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

N-type semiconductors are created by doping with donor impurities, such as phosphorus in silicon. These impurities introduce donor energy levels (\( E_d \)) situated slightly below the conduction band edge (\( E_c \)). When the temperature is above absolute zero (\( T > 0 \, \text{K} \)), electrons gain thermal energy and transition from \( E_d \) to the conduction band. This process results in electrons becoming the dominant charge carriers.

Step 1: Essential Characteristics

• Valence Band: Occupied by electrons up to the level \( E_v \).
• Conduction Band: Begins at \( E_c \). The energy gap (\( E_g \)) is the difference between \( E_c \) and \( E_v \) (e.g., 1.12 eV for silicon).
• Donor Level: \( E_d \) is located 0.01–0.05 eV below \( E_c \). The energy difference is \( E_c - E_d \approx 0.01 \text{ to } 0.05 \, \text{eV} \) (e.g., 0.045 eV for phosphorus in silicon).
• Fermi Level: \( E_F \) is positioned between \( E_d \) and \( E_c \), closer to \( E_c \).
• Charge Carriers: Primarily electrons in the conduction band, with a small number of holes in the valence band.

Step 2: Diagrammatic Representation

• Axes: A vertical energy axis is used, with units in eV.
• Valence Band: Depicted as a shaded region extending up to \( E_v \), clearly labeled.
• Band Gap: An unshaded region representing the band gap width (\( E_g \)).
• Conduction Band: An unshaded region above \( E_c \). Light shading indicates the presence of electrons.
• Donor Level: Represented by a dashed line at \( E_d \), which is 0.01–0.05 eV below \( E_c \). The energy difference \( E_c - E_d \) is labeled.
• Fermi Level: Shown as a dashed line at \( E_F \), located between \( E_d \) and \( E_c \), and closer to \( E_c \).
• Carriers: Electrons are indicated above \( E_c \), and a few holes are shown below \( E_v \).

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

P-type semiconductors are formed by doping with acceptor impurities, such as boron in silicon. These impurities create acceptor energy levels (\( E_a \)) positioned slightly above the valence band edge (\( E_v \)). At temperatures above absolute zero (\( T > 0 \, \text{K} \)), electrons from the valence band are captured by the \( E_a \) levels. This process leaves behind holes in the valence band, making holes the majority charge carriers.

Step 1: Key Features

• Valence Band: Contains holes up to \( E_v \).
• Conduction Band: Starts at \( E_c \), separated by the band gap \( E_g \).
• Acceptor Level: \( E_a \) is situated 0.01–0.05 eV above \( E_v \). The energy difference is \( E_a - E_v \approx 0.01 \text{ to } 0.05 \, \text{eV} \) (e.g., 0.045 eV for boron in silicon).
• Fermi Level: \( E_F \) is located between \( E_v \) and \( E_a \), closer to \( E_v \).
• Carriers: Predominantly holes in the valence band, with a small number of electrons in the conduction band.

Step 2: Diagram Description

• Axes: A vertical energy axis is displayed, with values in eV.
• Valence Band: Shown as a shaded region up to \( E_v \), with clear indications of holes as gaps within the shading.
• Band Gap: An unshaded area representing the band gap (\( E_g \)).
• Conduction Band: An unshaded region starting at \( E_c \). Minimal electron shading is present.
• Acceptor Level: Indicated by a dashed line at \( E_a \), 0.01–0.05 eV above \( E_v \). The energy difference \( E_a - E_v \) is labeled.
• Fermi Level: Represented by a dashed line at \( E_F \), positioned between \( E_v \) and \( E_a \), and closer to \( E_v \).
• Carriers: Holes are indicated below \( E_v \), and a few electrons are shown above \( E_c \).