables.
But the relationship between mobility and the diffusion coefficients is related as follows:
$$\frac{D_n}{\mu_n}=\frac{D_p}{\mu_p}=\frac{kT}{q}$$
Here k is Boltzmann’s constant, and the relation kT/q is called the voltage equivalent of temperature. For an electron at T = 300 K, kT/e = 0.026 volts.
Diffusion in p–n Junction:
Diffusion current plays a crucial role in p–n junction formation.
When p-type and n-type materials are joined:
-
Electrons diffuse from the n to the p side.
-
Holes diffuse from the p to the n side.
-
This diffusion creates a depletion region.
-
Built-in potential develops.
At equilibrium:
-
Diffusion current = Drift current
-
Net current = 0
Comparison: Drift vs Diffusion Current
| Property | Drift Current | Diffusion Current |
|---|---|---|
|
Cause |
Electric field |
Concentration gradient |
|
External voltage required |
Yes |
No |
|
Direction depends on |
Electric field |
Carrier concentration |
|
Dominant in |
Metals |
Semiconductors |
|
Formula (electron) |
$$J_d=nq\mu_nE$$ |
$$J_{diff}=qD_n\frac{dn}{dx}$$ |
Importance of Carrier Transport Mechanisms
Carrier transport explains the following:
- Electrical conductivity
- p–n junction current
- Diode equation
- Transistor action
- Integrated circuit behaviour
Important Examination Questions
Short Answer Questions
- What is drift current?
- Define diffusion current.
- State the Einstein relation.
Long Answer / Derivation Questions
- Derive expressions for drift current density of electrons and holes.
- Explain diffusion current and derive its expression.
- Derive the total current density equation in semiconductors.
⭐ Very frequently asked in university examinations
Conceptual Questions
- Can diffusion current exist without an electric field? Explain.
- Why does drift current depend on mobility?