One- and many-body effects on mirages in quantum corrals

Abstract

Recent interesting experiments used scanning tunneling microscopy to study systems involving Kondo impurities in quantum corrals assembled on Cu or noble metal surfaces. The solution of the two-dimensional one-particle Schrödinger equation in a hard wall corral without impurity is useful to predict the conditions under which the Kondo effect can be projected to a remote location (the quantum mirage). To model a soft circular corral, we solve this equation under the potential W δ(r − r0), where r is the distance to the center of the corral and r0 its radius. We expand the Green's function of electron surface states G 0 s for r < r0 as a discrete sum of contributions from single poles at energies ǫi − iδi. The imaginary part δi is the half-width of the resonance produced by the soft confining potential, and turns out to be a simple increasing function of ǫi. In presence of an impurity, we solve the Anderson model at arbitrary temperatures using the resulting expression for G 0 s and perturbation theory up to second order in the Coulomb repulsion U . We calculate the resulting change in the differential conductance ∆dI/dV as a function of voltage and space, in circular and elliptical corrals, for different conditions, including those corresponding to recent experiments. The main features are reproduced. The role of the direct hybridization between impurity and bulk, the confinement potential, the size of the corral and temperature on the intensity of the mirage are analyzed. We also calculate spin-spin correlation functions.

Mirages and many-body effects in quantum corrals

“…3 of Ref. [52]). Instead, the dependence on the impurity position R i should be affected by the relative strength of V b and V s (see next section).…”

“…The first one has been already used by us to study the mirage effect [38,39,52,53] and by one of us [39] and Shimada et al [80] to study the line shape of dI/dV in absence of the corral. The latter problem was also studied recently using the SBMFA [56], and to the best of our knowledge the results presented in section 8 are the first application of this technique to the mirage effect.…”

“…where r, θ are the polar coordinates on the plane and V conf is a dimensionless constant controlling the strength of the confinement potential [52,53]. The result is…”

“…For example the prediction of mirages out of the foci of elliptical corrals are somewhat hidden in the scattering approaches. Instead, mirages observed in a circular corral [13] were inspired by the extrema of the wave functions of the degenerate 37 th and 38 th conduction eigenstates of a hard-wall circular corral, and were calculated with our many-body approach for a circular corral with soft walls [52,53]. Having in mind the most studied case of the mirage effect: a Co impurity placed at one focus of an elliptical corral with e = 1/2 built on a Cu(111) surface [12], we have calculated the differential conductance dI/dV (R t , R i , V ) as a function of the tip position R t , for the impurity position fixed at the left focus [R i = (−0.5a, 0)] and voltage V = 10 mV.…”

Screening mechanisms in magnetic nanostructures

“…In this context, also the broadening of the energy levels due to a residual coupling of the nanostructure to the environment must be considered. [54][55][56][57] Another related question concerns particle-hole symmetry breaking, and the relation to hard-gapped problems, 58 where quantum-phase transitions between Kondo and unscreened states might also be accessible. To use NRG, we must map each channel α = e/o into the form of a Wilson chain.…”