| Bardeen's tunneling theory | |
|---|---|
| Name | Bardeen's tunneling theory |
| Author | John Bardeen |
| Field | Quantum mechanics |
| Introduced | 1961 |
| Related | Josephson effect, Scanning tunneling microscopy, Quantum tunneling |
Bardeen's tunneling theory
Bardeen's tunneling theory is a perturbative formalism for calculating tunneling currents between quantum systems separated by a potential barrier. Developed by John Bardeen in 1961, it provides a practical method to compute the tunneling matrix element and current for electronic junctions, laying groundwork for quantitative understanding of tunnel junctions, superconductor interfaces, and techniques such as scanning tunneling microscopy (STM). Its importance lies in connecting microscopic wavefunctions to measurable transport properties in mesoscopic and condensed matter systems.
Bardeen introduced his theory amid post-war advances in solid-state physics and the rising interest in electronic devices that exploit quantum effects. The work followed earlier conceptual developments in quantum tunneling and practical demonstrations of tunneling in metal–insulator–metal structures and Josephson junctions. Motivations included explaining tunneling between dissimilar electrodes where direct solution of the full barrier problem was intractable, and providing a bridge between band-structure calculations performed independently for each electrode. The formalism complemented contemporary research at institutions such as Bell Labs and universities where practitioners studied superconductivity (notably Bardeen's later work with BCS theory) and tunneling spectroscopy.
Bardeen's approach is perturbative and based on time-dependent perturbation theory applied to two weakly coupled subsystems. Each electrode is treated as an independent quantum system with its own set of eigenstates computed in the absence of coupling. The barrier is assumed sufficiently high or wide that direct overlap of eigenstates is small; coupling is introduced via a tunneling Hamiltonian. Key assumptions include the validity of single-particle descriptions for quasiparticles (electrons or Bogoliubov quasiparticles in superconductivity), negligible inelastic processes within the barrier, and conservation of energy in the tunneling event (elastic tunneling). The theory naturally accommodates mean-field descriptions such as BCS theory for superconductors and single-particle density functional theory derived states for normal electrodes.
Bardeen's formalism partitions the full Hamiltonian H = H_L + H_R + H_T, where H_L and H_R are the unperturbed Hamiltonians of the left and right electrodes and H_T is the tunneling term. The tunneling matrix element M_{mn} between a state ψ_m in the left electrode and ψ_n in the right electrode is given by a surface integral over a plane within the barrier region:
Bardeen's expression for M_{mn} is widely used to model transport in tunnel junctions, SIS junctions, and NIS junctions. In STM theory, the Tersoff–Hamann approximation adapts Bardeen's result by modeling the tip state as an s-wave probe and relating the tunneling current to the local density of states (LDOS) of the sample at the tip position; this connects to experimental STM imaging and spectroscopy performed in laboratories such as those at IBM Research and university nanoscale facilities. The framework underlies analysis of IETS, Andreev reflection, and spin-polarized tunneling measurements performed with magnetic tips and junctions. Practical computations often combine Bardeen matrix elements with electronic structure outputs from density functional theory packages and Green's function transport methods like the Non-equilibrium Green's function formalism.
Extensions of Bardeen's theory address inelastic tunneling, finite barrier dynamics, and strong coupling regimes. The tunneling Hamiltonian approach was formalized by others to include electron–phonon and electron–magnon interactions, and was integrated with Keldysh formalism and many-body techniques for non-equilibrium transport. Limitations arise when the barrier is thin or transparency is high, where perturbation theory breaks down and exact scattering approaches or Bogoliubov–de Gennes treatments are required. Developments in mesoscopic physics and quantum transport introduced Landauer–Büttiker and scattering-matrix perspectives that complement and sometimes supplant Bardeen-type perturbative calculations for ballistic and multi-channel conductors.
Quantitative comparisons between Bardeen-based calculations and experiments include tunneling conductance spectra of superconductors matching BCS-derived density of states, STM topography and spectroscopy resolving atomic-scale LDOS variations, and tunnel magnetoresistance in magnetic tunnel junctions consistent with spin-dependent matrix elements. Landmark experiments by groups at IBM, Stanford University, and University of Chicago provided high-resolution STM data and tunneling spectroscopy validating theoretical predictions. Observations such as the superconducting energy gap, coherence peaks, and subgap features from Andreev processes are routinely interpreted using Bardeen-derived tunneling formulas combined with many-body corrections, making the theory a cornerstone linking microscopic electronic structure to measurable tunneling phenomena.
Category:Quantum mechanics Category:Condensed matter physics