Quantum corral

Crommie M F, Lutz C P and Eigler D M 1993 Confinement of electrons to quantum corrals on a metal surface Science 262 218... 

Fig. 4. Atom manipulation by the scanning tunneling microscope (STM). Once the STM tip has located the adsorbate atom, the tip is lowered such that the attractive interaction between the tip and the adsorbate is sufficient to keep the adsorbate "tethered" to the tip. The tip is then moved to the desired location on the surface and withdrawn, leaving the adsorbate atom bound to the surface at a new location. The figure schematically depicts the use of this process in the formation of a "quantum corral" of 48 Fe atoms arranged in a circle of about 14.3 nm diameter on a Cu(lll) surface at 4 K.

Several striking examples demonstrating the atomically precise control exercised by the STM have been reported. A "quantum corral" of Fe atoms has been fabricated by placing 48 atoms in a circle on a flat Cu(lll) surface at 4K (Fig. 4) (94). Both STM (under ultrahigh vacuum) and atomic force microscopy (AFM, under ambient conditions) have been employed to fabricate nanoscale magnetic mounds of Fe, Co, Ni, and CoCr on metal and insulator substrates (95). The AFM has also been used to deposit organic material, such as octadecanethiol onto the surface of mica (96). New appHcations of this type of nanofabrication ate being reported at an ever-faster rate (97—99). 

STM tip (Reproduced from Ref. 3). (b) A quantum corral built by positioning iron atoms with an STM tip on a Cu(lll) surface. (Reproduced from Ref. 2). 

E. Heller, M. Crommie, C. Lutz and D. Eigler, Scattering and absorption of surface electron waves in quantum corrals, Nature 369, 4646 (1994). 

Invited 10. Qiang Miao, Chun-gen Liu, Hirohiko Adachi and Isao Tanaka (Nanjing University, Kyoto University) DV-Xa Study on Quantum Mirages and Quantum Corrals in Non-Metal Systems... 

Fig. 1.1. STM image of a quantum corral of 48 Fe atoms placed in a circle of 7.3 nm [IBM Research].

Figure 11. Spatial image of the eigenstates of a quantum corral made of 48 Fe atoms. Reprinted with permission from Ref. 98. Copyright (1993) AAAS.

Recall from chap. 2 that often in the solution of differential equations, useful strategies are constructed on the basis of the weak form of the governing equation of interest in which a differential equation is replaced by an integral statement of the same governing principle. In the previous chapter, we described the finite element method, with special reference to the theory of linear elasticity, and we showed how a weak statement of the equilibrium equations could be constructed. In the present section, we wish to exploit such thinking within the context of the Schrodinger equation, with special reference to the problem of the particle in a box considered above and its two-dimensional generalization to the problem of a quantum corral. 

Fig. 3.6. Successive steps in the construction of a quantum corral using the tip of a scanning tunneling microscope (courtesy of D. Bigler).

As a result of our separation of variables strategy, the problem of solving the Schrodinger equation for the quantum corral is reduced to that of solving two ordinary differential equations. In particular, we have... 

Fig. 3.7. Finite element mesh used to compute the eigenstates of the quantum corral (courtesy of Harley Johnson).

Fig. 3.8. Two different eigenfunctions for the quantum corral as obtained by using a two-dimensional finite element calculation (courtesy of Harley Johnson).

This expression will be used in Chapter 7 to calculate magnetic interactions and to discuss quantum corrals. 

Quantum corrals are fascinating structures, which consist of atoms positioned in a geometrical pattern on a noble metal (111) surface. They are built atom by atom using scanning tunneling microscopes (STM), which can then be used to study the properties of the corral as well. 

The corrals exhibit standing-wave patterns of electrons, and the first idea was to use the corrals to study quantum chaos [169, 170, 171, 172]. Unfortunately, the walls leaked too much for the electrons to bounce around long enough to detect any chaotic effects [173]. In 2000, Manoharan et a 1. [174] performed an experiment on an elliptic quantum corral, where a quantum mirage of a Co atom at one of the focus points of the ellipse was seen when a Co atom was placed at the other... 

Figure 7.17. A sketch from Wellenlehre, an 1825 book published by Ernst and Wilhelm Weber in Leipzig, showing the wave pattern of mercury waves when small amounts of mercury are dropped in one focus. The other focus looks identical, which means that from the point of view of the wave, the two foci are excited equally. The figure is shamelessly stolen from the excellent review on quantum corrals by Fiete and Heller [175].

The reason for this behaviour is the presence of Shockley surface states [176] on the noble metal surfaces. On these surfaces, the Fermi energy is placed in a band gap for electrons propagating normal to the surface. This leads to exponentially decaying solutions both into the bulk and into the vacuum, and creates a two-dimensional electron gas at the surface. The gas can often be treated with very simple quantum mechanical models [177, 178], and much research has been done, especially with regards to Kondo physics [179, 180, 181]. There has also been attempts to do ab initio calculations of quantum corrals [182, 183], with in general excellent results. 

The problems with performing ab initio calculations for quantum corrals is the very long computational time that is needed, especially if one would want to do big supercell calculations. Even with the help of the force theorem, and perturbative approaches to the problem, one has so far had to diagonalize very big matrices, which makes it hard to perform exhaustive searches for quantities of interest. There has also been interest in engineering quantum corrals to achieve specified electronic properties [184], and also here the problem of finding optimized quantum corral structures appears. 

Up until now, the ab initio approach has been to consider the quantum corral as... 

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