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Physical review. B, Condensed matter

Theory of tunneling of an atom interacting with a degenerate electron gas

A. Zawadowski, Gergely T. Zimányi

Published Jul 15, 1985• 20 citations
Publisher

Abstract

The tunneling of an atom in a double potential well is treated in the presence of interaction with conduction electrons. Screening and assisted tunneling processes are considered as well. In the spinless electron case the partition function is calculated and logarithmic interactions between tunneling processes are derived and expressed by a phase shift which scales to \ensuremath{\pi}/2 as the bandwidth is reduced.

DOI
10.1103/physrevb.32.1373

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Journal of the Physical Society of Japan•Jun 15, 1993•9 citations
The tunneling of an atom in a double potential well is treated in the presence of interaction with conduction electrons. Screening and assisted tunneling processes are considered as well. In the spinless electron case the partition function is calculated and logarithmic interactions between tunneling processes are derived and expressed by a phase shift which scales to \ensuremath{\pi}/2 as the bandwidth is reduced.
The interaction between a tunneling atom in a double potential well and the conduction electrons of different angular momenta results in an anisotropic two-channel orbital Kondo problem, but the Kondo temperature TK{\mathit{T}}_{\mathit{K}}TK​ found earlier has always been somewhat small. That difficulty can be resolved by including some of the excited states of the atom in the well. The TK{\mathit{T}}_{\mathit{K}}TK​ calculated is around 1--10 K in agreement with experiments. The same result can be obtained even if the direct electron assisted tunneling is ignored.
A one-dimensional Coulomb gas is considered in which the particles represent both charges and dipoles. The values of the charges and dipole moments and the fugacities of that gas have been obtained in the preceding paper by mapping the dynamics of a two-level system (TLS) interacting with a degenerate electron gas onto the present problem. The scaling procedure of Anderson, Yuval, and Hamann is applied to eliminate the close pair which corresponds to the short-time behavior of the TLS. The main contribution arises from pairs of zero total charge, but the role of other pairs is considered as well. The scaling equations show that the electron-assisted transitions of the TLS play a very important role if their value is different from zero for the starting Hamiltonian. In that case localization cannot occur in the case of a spin-(1/2 fermionic system. Localization may occur only if the fermion spin has an artificially large value with spin degeneracy Ns{N}_{s}Ns​>2. That clearly demonstrates that a particle moving in a double potential well may have different dissipative behavior depending upon whether it is coupled to a fermionic or a bosonic heat bath. The screening of the TLS by electrons is described by an arbitrarily large phase shift \ensuremath{\delta} which renormalizes towards the value \ensuremath{\delta}=\ensuremath{\pi}/2Ns{N}_{s}Ns​, but the scaling equations lose their validity as the fugacities corresponding to the TLS transitions become of the order of unity. The method developed is a very general one, and thus it can be applied to other similar one-dimensional gases as well.
This chapter is devoted to quantum tunneling, as a basic quantum process, essential for important applications. Tunneling between two wells, of the double well potential of a one particle system of interest, makes sense only when a second system, capable to distinguish between the presence of the particle in a well, or in the other well, is present. The potential of interaction between such two systems is called tunneling operator. We treat a few problems of interest as tunneling in a quasi-continuum of states, the energy shift in a well by the proximity of another well, dissipation effects, and tunneling spectrum.
Abstract For a non-superconducting system and in the spirit of the transfer or tunnelling Hamiltonian formalism, an expression for the electronic tunnelling current through an insulating barrier is calculated, explicitly taking into account the effects of electron-electron interactions that can persist across the barrier. In the presence of a single tunnelling barrier, the exact Hamiltonian is projected onto the subspaces of the ‘left’ and ‘right’ conducting leads. In the weak-tunnelling limit, the well-known tunnelling Hamiltonian is recovered, along with an additional term. This additional term originates from the projection of the electron-electron interaction onto each subspace and corresponds to a correlated or interaction-mediated tunnelling. It is shown that the tunnelling current in this approximation is related to—in addition to the single-particle density of states—the two-particle spin-spin and density-density susceptibilities. The signatures of these terms have been observed in scanning tunnelling microscope experiments on atomic spin chains.
A new formalism to treat the tunneling processes of a heavy particle coupled to degenerate electron systems has been presented. This simple and transparent method follows that of Caldeira and Leggett in a general framework, but the term representing the dissipation is given in terms of the imaginary part of the retarded density correlation function as far as the coupling is weak. Hence our method can incorporate the microscopic details of the electronic systems such as mutual interactions and randomness. It is found that for the Luttinger liquid, the dissipation can be either sub- or super-Ohmic depending on the type of interaction, while near the Anderson transition there exists a crossover from Ohmic to sub-Ohmic dissipation. In the case of sub-Ohmic dissipation, the tunneling rate is found to vanish discontinuously above some critical value as a function of the strength of dissipation.