Question

A quantum mechanical device known as the Josephson junction consists of two overlapping layers of superconducting metal (for example, aluminum at \(1.00 \mathrm{~K}\) ) separated by \(20.0 \mathrm{nm}\) of aluminum oxide, which has a dielectric constant of \(9.10 .\) If this device has an area of \(100 . \mu \mathrm{m}^{2}\) and a parallel plate configuration, estimate its capacitance.

Short answer

Question: Estimate the capacitance of a Josephson junction in a parallel plate configuration with a 100 µm² area, 20 nm distance between plates, and made of aluminum oxide with a dielectric constant of 9.10. Answer: The capacitance of the Josephson junction is approximately \(8.06 \times 10^{-14} F\).

Step-by-step solution

Convert the given area

The given area is 100 µm², so we need to convert it to meters squared. Recall that 1 µm = 10^(-6) m, so 100 µm² = 100 x 10^(-12) m².

Calculate the permittivity of aluminum oxide

To find the permittivity of aluminum oxide (\(\epsilon\)), we multiply the vacuum permittivity constant (\(\epsilon_0 = 8.85 \times 10^{-12} Fm^{-1}\)) with the dielectric constant (9.10). Thus, \(\epsilon = 9.10 \times 8.85 \times 10^{-12} Fm^{-1}\).

Calculate the capacitance

Now, we can use the formula for the capacitance of a parallel plate capacitor with a dielectric: \(C =\epsilon A / d\) Plug in the values we found: \(C = (9.10 \times 8.85 \times 10^{-12} Fm^{-1}) (100 \times 10^{-12} m^2) / (20.0 \times 10^{-9}m)\) \(C = 8.0559 \times 10^{-14} F\)

Estimate the capacitance

The capacitance of the Josephson junction is approximately \(8.06 \times 10^{-14} F\).

Key concepts

Quantum Mechanics

Quantum mechanics is the fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces complex concepts such as wave-particle duality, quantization, and entanglement.

The Josephson junction, an essential quantum mechanical device used in this exercise, exploits the quantum mechanical phenomenon of superconductivity to produce effects that can be applied in precision measurement and quantum computing. Quantum mechanics shows that electrons in superconductors can tunnel through an insulating barrier, leading to a measurable supercurrent that is sensitive to voltage, magnetic field, and temperature.

Superconducting Metals

Superconducting metals, such as the aluminum used in the Josephson junction from our exercise, have the extraordinary property of conducting electricity without resistance when cooled below a certain critical temperature. At about 1.00 K, aluminum transitions into this superconducting state.

Superconductors are characterized by the Meissner effect, where they expel magnetic fields, and the presence of Cooper pairs—bound states of electrons that move effortlessly through the lattice structure of the metal without the typical scattering that causes electrical resistance in conductors.

Dielectric Constant

The dielectric constant, also known as the relative permittivity, is a measure of how much a material can resist the electric field, reducing the electric force and potential energy between charged entities. A higher dielectric constant indicates a strong ability to store electrical energy in an electric field.

In the Josephson junction, the dielectric constant of aluminum oxide (\(9.10\)) dictates the effectiveness of this insulating layer in storing charge. It effectively enhances the junction's capacitance by allowing it to store more charge at a given voltage than it could in a vacuum.

Parallel Plate Capacitor

A parallel plate capacitor is a simple device consisting of two conductive plates separated by an insulating material—a dielectric. It's used to store electric energy in the form of an electrostatic field between the plates.

The capacitance of such a capacitor, represented by the symbol C, is determined by the formula \(C = \frac{\epsilon A}{d}\) where \(\epsilon\) is the permittivity of the dielectric material, \(A\) is the area of the plates, and \(d\) is the distance between them. In this exercise, the Josephson junction functions as a parallel plate capacitor with an enhanced capacitance due to the dielectric layer of aluminum oxide.