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Drift of Electrons and the Origin of Resistivity - Mobility of Electrons

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Estimated time: 11 minutes
CBSE: Class 12

Introduction

Mobility of electrons is an important concept in Current Electricity because it connects the electric field, drift velocity, and microscopic current flow in conductors and semiconductors. The source material defines mobility, derives its formula, relates it to current density, and notes its temperature dependence and use in semiconductor physics.

CBSE: Class 12

Definition: Mobility

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

CBSE: Class 12

Formula: Mobility

μ = \[\frac {∣v_d​∣}{E}\]​

where:

  • vd: drift velocity
  • E: electric field
CBSE: Class 12

Derivation of Mobility

Step 1: Force on an electron

When an electric field E is applied, an electron experiences a force:

F = eE

This force produces acceleration of the electron. 

Step 2: Acceleration of the electron

Using Newton’s second law,

a = \[\frac {F}{m}\] = \[\frac {eE}{m}\]

where m is the mass of the electron. 

Step 3: Drift velocity

If τ is the average time between two successive collisions, then

vd = aτ = \[\frac {eE}{m}\]τ

So,

vd = \[\frac {eτE}{m}\]

This is the expression for drift velocity. 

Step 4: Mobility formula

Mobility is the drift velocity per unit electric field. Therefore,

μ = \[\frac {∣v_d∣}{E}\]

Substituting the expression for vd,

μ = \[\frac {eτ}{m}\]

Hence, mobility depends on charge, relaxation time, and mass of the charge carrier. 

CBSE: Class 12

Physical Meaning

  • Mobility measures how quickly charge carriers respond to an applied electric field. 
  • If mobility is high, carriers move more easily, and drift velocity is larger for the same electric field. 
  • If mobility is low, the motion of carriers is more strongly hindered by collisions. 

Easy analogy

  • Imagine students walking through a corridor.
  • If the corridor is clear, they move easily; this is like high mobility.
  • If the corridor is crowded, motion is obstructed; this is like low mobility.
CBSE: Class 12

Relation with Current Density

From the microscopic current relation,

vd = \[\frac {j}{ne}\]

Using

μe = \[\frac {v_d}{E}\]

we get

μe = \[\frac {j}{neE}\]

This relation connects mobility with current density in a conductor. 

CBSE: Class 12

Units of Mobility

Quantity Expression Unit
Mobility μ = vd/E m2V−1s−1
Common semiconductor unit — cm2V−1s−1 

Important point

  • If the electric field E = 1 V m−1, then the numerical value of mobility equals the drift velocity.

CBSE: Class 12

Electron and Hole Mobility

In semiconductors, current may be carried by both electrons and holes. The source material notes that hole mobility is smaller than electron mobility. 

Feature Electron Mobility Hole Mobility
Carrier Electron Hole
Relative value Usually greater  Usually smaller 
Reason Electrons generally move more easily in many semiconductors Holes move less easily
CBSE: Class 12

Effect of Temperature

Important concept

  • Mobility decreases with increase in temperature. 
  • At higher temperatures, collisions become more frequent. 
  • Therefore, relaxation time τ decreases, and mobility also decreases because μ = eτ/m.

Quick chain

Temperature increases → collisions increase → relaxation time decreases → mobility decreases 

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