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| Landauer formula | |
|---|---|
| Name | Landauer formula |
| Caption | Schematic of quantum transport through a conductor connected to reservoirs |
| Field | Condensed matter physics, Mesoscopic physics, Nanotechnology |
| Introduced | 1957 |
| Introduced by | Rolf Landauer |
| Key concepts | Quantum conductance, Scattering theory, Transmission probability |
Landauer formula The Landauer formula expresses electrical conductance of a phase-coherent conductor in terms of quantum scattering, relating conductance to transmission probabilities between reservoirs. It bridges microscopic quantum mechanics and measurable macroscopic transport, influential across mesoscopic physics, nanoelectronics, and quantum information science. It underpins theoretical and experimental studies in fields ranging from semiconductor physics to molecular electronics, shaping research at institutions such as Bell Labs, IBM, MIT, and Cambridge University.
The Landauer formula originated in work by Rolf Landauer and was developed alongside contributions from researchers at IBM Research, Harvard University, and University of Tokyo. Influential contemporaries include David Thouless, Philipp Anderson, Pierre-Gilles de Gennes, and Leo Kadanoff. The formula reframes conduction problems involving contacts to reservoirs such as those studied at Bell Labs and Stanford University in the language used by Max Born and Douglas Hartree. It provided a paradigm shift for experiments at CERN-scale precision analogues in condensed matter laboratories and informed theoretical frameworks taught at Princeton University and Yale University.
The Landauer approach builds on quantum scattering theory developed by Werner Heisenberg, Erwin Schrödinger, and John von Neumann, and borrows mathematical tools used in work by John Bardeen, Walter Brattain, and William Shockley. It connects to the concept of conductance quanta observed in experiments inspired by techniques from Nobel Prize-winning laboratories and is related to the Kubo formula treatments by Ryogo Kubo and linear response methods advanced by Lev Landau. Formal apparatus involves Green’s functions similar to those used by Julian Schwinger and scattering matrices formalized by Vladimir Fock and Hendrik Casimir.
Derivations of the Landauer formula rely on scattering matrices (S-matrix) introduced in contexts associated with Enrico Fermi and Richard Feynman, and on transmission coefficients akin to analyses by Maxwell Planck in radiative contexts. The standard derivation treats reservoirs as thermal baths analogous to treatments by Ludwig Boltzmann and Josiah Willard Gibbs, and uses mode counting related to work by Hendrik Lorentz and Albert Einstein. Alternative derivations invoke non-equilibrium Green’s functions echoing techniques from Leonid Keldysh and many-body frameworks influenced by Julian Schwinger and Gerald D. Mahan.
The Landauer formula is applied in contexts including quantum point contacts studied in experiments at IBM Thomas J. Watson Research Center and Bell Labs, single-molecule junctions probed in setups like those at University of Cambridge and California Institute of Technology. It informs the design of devices in Intel Corporation-scale research, relates to topological insulator transport investigated at Microsoft Research and University of Oxford, and connects to mesoscopic superconductivity explored by groups at Argonne National Laboratory and Los Alamos National Laboratory. The formula is central to nanoscale thermoelectrics pursued in collaborations involving National Renewable Energy Laboratory and to quantum Hall effect studies dating to work at Grenoble and Bell Labs.
Generalizations extend to interacting systems via approaches developed by Kenneth Wilson and Phillip Anderson and to time-dependent driving influenced by techniques from Sin-Itiro Tomonaga and Julian Schwinger. Extensions connect to topological transport theories advanced by D. J. Thouless, F. Duncan M. Haldane, and Frank Wilczek, and to non-equilibrium steady-state frameworks associated with Göran Lindblad and Lars Onsager. Multi-terminal and spin-dependent generalizations draw on spintronics research from S. A. Wolf and Albert Fert, while thermoelectric and stochastic extensions relate to work by Rolf Landauer and later authors connected to Ilya Prigogine.
Quantized conductance steps corresponding to Landauer predictions were observed in quantum point contacts in experiments carried out at University of Cambridge and Bell Labs by groups influenced by Herbert Kroemer and Leo Esaki. Single-atom and single-molecule junction experiments at IBM and Max Planck Society laboratories have validated transmission-based conductance descriptions. Precision measurements in two-dimensional electron gas systems performed at Bell Labs, NIST, and University of Manchester corroborated the role of transmission channels in conductance, while cold-atom analogues of transport explored at MIT and Rice University have provided complementary tests.
Critiques focus on applicability when strong electron–electron interactions or inelastic scattering dominate, issues highlighted in debates involving researchers at Los Alamos National Laboratory and Argonne National Laboratory. The role of contact resistance and reservoir modeling—subjects of discussion at Bell Labs and IBM Research—challenges naive application in molecular junctions studied by groups at Columbia University and University of California, Berkeley. Alternative frameworks such as Kubo linear response and many-body perturbation theory developed at Harvard University and Princeton University are often invoked where Landauer assumptions break down.
Category:Condensed matter physics Category:Quantum transport