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Physical Review A

Tunneling in a high transverse magnetic field

Jainendra K. Jain, Steven Allan Kivelson

Published Oct 1, 1987• 16 citations
Publisher

Abstract

We have studied two simple model tunneling problems involving an electron in two spatial dimensions in the presence of a high magnetic field, both by numerical integration of Schr\"odinger's equation, and by a semiclassical evaluation of the coherent-state path integral. We find that the semiclassical approach is analytically tractable and produces results in good agreement with the numerical results.

DOI
10.1103/physreva.36.3467

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We have studied two simple model tunneling problems involving an electron in two spatial dimensions in the presence of a high magnetic field, both by numerical integration of Schr"odinger's equation, and by a semiclassical evaluation of the coherent-state path integral. We find that the semiclassical approach is analytically tractable and produces results in good agreement with the numerical results.
We have studied simple tunneling problems in two dimensions in the presence of a high transverse magnetic field both by numerical integration of the Schr\"odinger equation and by semiclassical evaluation of the path integral. We have chosen three model potentials: (i) asymmetric single well, (ii) symmetric double well, and (iii) quadruple well. We find that the semiclassical approach is analytically tractable and gives a very accurate description of the exponential and oscillatory behaviors of the tunneling matrix elements. A precise definition of the Aharonov-Bohm phase for the tunneling paths is given. In addition to the Aharonov-Bohm phase, there is also a geometrical phase coming from the fluctuation determinant, and we find that for every closed loop it is exactly \ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}.
We present a comprehensive study of the effect of a transverse magnetic field on the macroscopic quantum tunneling of two identical coupled particles with dissipation. Two coupled particles of identical masses but opposite charges in the plane, in the presence of a constant transverse external magnetic field and an external potential interacting with a bath of harmonic oscillators are studied. In this case, the problem cannot be mapped to a one-dimensional problem, it strictly remains two-dimensional. We calculate the exact effective action both for the case of linear coupling to the bath and without a linear coupling to the bath using imaginary time path integral and Leggett's prescription at T=0. In the limit of zero magnetic field we recover a two dimensional version of the result found in \cite{em1} for the case of two identical particles. We find that in the limit of weak and strong dissipation, the effective action reduces to a two dimensional version of the Caldeira-Leggett form in terms of the reduced mass and magnetic field. The case of Ohmic dissipation with the motion of the two particles damped by the Ohmic frictional constant η\etaη is studied in detail.
The spin coherent state path integral describing the dynamics of a spin-½ system in a magnetic field of arbitrary time dependence is considered. Defining the path integral as the limit of a Wiener regularized expression, the semiclassical approximation leads to a continuous minimal action path with jumps at the endpoints. The resulting semiclassical propagator is shown to coincide with the exact quantum mechanical propagator. A nonlinear transformation of the angle variables allows for a determination of the semiclassical path and the jumps without solving a boundary-value problem. The semiclassical spin dynamics is thus readily amenable to numerical methods.
We study the application of the semiclassical coherent-state path-integral technique to the problem of scattering. A numerical method based on initial-value trajectories and their stability information is proposed to solve the root search for the complex trajectories that determine the semiclassical propagator. Very good agreement between the numerically exact quantum and semiclassical results for the correlation function can be achieved for scattering from one-dimensional barrier as well as well-type potentials by using just one single complex trajectory at each time.
We apply the bosonization technique to the problem of tunneling in a Fermi liquid, and present a semiclassical theory of tunneling rates. To test the method, we derive and evaluate an expression for the tunneling current in the problem of two parallel Luttinger liquids. Next we study tunneling between parallel two dimensional Fermi liquids. In the absence of a magnetic field, the conductance I/VI/VI/V has a resonance peak at zero bias. Our estimate for the peak broadening agrees with the many-body theory results. In a magnetic field perpendicular to the plane, we find a tunneling gap linear in the field, in agreement with the experiment. In this regime the tunneling is spatially coherent over a distance of the order of the cyclotron radius. The coherence can be probed by a magnetic field parallel to the plane, which gives an Aharonov-Bohm phase to the tunneling amplitude. As a result, the current is an oscillatory function of the field.
The reflection and transmission of a wave on a two-dimensional barrier is studied quantum mechanically and semiclassically. The simple model consists of two degrees of freedom. One of them contains a barrier (of the Eckart type); the second allows only a bound motion in discrete quantum states. These two degrees of freedom are coupled. The transition probabilities for transitions between the different quantum states can, in the model considered here, be evaluated analytically, both quantum mechanically and semiclassically. Several of the so called uniform semiclassical approximations and also the initial value integral representation for the semiclassical transition amplitudes are considered and compared to the exact quantum mechanical results. Some new insight on the semiclassical treatment of the tunneling process is gained, specially concerning the proper choice of the integration path for the initial value integral representation of the transition amplitudes.