| Landauer–Büttiker formalism | |
|---|---|
| Name | Landauer–Büttiker formalism |
| Field | Condensed matter physics |
| Introduced | 1957 (Landauer); 1986 (Büttiker extension) |
| Developers | Rolf Landauer; Markus Büttiker |
| Related | Quantum transport, Scattering theory, Mesoscopic physics |
Landauer–Büttiker formalism
The Landauer–Büttiker formalism is a theoretical framework in Quantum transport that relates electronic conductance to quantum-mechanical scattering properties of a conductor. It provides a way to compute electrical conductance from transmission probabilities between reservoirs, explaining phenomena such as conductance quantization and nonlocal transport in mesoscopic systems. The formalism is foundational for understanding transport in nanostructures, quantum point contacts, and devices studied in mesoscopic physics and nanotechnology.
The formalism originated with Rolf Landauer's proposal in 1957 that conductance can be expressed in terms of transmission through a scatterer, later refined by several authors and generalized by Markus Büttiker in the 1980s to multi-terminal geometries. The Landauer picture unified ideas from Scattering theory and the physics of electronic reservoirs such as those realized in experiments at Bell Labs and later at institutions like IBM Research and Bell Labs-era collaborators. The Büttiker extension addressed coherent transport in multiterminal devices and incorporated concepts of phase coherence from studies of the Aharonov–Bohm effect and weak localization. Key early papers include Landauer's work and Büttiker's 1986 multiprobe theory, which together underpin modern treatments of quantum conductance.
The formalism rests on quantum coherent transport and the assumption that current flows between large thermal reservoirs described by Fermi–Dirac distributions. It combines single-particle Schrödinger equation scattering states with boundary conditions provided by reservoirs like those modeled in the Fermi gas approximation. Central assumptions include phase coherence over the conductor length, absence (in simplest form) of inelastic scattering inside the sample, and reservoirs that equilibrate carriers. The approach connects to the Landauer formula for two-terminal conductance and to the concept of transmission eigenchannels, linking to mathematical tools such as the S-matrix and Green's functions used in the Kubo formula and non-equilibrium Green's function (NEGF) methods.
In the scattering approach, conductance is obtained from the transmission matrix elements between incoming and outgoing modes. For a single-channel, spin-degenerate case at zero temperature, the two-terminal conductance equals G = (2e^2/h) T, where T is the total transmission probability; this explains the universal conductance quantum 2e^2/h observed in quantum point contacts and quantum Hall effect edge channels. The notion of conductance quantization links to experiments by B. J. van Wees and D. A. Wharam and theory for ballistic transport in quasi-one-dimensional channels. The S-matrix formalism also describes resonant tunneling and Fabry–Pérot–like interference in nanowire and molecular junction systems.
Büttiker generalized Landauer's idea to multi-terminal conductors by assigning chemical potentials to several reservoirs and expressing current–voltage relations via transmission coefficients between terminals. The resulting linear-response relation, often called the Büttiker formula, captures nonlocal voltages and Hall measurements without invoking local resistivity. This framework is essential for interpreting experiments in mesoscopic rings, multiterminal graphene devices, and topological insulators where edge-state transport and nonlocal signatures occur. The formalism naturally incorporates symmetries such as time-reversal and the role of magnetic fields in breaking them, linking to concepts from Berry phase and topological band theory.
The original single-particle formalism has been extended to include many-body effects, disorder, and proximity-induced superconductivity. Interaction effects such as electron–electron correlations are treated via methods like scattering theory with self-energies, NEGF, and functional renormalization group approaches; these extensions address phenomena like Coulomb blockade in quantum dots and Luttinger-liquid behavior in one-dimensional conductors. Disorder and localization are incorporated through ensemble-averaged S-matrices and scaling theory of localization developed by P. W. Anderson and others. Superconducting hybrids require incorporation of Andreev reflection, described by the Bogoliubov–de Gennes formalism and leading to modifications of conductance quantified by the Blonder–Tinkham–Klapwijk (BTK) approach and superconducting extensions of Büttiker theory.
The Landauer–Büttiker framework underlies interpretation of experiments in 2DEG systems, quantum point contacts, carbon nanotubes, semiconductor nanowires, and molecular electronics junctions. It is routinely applied in characterization of conductance quantization, shot noise, and thermoelectric transport in mesoscopic conductors. Applications extend to design and analysis of nanoscale electronic components in spintronics and quantum devices investigated at institutions such as CERN collaborations on quantum sensors and university laboratories worldwide. The formalism guides device modeling in scanning tunneling microscopy (STM) transport measurements and in modern cryogenic experiments probing Majorana zero modes in proximitized nanowires.
Mathematically, the formalism uses the scattering matrix (S-matrix), transmission eigenvalues, and mode counting. Computational implementations employ tight-binding models, recursive Green's function techniques, and NEGF solvers implemented in software packages used by condensed-matter theorists and computational materials scientists. Numerical tools compute transmission coefficients from Hamiltonians derived via density functional theory (DFT) or model Hamiltonians, enabling quantitative comparison with experiments. Analytical methods exploit random-matrix theory for statistical properties of S-matrices and semiclassical approaches to connect quantum transport with classical trajectories.
Category:Quantum mechanics Category:Condensed matter physics Category:Nanotechnology