### Topological Insulators and Stanene

February 24, 2013

The name of physicist, Enrico Fermi, is attached to many things, one of these being the Fermi level. The Fermi level, often symbolized as μ, is the energy level that would be occupied by electrons half the time at thermodynamic equilibrium. The probability f(ε) of an electron being at any particular energy level, ε, at temperature, T, is given by the Fermi-Dirac distribution,
where k is the Boltzmann constant. It's easy to see from this equation that a probability of one-half is obtained when the energy is the Fermi level energy; that is, when ε-μ = 0).

There's no requirement that a solid must have electrons at the Fermi level. It just describes how electrons at that level would behave. In the band theory of solids, the location of the Fermi level with respect to the energy bands of a material determines the material's electrical conductivity.

Insulators have a large band gap between the energy of their valence electrons, residing in a valence band, and conduction electrons, residing in a conduction band. The Fermi level sits within this gap, and there are few electrons with sufficient energy to jump the gap. In a metal, the Fermi level is within the conduction band, so there are electrons available for conduction. In semiconductors, the Fermi level is outside the conduction band, but it's close enough for electrons to be excited into the conduction band to allow some conduction.

This tutorial is prologue to a description of an interesting type of insulator, the topological insulator.[1-3] As the figure shows, these have a Fermi level between the valence band and the conduction band, as in other insulators. However, their surface electrons have a different trajectory in energy-momentum space, so these materials are bulk insulators and surface conductors at the same time.

Simplified band structure of a topological insulator.

(Modified image by "A13ean," via Wikimedia Commons.)

This is not that surprising, since atoms at a surface will have "dangling bonds," electron orbitals which would bond to adjacent atoms if any were present. These unemployed electrons (at least the anthropomorphic kind) are looking for a new job, and for the topological insulators this new job is conducting current along their surfaces. This strange property is a result of two quantum mechanical principles; namely, time-reversal symmetry and a spin–orbit coupling which exists in heavy elements such as mercury and bismuth.

In fact, the first demonstration of a topological insulator was in a compound of mercury. This was a thin layer of mercury telluride (HgTe) sandwiched between between layers of mercury cadmium telluride (HgxCd1–xTe). It was shown that the conductivity of the sandwich was independent of the width, confirming that the conductance was at the edges, only. Shortly thereafter, three-dimensional topological insulation was found in bismuth selenide (Bi2Se3) and bismuth telluride (Bi2Te3).

There's a lot of physics in topological insulators. The spins of the conducting electrons are aligned with the direction of current flow, so they're not easily scattered by defects or impurities. The interaction of an electron's spin and orbital angular momentum in heavy elements, such as Bi, is so strong that there's an inversion of the energy gap in topological insulators, with high energy states appearing below the gap. This results in metallic conduction at the surface.[3]

The same research team that predicted that these mercury and bismuth compounds would be topological insulators have recently made another prediction. Although tin (atomic number 50) is a lighter element than mercury (atomic number 80) or bismuth (atomic number 83), it's theorized that a single layer of tin (stanene), if produced, would be a topological insulator. It's further predicted that modification of the tin layer with fluorine will give the additional property that the surface electrons will mimic superconductivity at room temperature and higher temperatures.[4-8]

The team of theoretical physicists involved in this work is led by members of the SLAC National Accelerator Laboratory and Stanford University.[5] Other team members are from Tsinghua University (Beijing) and the Max Planck Institute for Chemical Physics of Solids (Dresden, Germany). Their first-principles calculations show that two-dimensional tin films are quantum spin Hall insulators with a bandgap for the bulk material of 0.3 eV.[4]

The properties of these films can be tuned by attachment of other chemical species to the tin atoms.[4] Calculations show that a layer of tin a single atom in thickness would be a topological insulator at and above room temperature, and attaching fluorine atoms to the tin atoms would extend its operating range to at least 100°C (see figure).[5] Attachment of hydrogen removes the topological property to produce an insulator, but topological insulators could be made by attachment of I, Br, Cl, F or OH.[4]

In this illustration, tin atoms (gray) are decorated with fluorine anions (yellow).

Fluorine is predicted to make this a superconducting topological insulator.

(Yong Xu/Tsinghua University; Greg Stewart/SLAC.)

Says team leader, Shoucheng Zhang, a professor of physics at Stanford and the Stanford Institute for Materials and Energy Sciences, a joint institute with SLAC,
"Stanene could increase the speed and lower the power needs of future generations of computer chips, if our prediction is confirmed by experiments that are underway in several laboratories around the world... Eventually, we can imagine stanene being used for many more circuit structures, including replacing silicon in the hearts of transistors... Someday we might even call this area Tin Valley rather than Silicon Valley."[5]

### References:

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