Conductance quantum
The conductance quantum, denoted by the symbol G0, is the quantized unit of electrical conductance. It is defined by the elementary charge e and Planck constant h as:
- <math>G_0 = \frac{2 e^2}{h}</math> = 7.748091729...×10−5 S.<ref group="Note">S is the siemens</ref><ref name="physconst-G0">"2018 CODATA Value: conductance quantum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.</ref>
It appears when measuring the conductance of a quantum point contact, and, more generally, is a key component of the Landauer formula, which relates the electrical conductance of a quantum conductor to its quantum properties. It is twice the reciprocal of the von Klitzing constant (2/RK).
Note that the conductance quantum does not mean that the conductance of any system must be an integer multiple of G0. Instead, it describes the conductance of two quantum channels (one channel for spin up and one channel for spin down) if the probability for transmitting an electron that enters the channel is unity, i.e. if transport through the channel is ballistic. If the transmission probability is less than unity, then the conductance of the channel is less than G0. The total conductance of a system is equal to the sum of the conductances of all the parallel quantum channels that make up the system.<ref>S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, 1995, ISBN 0-521-59943-1</ref>
Derivation
In a 1D wire, connecting two reservoirs of potential <math>u_1</math> and <math>u_2</math> adiabatically:
The density of states is <math display="block">\frac{\mathrm{d}n}{\mathrm{d} \epsilon} = \frac{2}{hv} ,</math> where the factor 2 comes from electron spin degeneracy, <math>h</math> is the Planck constant, and <math>v</math> is the electron velocity.
The voltage is: <math display="block">V = -\frac{(\mu_1 - \mu_2)}{e} ,</math> where <math>e</math> is the electron charge.
The 1D current going across is the current density: <math display="block">j = -ev(\mu_1-\mu_2) \frac{\mathrm{d}n}{\mathrm{d} \epsilon} .</math>
This results in a quantized conductance: <math display="block">G_0 = \frac{I}{V} = \frac{j}{V} = \frac{2e^2}{h} .</math>
Occurrence
Quantized conductance occurs in wires that are ballistic conductors, when the elastic mean free path is much larger than the length of the wire: <math>l_{\rm el} \gg L </math>[clarification needed]. B. J. van Wees et al. first observed the effect in a point contact in 1988.<ref>B.J. van Wees; et al. (1988). "Quantized Conductance of Point Contacts in a Two-Dimensional Electron Gas". Physical Review Letters. 60 (9): 848–850. Bibcode:1988PhRvL..60..848V. doi:10.1103/PhysRevLett.60.848. hdl:1887/3316. PMID 10038668.</ref> Carbon nanotubes have quantized conductance independent of diameter.<ref>S. Frank; P. Poncharal; Z. L. Wang; W. A. de Heer (1998). "Carbon Nanotube Quantum Resistors". Science. 280 (1744–1746): 1744–6. Bibcode:1998Sci...280.1744F. CiteSeerX 10.1.1.485.1769. doi:10.1126/science.280.5370.1744. PMID 9624050.</ref> The quantum hall effect can be used to precisely measure the conductance quantum value. It also occurs in electrochemistry reactions<ref>Bueno, P. R. (2020). "Electron transfer and conductance quantum". Physical Chemistry Chemical Physics. 22 (45): 26109–26112. Bibcode:2020PCCP...2226109B. doi:10.1039/D0CP04522E. PMID 33185207. S2CID 226853811.</ref> and in association with the quantum capacitance defines the rate with which electrons are transferred between quantum chemical states as described by the quantum rate theory.
