Question

The binding energy of an extra electron when As atoms are doped in a Si crystal may be approximately calculated by considering the Bohr model of a hydrogen atom. a) Show the ground energy of hydrogen-like atoms in terms of the dielectric constant and the ground state energy of a hydrogen atom. b) Calculate the binding energy of the extra electron in a Si crystal. (The dielectric constant of Si is about 10.0 , and the effective mass of extra electrons in a Si crystal is about \(20.0 \%\) of that of free electrons.)

Short answer

Answer: The binding energy of the extra electron in a Si crystal is 0.272 eV.

Step-by-step solution

a) Ground energy of hydrogen-like atoms

In the Bohr model of the hydrogen atom, the ground energy (n=1) can be found using the following formula: $$ E_1 = -\frac{e^2}{2a_0} $$ where \(a_0\) is the Bohr radius, which can be expressed as: $$ a_0 = \frac{\hbar^2}{m_ee^2} $$ In a hydrogen-like atom, we have a dielectric constant (\(\varepsilon\)) due to the surrounding medium and an effective mass (\(m^*\)), which is different from the mass of the actual electron (\(m_e\)). To find the ground energy of hydrogen-like atom, we need to relate the Bohr radius with the dielectric constant and the effective mass. We can write the modified Bohr radius as: $$ a^*_0 = \frac{\hbar^2\varepsilon}{m^*e^2} $$ Now, we can find the ground energy of the hydrogen-like atom by substituting \(a^*_0\) into the formula for \(E_1\): $$ E^*_1 = -\frac{e^2}{2a^*_0} = -\frac{e^2m^*e^2}{2\hbar^2\varepsilon} $$ Comparing the ground state energy of hydrogen-like atom with hydrogen atom, we have: $$ \frac{E^*_1}{E_1} = \frac{-\frac{e^2m^*e^2}{2\hbar^2\varepsilon}}{-\frac{e^2}{2a_0}} = \frac{m^*}{\varepsilon} $$ Hence, the ground energy of hydrogen-like atoms in terms of the dielectric constant and the ground state energy of a hydrogen atom is: $$ E^*_1 = -\frac{m^*}{\varepsilon}E_1 $$

b) Binding energy of the extra electron

We are given the dielectric constant of Si as \(\varepsilon = 10.0\) and the effective mass of extra electrons in the Si crystal as \(20.0 \%\) of that of free electrons. So, \(m^* = 0.20m_e\). Now, we can calculate the binding energy of the extra electron in the Si crystal using the formula obtained in part (a): $$ E^*_1 = -\frac{m^*}{\varepsilon}E_1 = -\frac{0.20m_e}{10.0}(-13.6\,\text{eV}) = \frac{13.6\,\text{eV}}{50} = 0.272\,\text{eV} $$ Hence, the binding energy of the extra electron in a Si crystal is \(0.272\,\text{eV}\).

Key concepts

Binding Energy

Binding energy is a fundamental concept that describes the energy required to remove an electron from a system, such as an atom, molecule, or solid. In the context of semiconductor physics, binding energy refers to the energy necessary to free an electron from a dopant atom within the semiconductor material. This concept is crucial when understanding how doping impurities like Arsenic (As) in Silicon (Si) affect the electrical properties of the material by adding additional carriers (electrons or holes).

The Bohr model allows us to approximate the binding energy of an electron in semiconductor materials by treating dopant-bound electrons as if they were in a hydrogen-like atom. However, unlike in a vacuum, the electron is influenced by the surrounding semiconductor crystal lattice. Factors such as the dielectric constant and the effective mass of the electron in the crystal must be taken into account to calculate the binding energy accurately.

Dielectric Constant

The dielectric constant, represented by \( \varepsilon \), is a measure of a material's ability to screen electric fields. It is an essential parameter in the study of semiconductors because it affects the force between charged particles, such as electrons and holes. The higher the dielectric constant, the more the material can reduce the effective electrostatic interaction between charges.

In the Bohr model calculation for a semiconductor, the dielectric constant is used to modify the electric field acting on the electron. This adaptation is necessary because the semiconductor environment has different electrical properties compared to the empty space where a real hydrogen atom exists. The dielectric constant effectively scales the Bohr radius and directly influences the energy levels of electrons, including their binding energies.

Effective Mass

Effective mass is an insightful concept used to describe the motion of electrons in a crystalline solid as if they were free particles. Since electrons in a semiconductor are subject to the periodic potential due to the atoms in the lattice structure, their behavior departs from that of free electrons. The effective mass incorporates these differences and allows us to use Newtonian mechanics to reason about the behavior of charge carriers in a semiconductor.

For the Bohr model calculations in doped semiconductors, we consider the effective mass \( m^* \) instead of the actual electron mass when computing binding energies. The effective mass determines how 'heavy' or 'light' an electron behaves within the crystal lattice. In Silicon (Si), the effective mass of the extra electrons introduced by the dopant is typically a fraction of the free electron mass, which significantly impacts the calculation of their energy states and binding energy.

Hydrogen-like Atom

A hydrogen-like atom refers to a simplified model system that consists of a nucleus with one electron orbiting around it, similar to a hydrogen atom. This model is beneficial for studying the behavior of electrons in more complex systems, such as doped semiconductors. By assuming the electron in the dopant atom behaves like the electron in a hydrogen atom, we can apply the well-understood Bohr model to determine properties such as energy levels and binding energies.

In semiconductor physics, comparing a dopant-bound electron to a hydrogen-like atom helps simplify the complex interactions within the solid. The modifications to the Bohr model using the dielectric constant and effective mass enable us to estimate the binding energy of an electron in the semiconductor, despite the differences between the actual semiconductor environment and the idealized conditions under which the original Bohr model operates.