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Rethinking the Electronics of Quantum Dots
15 June 2005
Quantum dots, tiny crystals consisting of a few hundred to a few
thousand atoms, sparkle with promise for uses ranging from tagging
proteins in living cells to foiling counterfeiters to enabling quantum
computers. The optics and electronics of these semiconductor
nanocrystals are dramatically different from the same materials in
bulk. But it turns out that one of the most important electronic
properties of quantum dots has been misunderstood for over a decade.
Theorists at the Department of
Energy's Lawrence Berkeley National Laboratory have shown that a quantum
dot's dielectric function (a term indicating how charge responds to an
electric field) does not depend on its band gap, as researchers long
believed. On the contrary, the dielectric function of a quantum dot,
measured on the microscopic scale, is virtually the same as that of the bulk
material -- except near the dot's surface.
"One of the interesting things about quantum dots is that their band gaps
are much larger than the same material in bulk. At the same time their
overall dielectric constants are much smaller," says Lin-Wang Wang of
Berkeley Lab's Computational Research Division. "Therefore it was natural to
assume that the size of the band gap in a quantum dot is what determines its
overall dielectric constant."
Recently French researchers led by Christophe Delerue of the Institut
Supérieur d'Electronique du Nord raised doubts about this assumed
relationship, however, basing their argument on approximate calculations. To
test the questions posed by the French group, Wang and postdoctoral fellow
Xavier Cartoixà performed, for the first time, ab initio ("from first
principles") microscopic studies of the dielectric function in quantum dots.
To do so they used PEtot, a quantum-mechanical electronic-structure program
developed by Wang, on the Seaborg supercomputer at the Department of
Energy's National Energy Research Scientific Computing Center (NERSC), based
at Berkeley Lab.
Wang and Cartoixá's results led them to devise a simple mathematical model,
the first that nanoscience researchers can use for quick, consistent
calculations of the dielectric function in nanocrystals.
Tunable band gaps and a rainbow of colors
One useful feature of
quantum dots is that the colors of light they absorb and emit can be tuned
simply by varying their size," says Wang. "This is because dots of the same
material but different sizes have different band gaps, which absorb and emit
different frequencies."
The band gap of a semiconductor like silicon or gallium arsenide is the
energy required to lift an electron from its valence band, filled with
electrons, to its conduction band, which is empty. For example, an incoming
photon whose energy matches or exceeds the band gap can boost an electron
into the conduction band, leaving behind a "hole" of opposite charge. This
is the principle that underlies photovoltaic cells, which generate
electrical current when stimulated by light.
Conversely, when an electron falls from the conduction band back down to the
valence band, eliminating a hole, the lost energy is emitted as light whose
color corresponds to the band gap -- this is the principle behind
light-emitting diodes, LEDs.
Each semiconductor has a characteristic band gap, but when the diameter of a
piece of the material is shorter than the quantum-mechanical wave function
of its electrons, the "squeezed" electron wave function makes the band gap
wider. For an electron to jump from the valence band to the conduction band
now requires more energy.
"In a classical picture this would be like the electron, which is free to
meander through the bulk material, suddenly being forced to speed up in a
confined space," Lin-Wang Wang says -- analogous to a circus motorcycle
rider moving faster inside a steel cage.
The smaller the quantum dot, the wider the band gap. The band gap of gallium
arsenide in bulk, for example, is 1.52 electron volts (eV), while a quantum
dot consisting of 933 atoms of gallium and arsenic has a band gap of 2.8 eV,
and a dot half as big, with 465 atoms, has a band gap of 3.2 eV -- about
twice that of the bulk material. Changing the band gap, and thus the color
of light a quantum dot absorbs or emits, requires only adding or subtracting
atoms from the quantum dot.
Enter the dielectric constant
The electron-hole pair formed when an incoming photon boosts an electron out
of the valence band into the conduction band is called an exciton. An
exciton's energy (which corresponds to the color of the quantum dot) is not
identical with the band gap; instead it depends on a number of other
factors.
Most important is the dielectric function inside the quantum dot, which
mediates how strongly the exciton's negatively charged electron and
positively charged hole attract each other. Calculating the dielectric
function is thus essential to understanding how excitons behave in a quantum
dot (including its exact color) and how its electronic states can be
manipulated -- for example by adding dopant atoms that seed the
semiconductor with extra electrons or holes.
In 1994 Wang, then at DOE's National Renewable Energy Laboratory, and his
colleague Alex Zunger found a consistent relationship between a quantum
dot's band gap and its overall dielectric constant, a relationship
suggestive of the observed scaling between a dot's size and its band gap. A
quantum dot's electric constant is the average of the dielectric function
inside the dot. Advances in computing now make it possible to calculate the
dielectric function on the microscopic scale -- virtually atom by atom.
In the recent study, Wang and Cartoixà calculated what would happen if a
single-electron "perturbation" -- caused by a dopant atom, for example --
were introduced into the center of a 933-atom quantum dot of gallium
arsenide. To replicate a realistic quantum dot, they "passivated" the atoms
on its surface with fractionally charged hydrogen-like atoms, mimicking
reactions between the dot and its surroundings.
Using the Seaborg supercomputer at NERSC, the researchers were able to
determine the electron charge density of the perturbation throughout the
dot, using an ab initio calculation technique called local density
approximation. In the presence of a weak electric field the results were
virtually identical to similar measurements of the bulk material -- at least
until the responses were measured near the surface of the dot.
quantum dot made of silicon. In the smaller dots, measurements near the
center of the dot were still similar to the bulk measurements -- but varied
significantly where the perturbation vanishes, near the surface.
A simple model
Measured microscopically, the dielectric function inside a quantum dot is
the same as it is in the bulk material; measurements near a perturbation in
the center of the dot show no significant difference, but in a small dot the
differences are large near the boundary. Averaging makes it appear that the
dielectric constant mimics size-dependent changes in the band gaps. But in
fact there is no direct relationship.
"Using many hours of supercomputer time, we calculated all the electronic
states in these quantum dots when they were perturbed by a single electron
in the middle," says Wang. "We found they were the same as in the bulk." The
electronic response of a quantum dot thus depends on where it is measured,
and on the dot's size.
"If the response of the dot had been different from the bulk, it would have
been hard to model," Wang says. "Instead we were able to devise a simple
model for calculating the dielectric function on the microscopic scale that
gives virtually the same results as ab initio calculations with a
supercomputer. This should be very useful in future calculations."
This work was published in
the June 17, 2005
issue of PRL.
Source: Berkeley Lab |
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