Question

The binding energy of an extra electron added when As atoms are used to dope a Si crystal may be approximately calculated by considering the Bohr model of a hydrogen atom. a) Express the ground energy of the hydrogen-like atoms in terms of the dielectric constant, the effective mass of an extra electron, and the groundstate energy of a hydrogen atom. b) Calculate the binding energy of the extra electron in a Si crystal. The dielectric constant of \(\mathrm{Si}\) 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 approximately -6.8 eV.

Step-by-step solution

Express the ground energy of hydrogen-like atoms

The ground energy of a hydrogen atom is given by the expression: \(E_0 = - \dfrac{e^2}{2 a_0}\) where \(e\) is the charge of an electron, and \(a_0\) is the Bohr radius. However, we are considering a hydrogen-like atom with a different dielectric constant \(\varepsilon\) (as opposed to vacuum in a hydrogen atom) and a different effective mass \(m^*\) (as opposed to the electron mass in a hydrogen atom). Hence, the energy will be affected by these two factors. Note that the dielectric constant appears in the denominator since it reduces the effective interaction between the electron and the nucleus. Similarly, the effective mass appears in both the numerator and the denominator. Thus, the ground energy of the hydrogen-like atom can be expressed as: \(E_1 = - \dfrac{e^2}{2 (\varepsilon a_0 m^*/m_e)}\) Now, we can express the ground energy \(E_1\) in terms of the ground state energy of a hydrogen atom \(E_0\): \(E_1 = E_0 \dfrac{m_e}{\varepsilon m^*}\)

Calculate the binding energy of the extra electron

Given the dielectric constant of Si (\(\varepsilon = 10.0\)), and the effective mass of extra electrons in a Si crystal is about \(20\%\) of that of free electrons (\(m^* = 0.2 m_e\)), we can find the binding energy of the extra electron (\(E_1\)), using the expression derived in the previous step: \(E_1 = E_0 \dfrac{m_e}{\varepsilon m^*} = E_0 \dfrac{m_e}{(10)(0.2 m_e)} = \dfrac{E_0}{2}\) We know that the ground state energy of a hydrogen atom is \(E_0 = -13.6\, eV\). Therefore, the binding energy of the extra electron in a Si crystal is: \(E_1 = \dfrac{-13.6\, eV}{2} = -6.8\, eV\) Thus, the binding energy of the extra electron in a Si crystal is approximately \(-6.8\, eV\).

Key concepts

Bohr Model

The Bohr model revolutionized understanding of atomic structure by introducing the concept of discrete orbits in which electrons circulate around the nucleus without radiating energy, contrary to classical physics. It predicts that the electron in a hydrogen atom resides in the orbit with the lowest energy called the ground state.

The energy level of an electron in the ground state is quantified by the formula:
\[\begin{equation}E_0 = - \frac{e^2}{2 a_0}\end{equation}\]
where e is the charge of the electron, and a_0 is the Bohr radius. This model, despite its simplicity, provides a foundational framework for understanding more complex atomic systems. It's especially useful in semiconductor doping, where impurity atoms are added to alter electrical properties. The effective mass and dielectric constant are crucial to adjusting the Bohr model for materials beyond hydrogen, like silicon in our exercise.

Dielectric Constant

The dielectric constant, symbolized as \(\varepsilon\), is a property that measures the ability of a material to reduce the effective force between electric charges. It's essentially a scaling factor in Coulomb's Law that impacts the energy levels within an atom when applied to modified versions of the Bohr model.

In the context of semiconductor doping, where an element like arsenic adds extra electrons to a silicon crystal, the dielectric constant modifies the electrical field interaction. The greater the dielectric constant of a material, the lower the energy between charges at a given distance. Therefore, when calculating the ground state energy of an electron in a doped semiconductor, this constant plays a critical role in determining the binding energy.

Effective Mass

When it comes to the physics of semiconductors, the effective mass (m^*) is far more than a simple extension of Newton's second law. It accounts for the response of an electron moving through a periodic lattice of atoms. Unlike a free electron, the electron in a semiconductor is influenced by the periodic potential of the crystal lattice, which modifies its dynamics.

The effective mass reflects how the electron accelerates under an applied force, thereby influencing its kinetic energy and other quantum properties such as the density of states. For our particular problem, the effective mass is a vital component, adjusting the Bohr model to predict the energy levels of electrons in silicon rather than in a vacuum.

Binding Energy

The term 'binding energy' in the context of an atom refers to the energy required to remove an electron from its orbit around the nucleus. It serves as an indicator of how strongly the electron is bound to the atom. A higher binding energy implies it is more difficult to free the electron.

In semiconductors, evaluating the binding energy of dopant electrons, like those from arsenic in a silicon lattice, is essential. It affects their mobility and hence their contribution to the conductivity of the material. The example problem demonstrates how sharpening the Bohr model with considerations such as the dielectric constant and the effective mass allows us to calculate the binding energy in a non-vacuum environment.

Ground State Energy

Electrons in atoms exist in various energy levels; the lowest of these is known as the ground state energy. It's the most stable, non-excitable state of the electron, and it has profound implications for chemical reactivity, ionization, and energy emission.

In a pure hydrogen atom, this ground state energy is well defined. For a doped semiconductor, however, calculating the ground state energy of additional electrons introduced via doping requires adjustments to the classical formula. Factors such as the dielectric constant and the effective mass come into play, altering the energy value. Understanding the adjusted ground state energy can provide insight into the electrical characteristics of the semiconductor material.