Quantum corral herds surface electrons into a fractal lattice

String theory notwithstanding, we live in three-dimensional space. But physics in other numbers of dimensions need not be a purely theoretical exercise. Atomically thin materials such as graphene are well described as 2D systems (see Physics Today, December 2010, page 14 ), and polymers and quantum wires have many 1D characteristics.

Sometimes changing the number of dimensions effects a qualitative change in a system’s properties. For example, the Ising model of coupled spins undergoes a phase transition at nonzero temperature in two or more dimensions, but not in 1D.

What if the dimensionality could be tuned continuously between 1 and 2? That’s not just a hypothetical question: Fractals, such as the Sierpinski triangle in figure 1, have fractional dimensionality. But although fractal-like shapes abound in the natural world—rugged coastlines and branched leaf veins, for example—few platforms exist for realizing microscopic physics in fractal geometry.

Figure 1.

Now Ingmar Swart, Cristiane Morais Smith, and colleagues at Utrecht University in the Netherlands have taken a step toward probing quantum physics in a real fractional-dimensional system. ¹ On a (111) surface of copper, they placed carbon monoxide molecules (shown as black indentations in the figure 1 inset) to corral the surface electrons into a simplified Sierpinski triangle. The electron density inside the triangle is an approximation of the fractal, just as a graphene sheet is an approximation of an infinitely thin plane. But like graphene, the surface-electron system inherits some of the dimensional properties of its mathematical idealization.

Physics in fractland

The basic definition of the dimension of a shape is the number of coordinates needed to specify a point within it. After allowing for topological transformations, whereby a donut is equivalent to a coffee cup, even a squiggly line has dimension 1, a crumpled plane has dimension 2, and so on. (Making that definition perfectly rigorous is surprisingly tricky, because it’s possible to construct an infinitely long, infinitely squiggly line that visits every point in a higher-dimensional space. But mathematicians have figured it out.)

A century ago Felix Hausdorff came up with a new way of defining dimension that could take noninteger values. In effect, it’s how the number of boxes needed to cover a shape scales with the size of the boxes. For most familiar shapes—points, lines, planes, and the like—the Hausdorff dimension is equal to the usual topological dimension. But certain infinitely intricate shapes, which wouldn’t be called “fractals” until Benoit Mandelbrot coined the term in 1975, have fractional Hausdorff dimension.

Wacław Sierpiński, a contemporary of Hausdorff, first wrote about his triangle in 1915, although versions of it had been used decoratively for hundreds of years before that. The shape is defined by its exact self-similarity: The large triangle, 1 unit on a side, is made up of three copies of itself, each 0.5 unit on a side. Self-similarity does not by itself a fractal make, though. Nonfractal lines and planes are self-similar too: A 1-unit line segment is made up of two 0.5-unit copies of itself, and a 1-unit filled-in square is composed of four 0.5-unit squares. In each case, the Hausdorff dimension is equal to the base-2 logarithm of the number of copies, so the line has dimension 1, the square has dimension 2, and the Sierpinski triangle has dimension log₂ 3 = 1.58.

Since the early 1980s, theorists have investigated the statistical and quantum physics of fractal lattices, ² and one of the first problems they tackled was whether spin systems such as the Ising model undergo phase transitions at nonzero temperature in dimensions between 1 and 2. As it turned out, the answer depends not on the dimension but on the so-called order of ramification, a measure of how much cutting is required to break the system into disconnected pieces. The Sierpinski triangle has a low order of ramification—one can isolate any of the smaller triangles from the rest of the lattice simply by snipping its three corners—and the Ising model on a Sierpinski-triangle lattice exhibits no phase transition. But other, more robustly connected fractals do show phase transitions, even if their Hausdorff dimension is the same or lower than the Sierpinski triangle’s.

Synthetic lattices

Limited only by their own imaginations, theorists can explore the physics of a system of any shape. But real atoms aren’t known to naturally arrange themselves into Sierpinski triangles, so experimenters have an extra step. The technology for realizing artificial lattices on metal surfaces dates back a quarter century, but only recently have researchers begun to exploit it for that purpose.

In a defect-free bulk crystal, translational symmetry dictates that electron wavefunctions take the form of delocalized Bloch waves. At the crystal surface, however, that symmetry is broken, and the Schrödinger equation admits additional solutions localized at the surface. On Cu(111) and certain other metal surfaces, the surface-state energies coincide with a bulk bandgap, so the surface and bulk states cannot mix: Some electrons are physically confined to the surface, where they behave like an ideal 2D electron gas.

