Depletion Region Width in PN Junctions Explained

How to Calculate Depletion Region Width in PN Junctions

The depletion region width is one of the most important concepts in semiconductor physics. It determines electric field strength, junction capacitance, and device behavior in diodes and MOSFETs.

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1. What is the Depletion Region?

When a p-type and n-type semiconductor are joined, electrons and holes diffuse across the junction. This leaves behind fixed ions and removes mobile carriers near the junction.

The region with no mobile charge carriers is called the depletion region.

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2. Total Depletion Width Formula

The total depletion width is:

W = xₙ + xₚ

Where:

• W = total depletion width
• xₙ = depletion width on n-side
• xₚ = depletion width on p-side

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3. PN Junction Depletion Width Formula

For an abrupt PN junction:

W = √[(2εₛ / q) × ((N_A + N_D) / (N_A N_D)) × (V_bi − V)]

Where:

Symbol	Meaning
εₛ	Permittivity of semiconductor
q	Electron charge
N_A	Acceptor doping concentration
N_D	Donor doping concentration
V_bi	Built-in potential
V	Applied voltage

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4. Effect of Doping

Doping concentration strongly affects depletion width.

• Higher doping → smaller depletion width
• Lower doping → larger depletion width

Heavily doped semiconductors have very narrow depletion regions.

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5. Effect of Reverse Bias

Applying reverse bias increases depletion width because it increases the effective barrier potential.

(V_bi − V) ↑ ⇒ W ↑

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6. One-Sided Junction Approximation

If one side is much more heavily doped:

N_A ≫ N_D ⇒ W ≈ xₙ

The depletion region extends mostly into the lightly doped side.

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7. MOS Capacitor Depletion Width

For MOS structures:

W_d = √[(2εₛ ψ_s) / (qN_A)]

Where:

• ψ_s = surface potential
• N_A = substrate doping

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8. Maximum Depletion Width

W_d,max = √[(4εₛ φ_F) / (qN_A)]

This occurs at strong inversion in MOS devices.

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9. Relationship Between xₙ and xₚ

Charge neutrality condition:

N_A xₚ = N_D xₙ

So:

xₙ = (N_A / (N_A + N_D)) W

xₚ = (N_D / (N_A + N_D)) W

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10. Physical Intuition

• Low doping → wider depletion region
• High doping → narrow depletion region
• Reverse bias → widens depletion region

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11. Junction Capacitance

C_j = εₛ A / W

As depletion width increases, capacitance decreases.

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12. Summary Table

Factor	Effect on W
Higher doping	Decreases W
Lower doping	Increases W
Reverse bias	Increases W
Forward bias	Decreases W

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Conclusion

The depletion region width depends on doping, applied voltage, and semiconductor properties. It is a key parameter in diodes, MOSFETs, and integrated circuits.

W = √[(2εₛ / q) × ((N_A + N_D) / (N_A N_D)) × (V_bi − V)]

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