Explain the concept of quantum Hall effect and its quantized conductance.

The classical Hall Effect describes the deflection of a charged particle's trajectory in the presence of a magnetic field, resulting in a voltage across the material perpendicular to both the current flow and the magnetic field. However, the Quantum Hall Effect occurs when the system's behavior deviates from classical expectations due to quantum mechanical effects.

The key features of the Quantum Hall Effect are:

Plateau in Hall Resistance: As the magnetic field strength increases, the Hall resistance (the ratio of the voltage across the material perpendicular to the current to the current itself) shows a quantized behavior. This means that the Hall resistance takes on distinct, discrete values that do not change even as the strength of the magnetic field varies. These quantized plateaus in the Hall resistance are highly accurate and robust, making them a precise standard for resistance measurements.

Zero Longitudinal Resistance: In the QHE regime, the longitudinal resistance (the resistance along the direction of the current flow) effectively becomes zero. This means that electrons can flow without experiencing any energy dissipation due to scattering, making it a potential candidate for creating extremely precise and stable electrical standards.

The quantized conductance refers to the quantization of the conductance in the Quantum Hall Effect, which is the reciprocal of resistance. The quantized conductance can be expressed using the formula:

=

=

2

ℎ

G=

V

I

=ν

h

e

2

Where:

G is the conductance.

I is the current flowing through the material.

V is the voltage across the material.

ν is a dimensionless factor that takes on discrete values.

e is the elementary charge.

ℎ

h is the Planck constant.

The integer values of

ν correspond to the quantized plateaus in the Hall resistance mentioned earlier. For example, when

=

1

ν=1, the conductance is quantized to

2

ℎ

h

e

2

, and for

=

2

ν=2, it's

2

×

2

ℎ

2×

h

e

2

, and so on. These quantized values of conductance are remarkably accurate and independent of the details of the material, making them a universal standard of metrology.

The underlying physics behind the Quantum Hall Effect is deeply rooted in the quantum mechanical behavior of electrons in a two-dimensional plane subjected to a strong magnetic field. This results in the formation of energy levels called Landau levels, which are quantized due to the magnetic field. The quantization of these energy levels leads to the quantization of the conductance and the emergence of the plateau in the Hall resistance. The behavior of electrons in these conditions is governed by the intricate interplay of quantum mechanics, electromagnetism, and condensed matter physics.