Question

Draw energy band diagrams of n-type and p-type semiconductors at temperature \( T>0 \, \text{K} \). Show the donor/acceptor energy levels with the order of difference of their energies from the bands.

Hint:

Donor levels in n-type semiconductors are close to the conduction band, while acceptor levels in p-type are close to the valence band, typically 0.01–0.05 eV apart, reflecting the ionization energies of impurities.

Answer 1

Part (a): Energy Band Diagram of an N-Type Semiconductor at \( T > 0\,\text{K} \)

N-type semiconductors are doped with donor impurities (e.g., phosphorus in silicon), introducing donor levels \( E_d \) just below the conduction band edge \( E_c \). At \( T > 0 \, \text{K} \), electrons are thermally excited from \( E_d \) to the conduction band, making electrons the majority carriers.

Step 1: Key Features

• Valence Band: Filled up to \( E_v \)
• Conduction Band: Starts at \( E_c \), with band gap \( E_g = E_c - E_v \) (e.g., 1.12 eV for silicon)
• Donor Level: \( E_d \) is 0.01–0.05 eV below \( E_c \):
  \( E_c - E_d \approx 0.01 \text{ to } 0.05 \, \text{eV} \) (e.g., 0.045 eV for phosphorus in silicon)
• Fermi Level: \( E_F \) lies between \( E_d \) and \( E_c \), closer to \( E_c \)
• Carriers: Electrons in the conduction band, few holes in the valence band

Step 2: Diagram Description

• Axes: Vertical energy axis (in eV)
• Valence Band: Shaded region up to \( E_v \), labeled
• Band Gap: Blank region of width \( E_g \)
• Conduction Band: Unshaded region above \( E_c \), with light shading to indicate electrons
• Donor Level: Dashed line at \( E_d \), 0.01–0.05 eV below \( E_c \), label \( E_c - E_d \)
• Fermi Level: Dashed line at \( E_F \), between \( E_d \) and \( E_c \), closer to \( E_c \)
• Carriers: Indicate electrons above \( E_c \), few holes below \( E_v \)

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Part (b): Energy Band Diagram of a P-Type Semiconductor at \( T > 0\,\text{K} \)

P-type semiconductors are doped with acceptor impurities (e.g., boron in silicon), introducing acceptor levels \( E_a \) just above the valence band edge \( E_v \). At \( T > 0 \, \text{K} \), electrons from the valence band occupy \( E_a \), creating holes in the valence band.

Step 1: Key Features

• Valence Band: Up to \( E_v \), with holes
• Conduction Band: Starts at \( E_c \), with band gap \( E_g \)
• Acceptor Level: \( E_a \) is 0.01–0.05 eV above \( E_v \):
  \( E_a - E_v \approx 0.01 \text{ to } 0.05 \, \text{eV} \) (e.g., 0.045 eV for boron in silicon)
• Fermi Level: \( E_F \) lies between \( E_v \) and \( E_a \), closer to \( E_v \)
• Carriers: Holes in the valence band, few electrons in the conduction band

Step 2: Diagram Description

• Axes: Vertical energy axis (in eV)
• Valence Band: Shaded region up to \( E_v \), with gaps indicating holes
• Band Gap: Blank region of width \( E_g \)
• Conduction Band: Unshaded region above \( E_c \), with minimal electron shading
• Acceptor Level: Dashed line at \( E_a \), 0.01–0.05 eV above \( E_v \), label \( E_a - E_v \)
• Fermi Level: Dashed line at \( E_F \), between \( E_v \) and \( E_a \), closer to \( E_v \)
• Carriers: Indicate holes below \( E_v \), few electrons above \( E_c \)