mesoscopic physics Mesoscopic physics is a subdiscipline of condensed matter physics that deals with materials of an intermediate size. These materials range in size between the nanoscale for a quantity of atoms (such as a molecule) and of materials measuring micr ...

, a quantum wire is an electrically conducting

wire Overhead power cabling. The conductor consists of seven strands of steel (centre, high tensile strength), surrounded by four outer layers of aluminium (high conductivity). Sample diameter 40 mm A wire is a flexible strand of metal. Wire is c ...

in which

quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...

effects influence the transport properties. Usually such effects appear in the dimension of nanometers, so they are also referred to as

nanowires A nanowire is a nanostructure in the form of a wire with the diameter of the order of a nanometre (10−9 metres). More generally, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less ...

Quantum effects

If the diameter of a wire is sufficiently small,

electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...

will experience

quantum confinement A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captur ...

in the transverse direction. As a result, their transverse energy will be limited to a series of discrete values. One consequence of this quantization is that the classical formula for calculating the

electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels ...

of a wire, :

R = \rho \frac,

is not valid for quantum wires (where

\rho

is the material's

resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...

l

is the length, and

A

is the cross-sectional area of the wire). Instead, an exact calculation of the transverse energies of the confined electrons has to be performed to calculate a wire's resistance. Following from the quantization of electron energy, the

electrical conductance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallel ...

(the inverse of the resistance) is found to be quantized in multiples of

2e^2/h

, where

e

is the

electron charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...

and

h

is the

Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...

. The factor of two arises from

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

degeneracy. A single

ballistic Ballistics may refer to: Science * Ballistics, the science that deals with the motion, behavior, and effects of projectiles ** Forensic ballistics, the science of analyzing firearm usage in crimes ** Internal ballistics, the study of the proc ...

quantum channel (i.e. with no internal scattering) has a conductance equal to this quantum of conductance. The conductance is lower than this value in the presence of internal scattering. The importance of the quantization is inversely proportional to the diameter of the

nanowire A nanowire is a nanostructure in the form of a wire with the diameter of the order of a nanometre (10−9 metres). More generally, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less ...

for a given material. From material to material, it is dependent on the electronic properties, especially on the effective mass of the electrons. Physically, this means that it will depend on how conduction electrons interact with the atoms within a given material. In practice,

semiconductor A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...

s can show clear conductance quantization for large wire transverse dimensions (~100 nm) because the electronic modes due to confinement are spatially extended. As a result, their Fermi wavelengths are large and thus they have low energy separations. This means that they can only be resolved at

cryogenic In physics, cryogenics is the production and behaviour of materials at very low temperatures. The 13th IIR International Congress of Refrigeration (held in Washington DC in 1971) endorsed a universal definition of “cryogenics” and “cr ...

temperatures (within a few degrees of

absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...

) where the thermal energy is lower than the inter-mode energy separation. For metals, quantization corresponding to the lowest

energy state A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The te ...

s is only observed for atomic wires. Their corresponding wavelength being thus extremely small they have a very large energy separation which makes resistance quantization observable even at room temperature.

Carbon nanotubes

Band structures computed using tight_binding_approximation_for_(6,0)_CNT_( tight_binding_approximation_for_(6,0)_CNT_(zigzag,_metal">zigzag.html"_;"title="tight_binding_approximation_for_(6,0)_CNT_(zigzag">tight_binding_approximation_for_(6,0)_CNT_(zigzag,_metallic),_(10,2)_CNT_(semiconducting)_and_(10,10)_CNT_(armchair_nanotube.html" ;"title="zigzag,_metal.html" ;"title="zigzag.html" ;"title="tight binding approximation for (6,0) CNT (zigzag">tight binding approximation for (6,0) CNT (zigzag, metal">zigzag.html" ;"title="tight binding approximation for (6,0) CNT (zigzag">tight binding approximation for (6,0) CNT (zigzag, metallic), (10,2) CNT (semiconducting) and (10,10) CNT (armchair nanotube">armchair, metallic) The carbon nanotube is an example of a quantum wire. A metallic single-walled carbon nanotube that is sufficiently short to exhibit no internal scattering (ballistic transport) has a conductance that approaches two times the

conductance quantum The conductance quantum, denoted by the symbol , is the quantized unit of electrical conductance. It is defined by the elementary charge ''e'' and Planck constant ''h'' as: :G_0 = \frac = It appears when measuring the conductance of a quantum p ...

2e^2/h

. The factor of two arises because carbon nanotubes have two spatial channels. The structure of a nanotube strongly affects its electrical properties. For a given (''n'',''m'') nanotube, if ''n'' = ''m'', the nanotube is metallic; if ''n'' − ''m'' is a multiple of 3, then the nanotube is semiconducting with a very small band gap, otherwise the nanotube is a moderate

. Thus all armchair (''n'' = ''m'') nanotubes are metallic, and nanotubes (6,4), (9,1), etc. are semiconducting.

References

{{refs Nanowire Quantum electronics Semiconductor structures Mesoscopic physics

Quantum effects

Carbon nanotubes

See also

References