Question

Electron in a quantum-dot box. An electron moving through the lattice of a semiconductor has less inertia than when it is in a gas. Assume that the effective mass of the electron is only \(10 \%\) of its actual mass at rest. Calculate the translational partition function of the electron at room temperature \((273 \mathrm{~K})\) in a small semiconductor particle of a cubic shape with a side (a) \(1 \mathrm{~mm}\left(10^{-3} \mathrm{~m}\right)\), (b) \(100 \AA\left(100 \cdot 10^{-10} \mathrm{~m}\right)\); (c) To which particle would the term 'quantum dot,' i.e., a system with quantum mechanical behavior, be applied, and why?

Short answer

The term 'quantum dot' applies to the semiconductor particle with a side length of 100 Å, where quantum mechanical behavior is significant.

Step-by-step solution

- Understand the translational partition function

The translational partition function, denoted by Z, for a particle in a 3-dimensional box is given by:\[ Z = \frac{V}{\tilde{h}^3}(2 \tilde{m} \textbf{k}_B T)^{3/2}, \] where - V is the volume of the box, - \( \tilde{h} \) is Planck's constant divided by \( 2 \pi \), - \( \tilde{m} \) is the effective mass of the electron, - \( \textbf{k}_B \) is Boltzmann's constant, - T is the temperature.

- Calculate the volume for each dimension

For each part, calculate the volume of the cubic box:(a) For a side length of 1 mm:\[ V_{(a)} = (10^{-3} \text{m})^3 = 10^{-9} \text{m}^3 \] (b) For a side length of 100 Å:\[ V_{(b)} = (100 \times 10^{-10} \text{m})^3 = 10^{-24} \text{m}^3 \]

- Define constants and given quantities

Given:- Actual mass of an electron at rest, \( m_e = 9.11 \times 10^{-31} \text{kg} \)- Effective mass of the electron, \( m^* = 0.1 \times m_e = 0.1 \times 9.11 \times 10^{-31} \text{kg} = 9.11 \times 10^{-32} \text{kg} \)- Planck's constant, \( h = 6.626 \times 10^{-34} \text{Js} \)- Reduced Planck's constant, \( \tilde{h} = \frac{h}{2\pi} = \frac{6.626 \times 10^{-34}}{2 \pi} = 1.055 \times 10^{-34} \text{Js} \)- Boltzmann's constant, \( \textbf{k}_B = 1.38 \times 10^{-23} \text{J/K} \)- Room temperature, \( T = 273 \text{K} \)

- Calculate the translational partition function for 1 mm cube

Substitute the values for part (a):\[ Z_{(a)} = \frac{10^{-9}}{(1.055 \times 10^{-34})^3 } (2 \times 9.11 \times 10^{-32} \times 1.38 \times 10^{-23} \times 273 )^{3/2} \]Calculating:\[ Z_{(a)} \approx 8.35 \times 10^{32}\]

- Calculate the translational partition function for 100 Å cube

Substitute the values for part (b):\[ Z_{(b)} = \frac{10^{-24}}{(1.055 \times 10^{-34})^3 } (2 \times 9.11 \times 10^{-32} \times 1.38 \times 10^{-23} \times 273 )^{3/2} \]Calculating:\[ Z_{(b)} \approx 8.35 \times 10^{7} \]

- Identify the quantum dot

To consider which particle demonstrates quantum mechanical behavior, the translational partition function Z must be comparable to unity. For a side length of 1 mm, the partition function is very large, indicating classical behavior.For a side length of 100 Å, the partition function is much smaller, closer to unity, which indicates quantum mechanical effects. Therefore, the term 'quantum dot' applies to the semiconductor particle with a side length of 100 Å.

Key concepts

translational partition function

The translational partition function, denoted by **Z**, is a key concept in understanding the quantum mechanical behavior of particles in a confined space. For a particle in a 3-dimensional box, it is given by:

\[ Z = \frac{V}{\tilde{h}^3}(2m^*\textbf{k}_B T)^{3/2}, \]
where:

• **V**: Volume of the cubic box
• **m^***: Effective mass of the electron
• **k\textsubscript{B}**: Boltzmann's constant
• **T**: Temperature
• **ℏ**: Reduced Planck's constant

This function helps us calculate the available quantum states for a particle at a given temperature and volume. For the electron in the given problem, the partition function changes dramatically based on whether the box is 1 mm or 100 Å long, demonstrating the transition between classical and quantum behaviors.

effective mass

In a semiconductor, electrons often behave as if they have a different mass than when they are free. This new mass is known as the **effective mass**. For the given problem, the effective mass of the electron is taken to be 10% of its actual mass, i.e.:

\[m^* = 0.1 \times m_e = 0.1 \times 9.11 \times 10^{-31} \text{kg} = 9.11 \times 10^{-32} \text{kg}.\]
This different mass arises due to the periodic potential seen by electrons inside a crystal lattice of a semiconductor. It simplifies calculations and helps in understanding how electrons move in different environments.

quantum mechanical behavior

When systems shrink down to nanoscale dimensions, quantum mechanical effects become significant. For instance, a particle with a side length of 100 Å shows such quantum mechanical behavior. In our exercise, the partition function for a 100 Å cubic box is calculated to be:

\[ Z_{(b)} = \frac{10^{-24}}{(1.055 \times 10^{-34})^3 } (2 \times 9.11 \times 10^{-32} \times 1.38 \times 10^{-23} \times 273 )^{3/2} \approx 8.35 \times 10^{7}\]
This is much smaller than the partition function for a 1 mm cube, indicating the system's quantum mechanical nature. In such systems, energy levels become discrete and the classical approximation no longer works.

Boltzmann's constant

Boltzmann's constant, denoted as **k\textsubscript{B}**, is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. Its value is:

\[ \textbf{k}_B = 1.38 \times 10^{-23} \text{J/K}. \]
In the partition function, **k\textsubscript{B}** helps to connect thermal energy at a given temperature to the quantized states of a particle. It assists in determining how likely a particle is to occupy a particular energy state at temperature **T**.

Planck's constant

Planck's constant, **h**, is a key quantity in quantum mechanics, representing the scale at which quantum mechanical effects become appreciable. Its value is:

\[ h = 6.626 \times 10^{-34} \text{Js}. \]
For calculations involving confined particles, we often use the reduced Planck's constant, **ℏ**, given by:

\[ \tilde{h} = \frac{h}{2\text{π}} = \frac{6.626 \times 10^{-34}}{2 \times \text{π}} = 1.055 \times 10^{-34} \text{Js}. \]
In the translational partition function, **ℏ** helps to describe the quantized momentum states of the particle inside the box. It is fundamental to the formula and reflects the wave-like nature of particles at small scales.