Wavefunction Engineering
a new paradigm in the design of
quantum
semiconductor devices
Wavefunction Engineering
a new paradigm in the design of
quantum
semiconductor devices
Wavefunction engineering refers to the unprecedented ability to specify the localization of carrier wavefunctions in quantum semiconductor heterostructures through control over the growth, geometry, and material composition. The work done at Quantum Semiconductor Algorithms has led to the development of computational methods based on finite element and tight-binding methods which allow us to explore basic issues in quantum mechanics and also to design new devices in which specific optical or electronic properties can be optimized.
This conceptual breakthrough has offered the freedom to ask fundamental questions about the behavior of nanometer (nm) scale semiconductor structures and their quantum mechanical behavior, and to develop simulations for the active response of quantum devices.
Semiconductor heterostructures are structures composed of
more than one semiconductor. When the defining geometries
of the individual parts are smaller than ~100 nm quantum
mechanical effects associated with Heisenberg's
uncertainty relation become important. To provide a
perspective, atomic dimensions are of the order of 1
Angstrom or 0.1 nm. The electronic and optical properties
of such "quantum" semiconductor structures are profoundly
altered from those of the bulk semiconductors. Such
structures can now be grown routinely, and their properties
tailored to specific needs. For example, a number of
structures have been grown using QSA's software to specify
the structures for developing lasers in the
3-5
These are benchmark results that have marked the progress of
wavefunction engineering.
In the late 1960's, technological advances in the
techniques for the growth of high quality crystalline
semiconductors had reached an unprecedented level of
sophistication. It became possible to to grow semiconductor
structures, in the laboratory, atomic layer by layer
in ultrahigh vacuum approaching that of interstellar space.
The process of growing such structures is known as
molecular beam epitaxy (MBE).
In the late 1960s, Esaki and Tsu suggested that it should
be possible to grow alternating layers of GaAs and
Aluminum-Gallium-Arsenide (AlGaAs) in a periodic array to
form a superlattice structure which would have
remarkably different electronic properties from those of
bulk GaAs or AlGaAs. These would be man-made crystals
with a super-periodicity, over and above atomic periodicity
of crystalline materials, corresponding to the layer
thickness of d = dGaAs
+dAlGaAs, with a binary modulation of the
material composition along the crystal growth
direction.
When a heterostructure is formed, its electronic energy
bands are altered from those of the bulk system. In order
to understand this, we look at the bulk band structures of
GaAs and of AlGaAs grown one on top of another. The choice
of GaAs and GaAlAs is a typical one, though any of the
compound semiconductors made from the Group III elements of
the Periodic Table, such as Ga, In, Al, and Group V
elements, such as Sb, As, or P, are possible materials.
When two materials such as GaAs and AlGaAs have a commn
interface, the band offset between the conduction bands
stops an electron near the bottom of the conduction band in
GaAs from propagating into the AlGaAs region. The band
offset acts as a quantum mechanical barrier. The interface
is said to be of Type I if the band gap of one of the
materials is located within the bandgap of the other. In a
Type II system, the bandgaps are situated with a
displacement in energy such that one energy bandgap does
not contain the other. This is shown below.
Click on the icon to see a larger picture of the quantum
well layered structure.
The raised energy of the electrons is a manifestation of
Heisenberg's uncertainty relation. With the position of the
electron determined to be within the quantum well, the
uncertainty in its momentum along the growth direction is
given by Heisenberg's relation. Hence its energy is
increased.
This leads to a larger effective enrgy bandgap for the
structure.
Correspondingly, when electrons and holes
recombine the energy of the emitted photon is larger than
the energy of a photon emitted in bulk GaAs when electron
and hole recombination occurs there. Moreover, the energy
gap in the quantum well can be controlled to some extent by
the layer thickness -- the smaller the layer thickness the
greater the bandgap.
This freedom to tailor the energy band structure in quantum
heterostructure led Capasso and co-workers to coin the
phrase "bandgap engineering". This concept was
central to the developments over the 70's and 80's, during
which time an expanding menu of materials and impurities
was brought to bear on the growth of a very large array of
composite quantum semiconductor structures for specific
purposes. For example, applications to quantum well
photodetectors, lasers, high-electron mobility structures,
modulators, electro-optic switches were all designed using
this fundamental idea.
When the barriers of a quantum well are doped with impurity
donor atoms, the donor-bound electrons fall into the
quantum well. This redistribution of carriers alters the
energy band-edge. A self-consistent solution of the
Schrodinger and the Poisson equation has to be implemented
in order to determine the new band-edge. This is a
nonlinear problem, and is considered to be a difficult one
to solve for an arbitrary layered structure. At
QSA, we have developed software to solve this problem in a
two-band model. A typical calculation requires about 30-40
iterations for convergence of the bound states in the
quantum well and the Fermi level.
