Quantum Point Contact - Properties

Properties

Geometrically, a quantum point contact is a constriction in the transverse direction which presents a resistance to the motion of electrons. Applying a voltage across the point contact induces a current to flow, the magnitude of this current is given by, where is the conductance of the contact. This formula resembles Ohm's law for macroscopic resistors. However there is a fundamental difference here resulting from the small system size which requires a quantum mechanical analysis.

At low temperatures and voltages, electrons contributing to the current have a certain energy/momentum/wavelength called Fermi energy/momentum/wavelength. Much like in a waveguide, the transverse confinement in the quantum point contact results in a "quantization" of the transverse motion—the transverse motion cannot vary continuously, but has to be one of a series of discrete modes. The electron wave can only pass through the constriction if it interferes constructively which for a given size of constriction only happens for a certain number of modes . The current carried by such a quantum state is the product of the velocity times the electron density. These two quantities by themselves differ from one mode to the other, but their product is mode independent. As a consequence, each state contributes the same amount per spin direction to the total conductance

.

This is a fundamental result; the conductance does not take on arbitrary values but is quantized in multiples of the conductance quantum, which is expressed through the electron charge and the Planck constant . The integer number is determined by the width of the point contact and roughly equals the width divided by half the electron wavelength. As a function of the width of the point contact (or gate voltage in the case of GaAs/AlGaAs heterostructure devices), the conductance shows a staircase behavior as more and more modes (or channels) contribute to the electron transport. The step-height is given by .

An external magnetic field applied to the quantum point contact lifts the spin degeneracy and leads to half-integer steps in the conductance. In addition, the number of modes that contribute becomes smaller. For large magnetic fields, is independent of the width of the constriction, given by the theory of the quantum Hall effect. An interesting feature, not yet fully understood, is a plateau at, the so-called 0.7-structure.