New sheet structures may be the basis for boron nanotubes
The triangular boron lattice has too many electrons, and the hexagonal lattice has too few. But a hybrid of the two is just right.
November 2007, page 20
Carbon nanotubes are interesting to materials scientists
for a variety of reasons, one of which is the nanotubes' electronic behavior. Depending on their
structure, they can be either conductors or semiconductors, so they have the potential to perform
many different functions in miniaturized electronic technology. The problem is that the structure
can't readily be controlled: Known methods of synthesis always yield a mixture of conducting and
semiconducting nanotubes, which need to be separated before their electronic properties can
be exploited in devices.
That challenge, along with the quest
for an even more diverse range of nanotube properties, has led some researchers to turn their attention
to nanotubes of other materials whose electronic properties may be both desirable and uniform.
Nanotubes have been synthesized of many inorganic materials, including molybdenum disulfide,
titanium dioxide, boron nitride (pictured in the article by Marvin Cohen, PHYSICS TODAY, June
2006, page 48), and pure boron.
Carbon's neighbor to the
left in the periodic table, boron poses theoretical challenges as a nanotube material that carbon
doesn't. Carbon's natural ground-state structure is graphite, a layered material whose sheets
(called graphene, and described by Andrey Geim and Allan MacDonald in PHYSICS TODAY, August 2007,
page 35) form the basis for carbon-nanotube structures. But elemental boron tends to form networks
of icosahedral clusters, not planar sheets that might be rolled up to form nanotubes. Since the
energetically preferred boron-sheet structure doesn't occur naturally, scientists have had
to look for it. Now, Yale University's Sohrab Ismail-Beigi and his student Hui Tang have made some
progress toward that goal.1 They performed theoretical calculations on a new class
of two-dimensional boron sheets and found a sheet that is lower in energy than any structure previously
considered.
The graphene-like hexagonal
lattice is far from the ideal structure for a boron sheet. It's the ideal structure for carbon, which
has four valence electrons per atom: exactly enough to fill all of the bonding (or stabilizing)
electronic orbitals but none of the antibonding (destabilizing) orbitals. But when the carbon
atoms are replaced by boron atoms, which have only three valence electrons, there aren't enough
electrons to fill all the bonding orbitals, and the structure is not stable.
At first glance, the triangular
boron lattice may look even worse. After all, if a boron atom doesn't have enough electrons to form
stable bonds with its three nearest neighbors in the hexagonal lattice, how can it possibly bond
to six nearest neighbors in a triangular lattice? The answer is that the triangular lattice allows
the kind of chemical bonds that boron forms best: bonds among three atoms rather than between two.
Such three-center, two-electron bonds have been recognized for decades as an important part of
boron's complex and diverse chemistry.
The ground state of the
triangular boron lattice is electronically degenerate. The degeneracy is lifted, and the energy
thus lowered, when the sheet buckles slightly. Before Tang and Ismail-Beigi's work, the buckled
triangular lattice was the 2D boron-sheet structure with the lowest known energy. Researchers
figured that boron nanotubes would be made of rolled-up pieces of the buckled triangular lattice,
with the buckles running either parallel to the length of the tube or in helices around it. Calculations
showed such structures to be stable.2
Tang and Ismail-Beigi
considered a new class of sheet structures, derived from the triangular lattice but with some atoms
removed to form hexagonal holes. They found that both the number and the arrangement of the hexagonal
holes affect the sheet's energy. The optimal density is one hole for every nine atoms in the original
triangular lattice, and the optimal hole arrangement tends to be as evenly spaced and far apart
as possible. Of all of Tang and Ismail-Beigi's structures, the lowest in energy is thus the one shown
in figure 1. But they found many other structures that were lower in energy than the buckled triangular
lattice. The researchers considered buckled versions of their structures too, but found the flat
versions to be lowest in energy.
To understand the greater stability
of the new structures, think of them as hexagon-doped triangular lattices. The hexagonal boron
lattice doesn't have enough electrons to form all the necessary bonds: Its Fermi energy (shown
as a black vertical line in the density-of-states plot in figure 2) lies below the boundary between
the bonding and antibonding orbitals (at which the density of in-plane states is zero). The sheet
thus acts as an electron acceptor. The triangular lattice, on the other hand, has a Fermi energy
that's too high: Some electrons are forced to occupy antibonding orbitals that have a destabilizing
effect, and the sheet tends to act as an electron donor. By combining the electron-donating triangles
with electron-accepting hexagons, Tang and Ismail-Beigi were able to tune the Fermi energy so
that all the bonding orbitals are filled and all the antibonding orbitals are empty.
Figure 2 reveals another
important fact about the sheets' electronic properties. For the boron hexagonal lattice, zeroes
in the in-plane and out-of-plane densities of states coincide. The same is true for graphene: The
total density of states at the Fermi energy is zero, but there is no bandgap between the occupied
and unoccupied orbitals. The electronic properties of carbon nanotubes derived from graphene
are thus very sensitive to the density of states right around the Fermi energy, which in turn depends
on the tube's structure. For the lowest-energy boron sheet, however, the in-plane density of states
is zero but the out-of-plane density of states is not. That means that the boron sheet, or any nanotube
made from it, should be an excellent electrical conductor via the out-of-plane orbitals.
Earlier this year researchers
from Rice University, led by Boris Yakobson, did some similar calculations on boron clusters,3
and their results are consistent with the doping interpretation. Yakobson and colleagues found
that boron can form highly stable cage-shaped clusters of 80 atoms each. The clusters have the soccer-ball
structure of C60 fullerenes, but with an additional boron atom at the center of each
hexagonal face. The cages are thus made up of triangles and pentagons, and they are probably stabilized
in the same way as Tang and Ismail-Beigi's lattices, made up of triangles and hexagons.
Knowing what structures
boron nanotubes are likely to have can be helpful in fabrication and synthesis efforts, according
to Lisa Pfefferle, Ismail-Beigi's Yale colleague who heads the only group so far to have synthesized
boron nanotubes.4 A commonly used technique for confirming the presence of nanotubes
in a sample is to look in the Raman vibrational spectrum for a characteristic low-frequency mode
that corresponds to the radial expansion and contraction, or "breathing," of the nanotube. That
frequency depends on both the tube's diameter and its structure.
Pfefferle and her coworkers
have some control over their nanotubes' diameters—they grow the tubes inside the parallel
pores of a mesoporous catalyst—so it's especially useful for them to know what structure
to expect and how that structure influences the breathing-mode frequency. There's also the possibility
that knowing what structures to aim for, and how those structures might be stabilized by a catalyst,
could help them to refine their synthesis process.
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