These carriers in the quantum well region can be made to
propagate along the quantum well layer. In the absence of
impurity ions in this layer due to modulation doping,
electron mobilities of the order of 106 cm2/Volt-sec
have been observed in GaAs systems. This is an order of magnitude
greater than the conventional bulk mobilities. The following
figure illustrates how this flow of carriers along the quantum
well layer can be manipulated to obtain transistor action.
When a layer of AlGaAs is sandwiched between two thick
layers of GaAs we have a single barrier structure. When the
structure has two barriers we have a double barrier
resontant tunneling diode (DBRTD) structure (this requires
5 layers). The following figure compares the action of a
DBRTD with a more convential p-ntunneling diode under the
action of an externally applied field.
Click on the icon to see a larger picture of the comparison
between a conventional tunneling diode and quantum resonant
tunneling double barrier structure under bias.
It was suggested by Sakaki that two dimensional confinement
of a carrier can be achieved by embedding a quantum wire of,
say, GaAs in AlGaAs. This has distinct advantages in that
effects of scattering are reduced by the reduction in
"phase space". Typically one does not grow quantum wires of
rectangular cross-section. Usually, v-grooves are
ion-etched on substrates and the wire material deposited in
these grooves. Then a cap layer is grown to enclose the
quantum wire. The additional confinement of carriers in
quantum wires allows laser action in these structures
to survive upto room temperature.
Click on the icon to see a larger picture of the quantum wire structure
representing GaAs wire embedded in AlGaAs.
Click on the icon to see a larger picture of the 2D "kitchen-sink"
potential well in a quantum wire structure.
Click on the icon to see a larger picture of a Checker-Board
Superlattice (CBSL), a bi-periodic array of GaAs/AlGaAs quantum
wires.
The band structure, the nonlinear optical properties of
such a structure were calculated by Ram-Mohan and Shertzer in 1990.
The actual wavefunctions of the carriers show considerable
structure. Two examples are given below. By changing the
geometry or the composition of such a structure it would be
possible to alter the wavefunction substantially.
Click on the icon to see a larger picture of the wavefunction
at the Gamma point in the CBSL.
--->
Click on the icon to see a larger picture of wavefunction
at the point midway to the Brillouin zone edges along
qx and qy in the CBSL.
--->
It has been found that quantum dots of semiconductors are
fairly easy to grow. The quantum dots are, at present,
usually of nonuniform size. With the right growth
conditions, strain-induced clumping of the deposited
material in an MBE chamber gives rise to pyramid-shaped
quantum dots. While the growth of such 3D confining
structures has been demonstrated, there are very few good
examples of modeling of the electronic properties of
quantum dots. At QSA, we have been making progress towards
developing software tools for such applications. The
figure shows a single pyramidal quantum dot grown on a
substrate.
Click on the icon to see a larger picture of a pyramidal quantum dot
structure.
Click on the icon to see steps in the mesh generation for quantum
dots of two different shapes.
With the advent of software tools it has now become possible to
address the next generation of issues in heterostructure design.
These issues becomes even more relevant when we start
considering quantum wires and quantum dot structures. The
modeling of such structures and their electronic properties
requires even more intense computations. At QSA, we have
been laying the groundwork for development of high quality
software for compute-intensive calculations. The next set
of computational, modeling, and visualization tools will be
essential for a thorough understanding of such 2-D and 3-D
confinement systems. The commercial development of devices
will require that the computational and modeling tools be
developed hand in hand in order to shorten the time from
the definition of novel ideas, the development of the
design of such new devices, and the growth and
characterization of such quantum devices. QSA intends
playing an important role in these developments.
We hope this brief and informal tour of typical quantum
heterostructures has provided the reader with a simple
pictorial introduction. More details can be obtained from
the publications, based on QSA software, listed in the
"Outcomes" section of these webpages.
To see examples of concrete results obtained using the software
Click on the QSA logo below
A Tutorial
An elementary tutorial on the concepts that are inherent in
Wavefunction Engineering is presented below.
What are quantum semiconductor heterostructures?
To see a schematic of an MBE chamber
click on the icon.
In an MBE chamber, atoms of elemental semiconductors such
as Silicon, or of compound semiconductors such as Gallium
Arsenide (GaAs) are evaporated out of ovens onto a
substrate semicoductor. The layers then take on the
crystalline structure of the substrate forming what is
known as a pseudomorphic crystalline structure.
Such structures were soon fabricated, and superlattices and
other quantum heterostructures have the novel properties
suggested initially.
Click on the icon to see a larger picture of the superlattice.
Electronic properties of layered heterostructures
Click on the icon to see a larger picture of the bulk band structure.
