topological insulator

Magnetic solitons and topological insulators have “opposite” protected regimes

A cute thought occurred to me today. In the study of magnetic textures, we can have topologically protected spin configurations which belong to nontrivial classes of some homotopy group. Generally speaking, such a configuration assigns a vector on the Bloch sphere to each point in space, m : R² → S², which is equivalent to an endomorphism on S² (by compactifying through the point at infinity). Such a map m, then, is classified by the second homotopy group π2(S²). A [baby] skyrmion belongs to the charge ±1 sector of this group.

Topological protection is so-called because the integral classification of a topological phenomenon cannot be continuously modified by a weak perturbation. What seems to follow: once in a topologically nontrivial sector, always in a topologically nontrivial sector.

But we can of course create and destroy skyrmions in the lab. How is this possible? This question was (mythically) answered by Alexander III of Macedon, namely Alexander the Great, in the fourth century BCE. When faced with the problem of untying the untieable Gordian knot in ancient Phrygia, he famously slashed the knotted rope in half with his sword making him, by a retroactive prophecy, the King of Asia.

Alexander’s key insight was that the topological protection of the Gordian knot could be circumvented by confronting a key assumption: that the rope is a continuous manifold, a prerequisite condition for homotopical classification. At the length scale of a blade’s edge, and the energy scale of human strength, this assumption about rope breaks down, and so Alexander could remove the knot without, strictly speaking, untying it.

This same principle carries over to skyrmions. When the continuum approximation is respected (the Gordian regime), skyrmions are indeed topologically protected. But when sufficiently energetic perturbations occur at small enough length scales (on the order of the lattice constant), the continuum approximation fails to accurately describe our spin system. In this Alexandrian regime, there is no notion of topological protection, and it is here that skyrmions may be created or destroyed.

As promised, I had a cute thought today, and here it is: in a certain sense, the Alexandrian and Gordian take on opposite roles in two main schools of condensed matter topology. I have outlined a bit about spin textures above, which are conserved due to a real space configuration. But the more famous topological insulators are protected by homotopies of the d-torus, which arises as the [crystal] momentum space. In that case, it turns out that the discretization of the lattice is a necessary precondition for nontrivial topology: if the lattice constant were taken to zero, the indices labeling the atomic orbitals would end up being real valued rather than integer valued. Instead of the torus, then, the we would end up with a momentum space given by the Pontryagin dual to the reals—which is just the always uninspiring real numbers themselves, where there are no circles to wind and no homotopy to classify.

Perhaps the inverted role of the continuum approximation in these two cases has something to do with the inverted physical units or the conjugate nature of real and momentum space, but I’ve not figured it out quite that far quite just yet.

Physicists announce graphene’s latest cousin: stanene

Two years after physicists predicted that tin should be able to form a mesh just one atom thick, researchers say that they have made it. The thin film, called stanene, is reported on 3 August in Nature Materials. But researchers have not been able to confirm whether the material has the predicted exotic electronic properties that have excited theorists, such as being able to conduct electricity without generating any waste heat.

Stanene (from the Latin stannum meaning tin, which also gives the element its chemical symbol, Sn), is the latest cousin of graphene, the honeycomb lattice of carbon atoms that has spurred thousands of studies into related 2D materials. Those include sheets of silicene, made from silicon atoms; phosphorene, made from phosphorus; germanene, from germanium; and thin stacks of sheets that combine different kinds of chemical elements (see ‘The super materials that could trump graphene’).

Many of these sheets are excellent conductors of electricity, but stanene is — in theory — extra-special. At room temperature, electrons should be able to travel along the edges of the mesh without colliding with other electrons and atoms as they do in most materials. This should allow the film to conduct electricity without losing energy as waste heat, according to predictions2 made in 2013 by Shou-Cheng Zhang, a physicist at Stanford University in California, who is a co-author of the latest study.

That means that a thin film of stanene might be the perfect highway along which to ferry current in electric circuits, says Peide Ye, a physicist and electrical engineer at Purdue University in West Lafayette, Indiana. “I’m always looking for something not only scientifically interesting but that has potential for applications in a device,” he says. “It’s very interesting work.”

Stanene is predicted to be an example of a topological insulator, in which charge carriers (such as electrons) cannot travel through a material’s centre but can move freely along its edge, with their direction of travel dependent on whether their spin — a quantum property — points ‘up’ or ‘down’. Electric current is not dissipated because most impurities do not affect the spin and cannot slow the electrons, says Zhang.

But even after making stanene, Zhang and his colleagues at four universities in China have not been able to confirm that it is a topological insulator. Experimentalists at Shanghai Jiao Tong University created the mesh by vaporizing tin in a vacuum and allowing the atoms to waft onto a supporting surface made of bismuth telluride. Although this surface allows 2D stanene crystals to form, it also interacts with them, creating the wrong conditions for a topological insulator, says Zhang. He has already co-authored another paper examining which surfaces would work better.

Ralph Claessen, a physicist at the University of Würzburg in Germany, says that it is not completely clear that the researchers have made stanene. Theory predicts that the 2D tin lattice should form a buckled honeycomb structure, with alternate atoms folding upwards to form corrugated ridges; Zhang and his team mostly saw only the upper ridge of atoms with their scanning tunnelling microscope, except in a small spot where that ridge disappeared and a lower layer of tin atoms was exposed. However, they are confident that they have created a buckled honeycomb, partly because the distance between upper and lower layers matches predictions.

Claessen says that he would need to see direct measurements of the lattice’s structure — from X-ray diffraction — to be confident that the team has made stanene, and not some other arrangement of tin. This would require larger amounts of the material than Zhang and his co-authors have grown.

Yuanbo Zhang, a physicist at Fudan University in Shanghai, China, who was not involved in the study, is more convinced. “I think the work is a significant breakthrough that once again expands the 2D-material universe,” he says. “It’ll be exciting to see how the material lives up to its expectations.”

And Guy Le Lay, a physicist at Aix-Marseille University in France who was among the first to produce both silicene and germanene, preaches optimism in the attempt to verify stanene’s electronic properties. “It’s like going to the Moon,” he says. “The first step is the crucial step.”

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