page_title: Quantum point contact
section_title: Properties
hierarchical_section_title: Quantum point contact / Properties
caption_attribution_description: English: Balistic Transport
context_page_description: A quantum point contact is a narrow constriction between two wide electrically conducting regions, of a width comparable to the electronic wavelength.
  The importance of QPC lies in the fact that they prove quantisation of ballistic conductance in mesoscopic systems. The conductance of QPC is quantized in units of, the so called conductance quantum.
  Quantum point contacts were first reported in 1988 by a Dutch team from Delft University of Technology and Philips Research and, independently, by a British team from the Cavendish Laboratory. They are based on earlier work by the British group which showed how split gates could be used to convert a two-dimensional electron gas into one-dimension, first in silicon and then in gallium arsenide
  This quantisation is reminiscent of the quantisation of the Hall conductance, but is measured in the absence of a magnetic field. The zero-field conductance quantisation and the smooth transition to the quantum Hall effect on applying a magnetic field are essentially consequences of the equipartition of current among an integer number of propagating modes in the constriction.
context_section_description: 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. 
   
  It is most common to study QPC in two dimensional electron gases (2DEG). This way the geometric constriction of the point contact turns the conductance trough the opening to a 1-dimensional system. Moreover, it requires a quantum mechanical description of the system that results in the quantisation of conductance. Quantum mechanically, the current through the point contact is equipartitioned among the 1D subands, or transverse modes, in the constriction. 
  It is important to state that the previous discussion does not take into account possible transitions among modes. Landauer's formula can actually be generalised to express this possible transitions , where  is the transition matrix which incorporates non-zero probabilities of transmission from mode n to m.  
  At low temperatures and voltages, unscattered and untrapped 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 waveguide analogy is applicable as long as coherence is not lost through scattering, e.g., by a defect or trapping site. The electron wave can only pass through the constriction if it interferes constructively, which for a given width 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 .
  On   increasing   the   temperature,   one   finds   experimentally   that   the   plateaux acquire a finite slope until they are no longer resolved. This is a consequence of the  thermal  smearing  of  the  Fermi-Dirac  distribution.    The  conductance   steps should disappear for T >∼ ∆E/4kB ∼ 4 K (here ∆E is the subband splitting at theFermi level). This is confirmed both by experiment and by numerical calculations. 
  
  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.
caption: Classical ballistic transport through a point contact induced by a concentration difference.
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