In the above figure, the layered structure is shown with
the growth direction along the horizontal axis. The energy
E versus wavevector k plots of the electron
propagating in the two materials are shown. These curves
are called energy dispersion curves for the electrons. It
is only along these curves that the electrons can have
allowed energy values. The energy band gap between the
conduction and the valence energy bands is a forbidden
energy gap. This bandgap is different for GaAs and for
AlGaAs. They are also juxtaposed (in an absolute energy
scale) such that their valence and conduction bands are
shifted. The offset in energy at the bottom of the
conduction bands is called the conduction band offset, and
is denoted here by Vs.
Click on the icon to see a larger picture of the band offset.
When a layer of GaAs is sandwiched between two "infinite"
layers of AlGaAs, the carriers in GaAs are trapped in the
GaAs layer along the growth direction. This leads to a
confinement of the electrons in the conduction band and of
the elementary excitations of carriers (called "holes") in
the filled valence bands. This leads to a quantum well
structure. The carriers in the quantum well are free to
move in the in-plane direction.
The confinement along the growth direction characterizes
quantum wells as structures that have "1-D
Confinement" for charged carriers.
For a quantum well, the energy levels in
the well are raised in the conduction band for the
electrons, and lowered in the valence bands for holes.
Click on the icon to see a larger picture of the quantum well energy level
diagram.
The confined electrons in the quantum well occupy discrete
energy levels as defined by solving Schrodinger's equation
for a one-dimensional quantum well. From each of these
"sub-bandedges" we still have energy dispersion in the
in-plane direction, and a continuum of energies associated
with motion in the plane. Transistors and Resonant Tunneling Structures
Click on the icon to see a larger picture of the lateral tunneling
quantum well structure.
The negative differential resistance shown by the I-V curve
is a characteristic required for transistor action. With
multiple resonant states, in the quantum well between the
double barriers, we will have one peak for each resonant
state. It has been shown by Capasso that two DBRTDs can
replace 24 conventional transistors in logic
circuits. Thus, we gain in functionality as well as in
size, and hence in the speed of the device. Electronic properties of quantum wire heterostructures
The above figure shows a quantum wire of rectangular cross-section.
The potential well in such an idealized well will be like the
kitchen-sink potential well shown below.
The symmetry properties of a square quantum well were explored
by Shertzer and Ram-Mohan in 1990.
It is possible to envisage theoretically a bi-periodic array
of rectangular quantum wires of GaAs and AlGaAs. Such an
array would, of course, be difficult to grow at the present
time. However, such structures are straightforward to
model within the framework of the Finite Element Method.
The potential barriers experience by a carrier in the the
conduction band of such a structure would be as shown below.
Click on the icon to see a larger picture of the potential energy
barrier in one period of a CBSL structure.
The wavefunction engineering of such 2D confining structures could
lead to remarkable optical and transport properties in them. The aim
of the software tools being developed at QSA is to provide precisely
the type of freedom, in designing and modeling such structures, that
would lead to exploring new mechanisms in electronic transitions and
in electronic localization. The potential for device applications is
very promising.
Click on the icon to see the sequence of computations:
2D mesh generation, FEM application, constructing
wavefunctions. Finally, optical matrix elements, etc., can
also be calculated to obtain optical absorption
coefficient, dielectric properties, etc.
The transport properties of 2D quantum waveguides have been the focus
of considerable recent research. The quantization of resistance, the
formation of "scarred quantum states" in 2D resonators, and the Bohm
Aharanov effect in complex elctron waveguides --- these are the type
of effects that can manifest themselves in such structures. The
potential applications for high speed transistors using 2D waveguiding
of electrons can be very easily modeled using the Finite Element
Method. A suite of software tools are being created at QSA to perform
such modeling. The diagram below shows a simple "stadium"-like
reservoir in a quantum waveguide.
Click on the icon to see a larger picture showing examples of
a 2D electron waveguide.
Electronic properties of quantum dots
The steps in modeling a quantum dot include meshing the physical
volume, given the surface, and applying the FEM to include effects
of physical conditions such as external fields, internal strain.
The results from the FEM are post-processed to derive quantities of
physical interest. The finite element method provides the flexibility
to model the physical shapes, include the correct boudnary conditions,
incorporate external fields and builtin strains, and provide a
very effective, flexible, and stable, working environment for
3D modeling of quantum systems.
Aspects of Wavefunction Engineering
It is the ability to answer such questions through the
selection materials, the application of external
perturbations such as external magnetic fields or electric
fields, and the like, in a simulation software that
represents the essential aspects of wavefunction
engineering. In this we are clearly going beyond bandgap
engineering into shaping the basic, fundamental, and
dynamic properties inherent in the wavefunctions. Through
wavefunction visualization choices we have to make for
optimizing specific properties also become transparent.
developed by QSA, click here for
Outcomes
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