The energy levels and wavefunctions of electrons in an insulator are
input to an understanding and computation of the linear and nonlinear
susceptibilities and other optical properties that are of interest
experimentally and technologically. There are several important
concerns in implementing calculations based on the Local Density
Approximation (LDA) treatment of band energies and wavefunctions.
the LDA description of band states the electron-electron interaction
effects are approximated by the effective potential that depends on the
local density. This fails to accurately represent the conduction
(i.e., the unfilled) states. The next best approximation is believed
to include an energy-dependent (self-energy) shift in the conduction
electron energies. This so-called GW approximation -- involving the
dynamical screened Coulomb interaction -- requires 1-2 orders of
magnitude more computing. We are actively exploring three
implementations of this `GW' approach.
- Local field effects.
When an external field encounters the periodic lattice, it induces
charge densities that give rise to "local fields" that vary on the
atomic scale. These local fields must be included in computing the
- First-order: always reduces; on few to 10% scale.
- Second-order: either sign; typically 10-50%.
- Rotatory power: can change sign and value by factor of 10.
- Importance of accurate band gaps, E_gap.
The susceptibilities scale as 1/E_gap (1st order), 1/E_gap^3 (2nd order)
LDA typically underestimates the band gap by 0.5 to 1.0 eV. We use
a `scissors operator' to shift conduction bands to match the experimental
gap or a gap estimated by many-body theory (see below).
- Systematics across related and different materials. The
`scissors operator' works well for small and medium-gap
semiconductors. But for wide-gap insulators we find empirically that a
smaller or no shift of the conduction bands yields results closer to
experiment. This is not understood.
Central to faster computing hardware is producing increasingly smaller
line resolution in silicon chips. This is done by implanting ions such
as boron. After the implantation, the material is annealed to restore
the crystal structure and the high mobility. Unfortunately, the boron
dopants show transient enhanced diffusion (TED) in which the boron
diffusion `constant' is 1000 times its value in bulk for a transient
but significant time in the annealing. Understanding this process has
been identified a key step on the `highway' to continued improvement in
- Momentum space. Most widely used,
suitable for bulk calculations with a few atoms in unit cell and
periodic boundary conditions. Applications: superlattices
and simple defects.
- Real space. Most suitable for clusters
and defects, both simple and extended. There is only one version of
this code, due to Godby and recently parallelized.
- Localized description. Designed for
very large unit cells and utilizing localized wavefunction of the
An extended defect in Si -- called the (311) defect due to its habit
plane -- has been identified as a possible source/sink of Si
interstitials believed to dominate boron diffusion. We are engaged in
large-scale parallel simulations of extended defects in silicon.
Essential to the treating 1000s of atoms is the use of the simpler,
tight-binding Hamiltonian. Even so it is necessary to time-evolve a
defect for a picosecond at 300-600 K to avoid being trapped in local
minima. Then we fully relax the atomic positions by `steepest
descent' until the atomic force on each atom is less than 0.1
eV/Angstrom and the effective temperature of the lattice is less than
0.1 K. Initial progress in identifying possible forms of the <311>
defect has raised as many new questions as it offered possible answers
to the original concerns that provoked the research.
Conventional electronic structure methods (LDA-based) fail to explain
the magnetic transitions of the the 3d elements (Cr, Mn, Fe, Co, Ni).
They both dramatically overestimate the transition temperature and
fail to `predict' differing magnetic states for differing crystal structure
of the same element. The so-called LDA + U method is a first attempt to
add correlation effects to the band structure work. We have extended
this approach to compute the temperature depended magnetic
susceptibility with an effective electron-electron interaction computed
with real electronic functions and including the effects of
particle-particle and particle-hole (screening) scattering.
The quantitative understanding of atoms and molecules has continued to a
theoretical challenge as the experimental resolution has always stayed
ahead of theory. Mille-electron-volt resolution sets very high
standards. The classic chemical approach of configuration interaction
-- adding increasing numbers of excited states (as Slater determinant)
to achieve increased accuracy. Our concerns are different from those of
the chemists. We wonder whether these accurate approaches can be used
to produce data set of correlation energies for atoms and molecules that
can be used to produce a better LDA approximation for the effect of
electron-electron interaction? Can we find faster ways of doing these
calculations so as to permit their use in small clusters as models for
The configuration-interaction approach suffers from the fact that
computational time grows exponentially with the number of electrons.
