Question

If μ_e and μ_h represent the electron and hole mobility respectively, n represents the electron density, e represents the charge of an electron and E represents the electric field, which of the following is the equation for current density in an intrinsic semiconductor?
1. n2eE2(μe - μh)
2. n2eE2(μe + μh)
3. neE(μe + μh)
4. neE(μe - μh)

Answer 1

Correct Answer - Option 3 : neE(μe + μh)

The correct answer is option 3) i.e. neE(μe + μh)

CONCEPT:

• The drift velocity (vd) of the electron in a conductor is given by:​

\(v_d =\frac{I}{NAq}\)vd=InAqvd=InAq

Where I is the current flowing through the conductor, A is the area of cross-section of the conductor, q is the charge on the electron and N is the number of electrons.

• Mobility is defined as the magnitude of drift velocity per unit electric field. 

Mobility, \(μ = \frac{v_d}{E}\)

• Current density is defined as the amount of electric charge flowing across a unit cross-sectional area in unit time. The electric current density (J) for a given conducting material is given as:

\(J = \frac{I}{A}\)

Where I is the current flowing and A is the cross-sectional area.

EXPLANATION:

• We know that drift velocity, \(v_d =\frac{I}{NAq}\)       ----(1)
• Current density, \(J = \frac{I}{A}\)      ----(2)

​Substituting (2) in (1),

\(⇒ v_d =\frac{J}{Nq}\)      ----(3)

• Mobility of charge, \(μ = \frac{v_d}{E} \Rightarrow v_d = \mu E\)      ----(4)

From (3) and (4) we get,

J = Nqv_d

⇒ J = NqμE

• In an intrinsic semiconductor, the number of holes and free electrons is the same. So, the number of electrons and holes per m³ (charge density) will be the same. It is given as 'n' in the question.

⇒ J = nqμE

• The magnitude of charge 'q' of the electron and hole is the same. It is given as 'e' in the question.

⇒ J = enμE

• The total current density (J) in the semiconductor will be the sum of current density due to electrons (Je) and holes (Jp).

⇒ J = J_e + J_h

⇒ J = (enμ_eE + enμ_hE)

⇒ J = neE(μe + μh)