Hong Life CalculatorsUnderstanding the electrical mobility calculator — concept and principle
This calculator finds the electrical mobility of charge carriers, the drift speed they gain per unit electric field, from drift speed and field, from conductivity and carrier density, or from the diffusion coefficient. It is used in semiconductor physics, electrochemistry and aerosol and ion physics.
How it is calculated
μ = v_d ÷ E
σ = n z e μ, so μ = σ ÷ (n z e)
Einstein relation: D = μ k_B T ÷ (z e)
- E is entered in V/cm (1 V/cm = 100 V/m), n in carriers per cm³ (1 cm⁻³ = 10⁶ m⁻³), D in cm²/s.
- z is the charge number, T the temperature in kelvin; k_B T/e ≈ 25.85 mV at 300 K.
- Results are given in cm²/(V·s) and m²/(V·s).
The principle
In a solid or gas, carriers are constantly knocked about by collisions. A field gives them a small average drift on top of this random motion, and in many materials the drift speed is proportional to the field; the constant of proportionality is the mobility. The current density is then the number of carriers times their charge times their drift speed, which gives σ = nzeμ. Diffusion and drift are both limited by the same collisions, which is why Einstein could link D and μ through the thermal energy k_B T.
Worked example
Default inputs: drift speed 14,000 m/s, field 1,000 V/cm, T = 300 K, z = 1.
- E = 1,000 V/cm = 100,000 V/m.
- μ = 14,000 ÷ 100,000 = 0.14 m²/(V·s) = 1,400 cm²/(V·s).
- Thermal voltage k_B T/e at 300 K = 25.852 mV, so D = 1,400 × 0.025852 = 36.19 cm²/s.
- At 10¹⁶ cm⁻³: σ = 10²² × 1.602 × 10⁻¹⁹ × 0.14 = 224.3 S/m. Drift speed at the field entered = 14,000 m/s.
| Electric field (V/cm) | Mobility (cm²/(V·s)) |
|---|---|
| 250 | 5,600 |
| 500 | 2,800 |
| 750 | 1,867 |
| 1,000 | 1,400 |
| 1,250 | 1,120 |
| 1,500 | 933.3 |
| 2,000 | 700 |
How to read the result
A mobility around 1,400 cm²/(V·s) is of the order quoted for electrons in lightly doped silicon at room temperature; holes in silicon are several times slower. Higher mobility means faster devices and lower resistance for the same doping. Mobility falls as temperature and doping rise, because carriers collide more often.
Common mistakes
- Mixing V/cm and V/m, which changes μ by a factor of 100.
- Applying μ = v_d/E at very high fields, where drift speed saturates.
- Using a single-carrier formula when electrons and holes both conduct; then σ = e(nμₙ + pμₚ).
- Using the Einstein relation for degenerate (very heavily doped) carriers.
Frequently asked questions
What units is mobility given in?
Usually cm²/(V·s) in semiconductor work; 1 cm²/(V·s) = 10⁻⁴ m²/(V·s).
Why is electron mobility higher than hole mobility in silicon?
Electrons in silicon have a smaller effective mass in the directions that matter for conduction.
What is the thermal voltage?
k_B T/e, about 25.85 mV at 300 K; it sets the scale in the Einstein relation and in diode equations.
Sources: H. Föll, Semiconductors (lecture script, University of Kiel), section 2.3.4 · OpenStax, University Physics Volume 2 (2016), section 9.2 "Model of Conduction in Metals"