J R Barker
The Boltzmann-Bloch electron transport picture which is widely used in device modelling pre-supposes the existence of electron trajectories. Not only does this aid physical intuition on transport theory but it also opens the door to powerful computational techniques from hydrodynamic models to ensemble Monte Carlo. Quantum transport theory has similarly blossomed when some kind of trajectory picture has been made possible, for example with the existence of edge states in high magnetic field, or in quadratic potential fields for which the Wigner distributions provide classical flowlines. Generally, however, the existence of interference, diffraction and tunnelling phenomena brings out the wave- or rather field-theoretic nature of quantum transport and the simplifying classical picture is lost. In hot electron theory this has been particularly acute for phenomena such as the intra-collisional field effect and other consequences of extended collisions. Although the Feynman formulation of quantum mechanics involves paths for electrons, it is their integration into a final field which is significant and the existence of the uncertainty relations and more seriously the known non-locality of quantum mechanics would seem to rule out the existence of a genuine phase-space trajectory for an electron. In fact this is not at all obvious; as first pointed out by Bohm an interpretation of quantum mechanics is possible in which electron trajectories not only exist but evolve deterministically under classical forces plus a quantum force which derives from a complicated dependence on the intensity of the wavefunction (itself represented as a real physical field-the pilot field) .
In the Bohm picture the electron momentum at any position in the pilot field is given by the gradient of the phase of the field.What makes the Bohm picture of current interest is the new possibility for manipulating individual electrons within tightly controlled geometries with injection from single-electronics systems based on ultra-small capacitative structures. This has led to renewed interest in conceptual tools for describing individual electrons. Our studies concern a critical re-evaluation of the Bohm picture as a transport theory and show that although it can be significantly reconciled with orthodox quantum transport theory it still contains serious flaws in its description of the initial value problem, non-integrable phase problems, stationary bound states, deterministic trajectories and the handling of electron transit/tunnelling times. The root cause of the problems is the initial condition in the Bohm picture: to determine an electron trajectory one must select its initial location with a probability given by the conventional quantum probability density for locating the electron in space; thereafter, the electron dynamics evolves deterministically. We have shown that a re-interpretation of the Bohm picture exists, based on a different deconstruction of the Schrodinger equation, for which these problems appear to be resolved.Individual electron trajectories occur, but they are stochastic and Bohms result represents the mean trajectory.The main problem with these approaches is that in general they involve post-processing: the trajectories may be calculated only after a full quantum state is computed.There are situations where this can be avoided. For instance the drift-diffusion theory of classical device modelling may be extended with the aid of Poisson's equation coupled to the quantum potential to obtain direct solutions of some problems. Where such approaches succeed is when the phase of the wavefunction is constant. Otherwise the phase contributes quantised anular momentum states, vortex flows and other phenomena that cannot be constructed from drift-diffusion-quantum potential (or density gradient) schemes.
In most cases of interest the Bohm picture is better understood in terrms of quantum hydrodynamics where the density-current density formalism leads to flowlines rather than particle trajectories. Quantum hydrodynamic flows are most rigorousl obtained from non-equilibrium (Keldysh) Green's function methods.
This work is intended to provide a fundamental basis for modelling single electrons in single electronic and granular limit devices.
J.R. BARKER,Quantum phase space distributions with compact support, Physica E 42, 491-496 (2010).
J.R. BARKER AND A. MARTINEZ, Vortex flows in semiconductor device quantum channels: time-dependent simulation, J. Computational Electronics, 3, 401-405, (2004)
J.R. BARKER,Bohm trajectories in quantum transport, Chapter in "Progress in Nonequilibrium Green's Functions II", M. Bonitz and D. Semkat (eds.), World Scientific Publ., Singapore, 198-213 (2003)
J.R. BARKER,Normal vortex states and their application in mesoscopic semiconductor devices, Microelectronic Engineering 63 223-231 (2002)
J.R. BARKER,On the Completeness of Quantum Hydrodynamics: Vortex Formation and the Need for Both Vector and Scalar Quantum Potentials in Device Simulation, Journal of Computational Electronics 1 17-21 (2002)
J.R. BARKER, On the Current and Density Representation of Many-Body Quantum Transport Theory, Journal of Computational Electronics 1 23-26 (2002)
J.R. WATLING, J.R. BARKER, S. ROY, Quantum Potential Corrections for Spatially Dependent Effective Masses with Application to Charge Confinement at Heterostructure Interfaces,Journal of Computational Electronics 1 279-282 (2002)
J. R. BARKER, A simple model for the quantum hydrodynamic simulation of electron transport in quantum confined structures in the presence of vortices, VLSI Design, 13, 237-244 (2001)
J. R. BARKER and J. WATLING, Simulation of enhanced interface trapping due to carrier dynamics in warped valence bands in SiGe devices, VLSI Design, 13 453-458 (2001)
J. R. BARKER , D. K. FERRY, AND R. AKIS, On the use of Bohm trajectories for interpreting quantum flows in quantum dot structures, Superlattices and microstructures, 27, 319-325 (2000)
D.K. FERRY and J. R. BARKER, Issues in general quantum transport with complex potentials, Applied Physics Letters 74 582-4 (1999)
J. R. BARKER, Trajectory-based representations of quantum transport theory and their connection with semi-classical physics, Extended Abstracts, 1998 Sixth International Workshop on Computational Electronics (IWCE-6) IEEE Catalog No. 98EX116,1-4, (1998).
J. R. BARKER, Electron logic devices and nano-instrumentation based on laterally patterned interaction-free quantum measurement structures. Semiconductor Science and Technology, 13 A93-96 (1998)
J. R. BARKER AND D K FERRY. On the validity of quantum hydrodynamic and quantum kinetic frameworks for describing anti-dot array devices. Semiconductor Science and Technology,13 A135-139 (1998).
D K FERRY AND J R BARKER. Open problems in quantum simulation in ultra submicron devices, VLSI Design, 8 165-172 (1998)
BARKER, J.R. Trajectories In Quantum Transport, in Quantum transport in ultrasmall devices, ed D K Ferry, Plenum Press 171-180 (1995).
BARKER, J R , BROUARD, S, GASPARIAN, V, IANNACCONE, G, JAUHO, J P, LEAVENS, C R, MUGA, J G, SALA, R, AND SOKOLOVSKY D. Report on the first European Workshop on Tunnelling Times, Phantoms Newsletter 7 5-10 (1994)
BARKER, J.R. On the pilot-field representation of quantum transport theory, Semiconductor Sci.Tech. 9 911-917 (1994).
BARKER, J.R., ROY, S., BABIKER, S. Trajectory representations, fluctuations and stability of single electron devices', Science and Technology of Mesoscopic Structures', Namba, S., Hamaguchi, C., Ando, T., eds., (London:Springer Verlag, Ch 22, 213-231 (1992)
BARKER, J.R. Fundamental aspects of quantum transport theory, in Handbook on Semiconductors, volume 1 Basic Properties of Semiconductors , Landsberg, P., ed., Second Completely Revised Edition, (:Elsevier-North Holland, Ch 19, 1079-1128( 1992).