Arsenic has five valence electrons. If arsenic is used as a dopant in silicon, compute (a) the ionization energy and (b) the orbit radius of the fifth arsenic electron. The effective mass for electrons in silicon is 0.2 m. (c) What is the ratio of the ionization energy of the fifth electron to the energy gap in silicon?
Gallium has three valence electrons. If gallium is used to dope germanium, compute (a) the ionization energy of the hole and (b) the orbit radius of the hole. The effective mass of holes in germanium is 0.34 m.
What type of semiconductor is obtained if silicon is doped with (a) aluminum and (b) phosphorus? (See Appendix C for the electron configurations of these elements.)
The correct answer and explanation is :
Arsenic Doping in Silicon:
Arsenic has five valence electrons, and when used as a dopant in silicon (Si), it introduces an extra electron, which is typically in the conduction band. This type of doping creates an n-type semiconductor because the majority charge carriers are electrons.
(a) Ionization Energy of the Fifth Arsenic Electron:
The ionization energy for the extra electron in arsenic can be approximated using the Bohr model. The formula for the ionization energy $E$ is given by:
$$
E = \frac{13.6 \, \text{eV} \cdot m_{\text{eff}}}{m_e} \cdot \left(\frac{Z^2}{n^2}\right)
$$
Where:
- $13.6 \, \text{eV}$ is the Rydberg energy for hydrogen.
- $m_{\text{eff}}$ is the effective mass of the electron in silicon.
- $m_e$ is the mass of the electron.
- $Z$ is the atomic number of arsenic (which is 33 for arsenic).
- $n$ is the principal quantum number (for the outermost electron in arsenic, $n = 1$).
In the case of doping, the effective mass is $m_{\text{eff}} = 0.2 m$ where $m$ is the free electron mass.
(b) Orbit Radius of the Fifth Arsenic Electron:
The orbit radius can be computed using the Bohr model for an electron in a hydrogen-like atom:
$$
r = \frac{4 \pi \epsilon_0 \hbar^2}{m_{\text{eff}} e^2} \cdot \frac{n^2}{Z}
$$
Where:
- $\epsilon_0$ is the permittivity of free space.
- $\hbar$ is the reduced Planck’s constant.
- $e$ is the elementary charge.
For the electron in arsenic, we substitute the relevant values to find the orbit radius.
(c) Ionization Energy to Energy Gap Ratio:
The energy gap for silicon is approximately 1.1 eV. The ratio of ionization energy to energy gap is:
$$
\text{Ratio} = \frac{\text{Ionization Energy}}{\text{Energy Gap}}
$$
Gallium Doping in Germanium:
Gallium has three valence electrons, and when it is used as a dopant in germanium, it creates “holes” (missing electrons) that act as positive charge carriers. This type of doping produces a p-type semiconductor.
(a) Ionization Energy of the Hole:
Similar to the arsenic case, the ionization energy of the hole in germanium can be approximated using a similar model. The ionization energy of a hole is related to the effective mass of the hole, which in the case of germanium is $0.34 m$. The formula is similar to the one used for the electron in arsenic.
(b) Orbit Radius of the Hole:
For holes, we can use the Bohr model to estimate the orbit radius, substituting the effective mass of the hole in germanium.
Semiconductor Types for Silicon Doping:
- Aluminum Doping in Silicon (p-type Semiconductor):
Aluminum has three valence electrons. When aluminum is used to dope silicon, it creates an acceptor level, leaving a hole in the silicon lattice. Since holes behave as positive charge carriers, this doping results in a p-type semiconductor. - Phosphorus Doping in Silicon (n-type Semiconductor):
Phosphorus has five valence electrons. When phosphorus is used to dope silicon, it introduces an extra electron into the conduction band. This results in an n-type semiconductor, where electrons are the majority charge carriers.
Conclusion:
- Arsenic in silicon creates an n-type semiconductor by introducing extra electrons.
- Gallium in germanium creates a p-type semiconductor by introducing holes.
- Aluminum in silicon results in a p-type semiconductor, while phosphorus in silicon creates an n-type semiconductor.
These dopants alter the electrical properties of the semiconductors, allowing control over conductivity, which is fundamental in electronic devices like transistors.