A simple interpretation of quantum mirages

Abstract

In an interesting new experiment the electronic structure of a magnetic atom adsorbed on the surface of Cu(111), observed by STM, was projected into a remote location on the same surface. The purpose of the present paper is to interpret this experiment with a model Hamiltonian, using ellipses of the size of the experimental ones, containing about 2300 atoms. The charge distribution for the different wavefunctions is analyzed, in particular, for those with energy close to the Fermi energy of copper EF . Some of them show two symmetric maxima located on the principal axis of the ellipse but not necessarily at the foci. If a Co atom is adsorbed at the site where the wavefunction with energy EF has a maximum and the interaction is small, the main effect of the adsorbed atom will be to split this particular wavefunction in two. The total charge density will remain the same but the local density of states will present a dip at EF at any site where the charge density is large enough. We relate the presence of this dip to the observation of quantum mirages. Our interpretation suggests that other sites, apart from the foci of the ellipses, can be used for projecting atomic images and also indicates the conditions for other non magnetic adsorbates to produce mirages. The electronic structure of systems with adsorbed impurities on noble metal surfaces should depend both on the type and the geometric arrangement of the adsorbed atoms. In particular, the effect of single transition-metal impurities and also of dimers on the (111) surface of gold has been studied recently with scanning tunneling microscopy [1][2][3]. Previously, closed loops of impurities had been shown to confine the surface state wavefunctions of Cu (111), producing what are now called "quantum corrals" [4]. In an interesting new experiment the electronic structure surrounding an adsorbed magnetic atom could be projected into a remote location on the same surface [5]: a Kondo "signature" in the tunneling spectrum of a cobalt atom located on one focus of an elliptical corral was observed as a mirage on the other focus. This possibility raises many theoretical questions, the first one being the relation between this quantum system and classical image projection. Other questions regarding the experimental setup are: How does the appearance of a mirage depend on the surface type, on the adsorbed atom type, on the shape of the corral, etc? The fact that the (111) surface of noble metals presents a symmetry gap at the Γ point and a surface state at about 0.5 eV below the Fermi energy, uncoupled from the bulk states, seems of course fundamental. For the other questions one must take a closer look and this is the purpose of the present paper. It must be emphasized, however, that we are not attempting a full theoretical description of the experiment but rather a simplified approach to the previous questions.We start by assuming a model Hamiltonian for the interaction of one adsorbed atom with the surface state of the metal, using the following approximations:1. The ...

Colloquium: Theory of quantum corrals and quantum mirages

“…Agam and Schiller (2001);Porras et al (2001) and Weissmann and Bonadeo (2001) have also developed theories for the quantum mirage based on a single-particle picture. More recently, Aligia (2001) and Shimada et al (2002) has developed a many-body theory of the quantum mirage.…”

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

“…The latter have the disadvantage that many-body effects are very hard to include. In addition, some features which are clear from the hard wall eigenstates, like mirages out of the foci [5,7,8,9], are somewhat hidden in the scattering approaches. On the other hand, while the eigenstates inside a hard wall corral are perfectly defined, in the actual experiments the boundaries of the corrals are soft and the eigenstates become resonances with finite width δ i [2].…”

“…Some theories have assumed that δ b ∼ δ s , while others have taken δ b = 0 [7,8,9,10,11]. Clearly, δ s = 0, because otherwise ∆dI/dV would be independent of the 2D conduction eigenstates in contrast to the experiment [4].…”

Interaction between Kondo impurities in a quantum corral

“…In Ref. [9] the Kondo effect is absent, and perturbation theory in the Coulomb repulsion U [10,11] is restricted to small values of U .…”