Furthermore the the number of configuration grows exponentially with the
number of one-electron basis functions used to generate
configurations. These problems are addressed in two ways.
(i)Partition of the basis. The partitioning takes advantage
of so-called natural orbits to develop a basis set each of set of which
as an occupancy an order of magnitude smaller than the next.
(ii)Many-electron state selection. The one-electron basis
partitioning leads a division of the state space into an external and
internal subspaces. A perturbative calculation of the external
subspace can be combined with an exact treatment of the internal
subspace to yield results comparable to those using the full state
Typically the basic set reduction approach would reduce the 10^8
number of states in full CI to 10^5. This approach produces the same
answer with 10^2 states.
Basis reduction allows an accurate calculation of the correlated state
in a sufficiently short time that it is feasible to consider the time
evolution of the state. An atoms in a femtosecond pulse could be
computed at subfemtosecond intervals to accurately track the
evolution of various excited states.
- The orthogonal basis functions in each determinant are different
so that determinants are not orthogonal. This severely slows down the
calculation of matrix elements.
- The small number of determinants making up the state allows
for a better understanding of the correlations.
More interestingly, almost all the object-oriented software can be
adapted to study semiconductor quantum dots in a laser pulse. It turns
out that one can readily identify effects demonstrating important
correlations in the dots -- that is, easy to measure experimentally.
This is such a new area that it is difficult to predict where it is
going. But effective probes of correlation in strongly interacting
systems are few. This could be an important new probe.
The technology responsible for high-speed computing allows physicists
to create structures more often seen in quantum mechanics text books:
tunneling between two quantum wells, a periodic array of square wells.
This ideality permits comparison between analytic calculations and
actual measurements. Three areas of interest are described below.
Others are in Many-body physics section.
A long term development in dilute magnetic semiconductors has entered
the nanostructure era. The groups Awschalom at UCSB and Samarth at Penn
State make and measure layers of magnetic atoms (Mn) by probing the
decay of a gas of magnetic excitons created with a fast laser pulse.
The magnetization and decay in an applied magnetic field exhibit a range
of phenomena never seen before. We are developing models to
qualitatively and quantitatively understand these effects.
The dynamics in a one dimensional array of potential wells -- forming a
so-called miniband -- offers a range of predicted phenomena: Bloch
oscillations, Stark shifts, dynamic localization. The latter
corresponds to a localization of the wavefunction when the quantity,
(applied electric field)(well spacing)/frequency, is equal to zeros of
the first Bessel function. We are calculating properties that would
most dramatically exhibit this effect.
The example above are just a few of the possible femtosecond effects.
We are developing a formalism for treating electron-electron and
electron-phonon effects in these short time scales. Other
non-equilibrium formalism are devised to work in the steady state or
long-time limit. Here we want to describe the initial phenomena before
the long-time decay process set. This is the relevant time regime of
Since the 60's physicist have been studying strongly interacting
systems. The initial success in study translational invariant models
for 3D systems has encouraged efforts to study correlations in real
materials and in materials of restricted dimensionality. For example,
in one dimension, the interactions between electrons are strongly
modified by the restricted motion: one particle cannot get around
A very early prediction that the electron-electron scattering rate at
low temperature of T^2 in 3D would change to T^2 log(T) in two
dimensions. Only recently in layered electron gasses has experimental
verification been possible. Deviation from the prediction have sent us
back to considered contribution to the rate that cannot be understood in
the terms of the Fermi Golden Rule. These so-called vertex corrections
should be larger due to the restricted motion in lower dimensions and
should effect them magnitude of the rate.
A one-dimension periodic array of atoms in a half-filled band is
unstable to dimerization: alternating short-long atomic spacings which
opens a gap in the spectra. This is called the Peierls effect.
Measurement of the absorptivity do not see this gap clearly, the
absorptivity turns on over smoothly. However, in one dimension the
zero-point motion is enhanced; so much so that the atoms provide a
disorder potential to scatter the electrons. Including this idea in
the conductivity lead to good agreement with published data. More
interesting is the possibility of unusual states in the Peierls gap:
soliton and polarons. Since the detection of solitons was strong
evidence for the Peierls transition, it is important to predict the
absorptivity spectrum of solitons in the zero-point-motion disordered
state to see if the experimental shapes can be explained.
Your comments and
suggestions are appreciated.
[College of Mathematical and
[Center for Materials
Edited by: email@example.com [October 1996]