In 1993 Donald Eigler and colleagues at IBM’s Almaden Research Center showed that they could corral those surface electrons with a ring of iron atoms placed on the surface with the tip of a scanning tunneling microscope (STM). ³ (See Physics Today, November 1993, page 17 .) The adsorbed atoms have their own electrons bound to their nuclei, and repulsive interactions create an energy barrier—a fence—for the surface electrons, which are otherwise free to move around inside their corral. The STM serves a dual purpose: It not only moves the atoms into place but also maps the spatial density of the corralled electrons.

In 2012 Hari Manoharan and colleagues at Stanford University created the first quantum corral lattice, artificial graphene, ⁴ shown in figure 2a. The densely packed CO molecules leave the surface electrons confined to a honeycomb network of thin paths. The synthetic lattice has a couple of advantages over real graphene. It’s several times larger, so the electron wavefunctions can be more readily mapped in real space. And it’s tunable: Tweaking the lattice structure introduces effects analogous to charge-carrier doping and applied magnetic fields.

Figure 2.

Figures 2b and 2c show some of the other lattices that have been created: The Lieb lattice and square lattice made by Swart, Morais Smith, and colleagues ⁵ and the quasicrystalline lattice assembled by Kenjiro Gomes and colleagues at the University of Notre Dame. ⁶ (For more on quasicrystals, see Physics Today, December 2011, page 17 .) Importantly, the newer lattices break free of the underlying hexagonal symmetry of Cu(111). Together they show the potential of quantum corralling as a platform for simulating one quantum system—electrons in a solid, usually—with another. The current leading method for quantum simulation uses ultracold atoms in arrays of optical traps (see Physics Today, October 2010, page 18 ; August 2017, page 17 ; and the article by Victor Galitski, Gediminas Juzeliūnas, and Ian Spielman on page 38 of this issue). Corralled surface-electron systems can offer a complementary set of capabilities, such as having particles that are charged, not neutral.

Although the STM technology is mature, the experiments remain extremely difficult. To build and probe a lattice of dozens or hundreds of atoms takes about a week, during which time the STM tip must be held near the surface, with atomic precision, without ever touching it. (A single tip crash, a common user error even in well-automated experiments, will destroy an entire lattice.) Fortunately, Marlou Slot, Swart’s PhD student at the fore of the experimental effort, was up to the task. “She is an outstanding STM professional,” says Morais Smith, “and built our largest lattice in only one weekend.”

Building the triangle

Inspiration for the Sierpinski lattice came from two directions. First was Morais Smith’s past theoretical work on electrons in multilayer graphene, which behave in many ways that are intermediate between 2D and 3D. Second was her recent exploration of fractals to try to get her stepdaughter interested in math. A fractal quantum corral, she realized, could open up the largely uncharted experimental world of electrons between 1D and 2D.

It was a nontrivial task to find just the right arrangement of CO molecules to create the Sierpinski-like corral. Fortunately, the trial and error could be done computationally, saving the experimental team the effort of building too many lattices that wouldn’t work. Calculations by Morais Smith’s student Sander Kempkes settled on the arrangement shown in the inset in figure 1, and that’s the one they realized experimentally.

The researchers used the box-counting method to calculate the Hausdorff dimension of the wavefunction inside the triangle: How many circles of radius r does it take to cover the wavefunction, and how does that number scale with 1/r? With the true, infinitely detailed Sierpinski triangle, the power-law dependence with exponent 1.58 extends all the way to the limit of r = 0. For the quantum corral wavefunction, that’s not the case, but the power-law relation still extends over more than an order of magnitude in r—similar to other fractal-like shapes in the physical world. Says Morais Smith, “We got shiny eyes when we realized the electrons were really living in 1.58 dimensions.”

The proof-of-principle experiment is not yet at the point of revealing any new physics. One of the most intriguing open questions is how an ensemble of interacting electrons behaves in noninteger dimensions. Fermi liquid theory, which describes interacting electrons in two or more dimensions, breaks down in 1D. And Luttinger liquid theory, which replaces it, has some unusual features, such as the independent propagation of charge and spin waves (see Physics Today, September 1996, page 19 ). What happens in 1.58 dimensions is not known. Unfortunately, the surface electrons of Cu(111) don’t interact strongly enough to find out. Surfaces of other materials, however, might allow the researchers to study interactions in fractal geometry.