Question
A non-uniformly doped p-type semiconductor has length L. The energy band diagram is shown below.
Given:
- Energy difference = Δ
- Length = L
Find:
Electric field inside the semiconductor.
Solution
Step 1:
The semiconductor is at equilibrium, so the Fermi level remains constant throughout the material. \[ E_F=\text{constant} \] A non-uniform acceptor concentration causes the valence band to slope, producing an internal electric field.
The semiconductor is at equilibrium, so the Fermi level remains constant throughout the material. \[ E_F=\text{constant} \] A non-uniform acceptor concentration causes the valence band to slope, producing an internal electric field.
Step 2:
For a uniform electric field, \[ E=V/L \] \[ \Delta V=EL \] where
For a uniform electric field, \[ E=V/L \] \[ \Delta V=EL \] where
- \(E\) = electric field
- \(L\) = semiconductor length
Step 3:
The band diagram shows an energy difference (eV) \[ \Delta \] Energy and voltage are related by \[ \Delta=q\Delta V \] Substitute \[ \Delta V=EL \] to obtain \[ \Delta=qEL \]
The band diagram shows an energy difference (eV) \[ \Delta \] Energy and voltage are related by \[ \Delta=q\Delta V \] Substitute \[ \Delta V=EL \] to obtain \[ \Delta=qEL \]
Step 4:
Rearrange to obtain the electric field: \[ \boxed{E=\frac{\Delta}{qL}} \]
Rearrange to obtain the electric field: \[ \boxed{E=\frac{\Delta}{qL}} \]
Final Answer
\[
\boxed{E=\frac{\Delta}{qL}}
\]
Summary
- Non-uniform doping → hole diffusion.
- Diffusion creates an internal electric field.
- At equilibrium, the electric field balances diffusion.
- The energy drop across the semiconductor is \( \Delta=qEL \).
- Hence, \(E=\Delta/(qL)\).