We find that variation in the configuration and stoichiometry of the interfacial roughness leads to significant fluctuations in the transmission properties. These fluctuations are reduced by an attractive impurity placed near the center of the dot.
List of Figures
94 4.6 Resonant splitting versus in-plane pulse 95 4.7 Wave function magnitude for two impurities 96 4.8 Resonant splitting calculated in different dimensions 98. GaAs/AlGaAs double-barrier structure with alloy disorder, 1.119. part 5.6 GaAs/AlGaAs double-barrier structure with interference alloy, part
List of Tables
Chapter 1
Introduction
Overview and Motivation
The qualitative features of the current-voltage characteristics of the double-barrier resonant tunnel structure are understood [2, 10], but a good quantitative agreement with experiment is still lacking. Among those that may play a role in the operation of the double barrier and other nanostructures are electron-electron interactions, electron-phonon interactions, band structure effects, and structural and compositional imperfections.
Background
- Early Investigations
As the bias is further increased, the quasi-bound level drops below the conduction band edge in the emitter and electrons can no longer tunnel resonantly, leading to a reduction in current and negative differential resistance (see Figure 1.1). The first calculation of current-voltage characteristics in the double barrier resonant tunnel structure is due to Tsu and Esaki [2].
JNDRI+
Electron-Electron Interactions
The accumulation of space charge in the electrodes and in the well causes shielding of the applied bias and can lead to hysteresis and tristability, which we discuss below. Therefore, at zero bias, the conduction band edge in the well lies above the conduction band edge in the emitter (see Figure 1.2).
Load Line
Because hysteresis depends on the accumulation of significant charge density in the well, it is most prevalent in asymmetric structures, as described previously. Since we do not account for structures where, at zero bias, the collector barrier is significantly higher or thicker than the emitter barrier, this is the effect of space charge in the well.
Hysteresis
Electron-Phonon Interactions
Electrons in the double barrier can interact with acoustic phonons (distortion potential) and with optical phonons (polarization field). When the applied bias is so high that the quasi-bonded level in the well is below the edge of the emitter conduction band, electrons cannot resonantly tunnel through the double barrier, directly from the Fermi sea into the emitter.
Phonon-Assisted Tunneling
Fermi Energy
Emitter
Quasi bound Level
Collector
Band Structure
Local minima in the GaAs conduction band occur at the r-point (k = 0) as well as at the X and £ points. The r-point minimum is the lowest, about 250 meV below the L-point minimum and 400 meV below the X-point minimum.
GaAs Band Structure
Brillouin Zone
Elastic Scattering
For example, perturbation theory only allows the investigation of the weak scattering limit and important effects such as multiple scattering and apparent. In addition, correlations in imperfections such as clustering and ordering are neglected, and the models cannot adequately handle fluctuations.
Supercell Model
More importantly, the models to date are unable to treat transport in low-dimensional structures such as quantum wires and dots.
Bulk Electrode
Supercell
Quantum Dot Layer
When a small bias voltage is applied along a quantum wire, the conductance is quantized as a function of the Fermi energy in the electrodes, Ep, in multiples of 2e2 j h. The higher the Fermi energy in the electrodes, the more subbands are available to carry current.
Quasi-1 D Structures
Summary of Thesis and Results
Our numerical technique is based on a new method for calculating quantum transport in the tight-binding model. We find that varying the surface roughness of the quantum dots leads to significant fluctuations in the transmission properties.
Bibliography
Chapter 2
The Supercell Model
Formalism
{In)} are orbitals localized at the lattice sites, {En} are the site energies, and {tn,m} are hopping matrix elements. This together with the dependence of the onsite energy of the hopping matrix elements to the nearest.
Here N is the number of locations in the lattice r, representing the discretization of the Schrödinger equation. In the supercell method, the in-plane translation symmetry is reduced (by the period of the supercell) and we must choose a new basis.
Plane a
Physical Quantities
- Thansmission Coefficient
- Electronic Wave Function
- Probability Current Density
We derive below an expression for J which represents the flow of probability density from place to place. J is then a vector field which represents the flow of probability density from place to place.
- Current-Voltage Characteristics
In addition, at OK, the second integral vanishes in forward bias, because there are no empty states available in the emitter that can be filled by those tunneling from the collector. In the second case, T(E, k11) is approximated as independent of . 2.38) where m is the effective mass in the electrodes, E0 is the edge of the conduction band in the emitter, and f.l is the Fermi level in the emitter.
2. 3 Extensions
Chapter 3 Numerics
- Overview
- Solving Linear Systems
- Introduction
- Direct Methods
- Iterative Methods
- Storage Modes
- Overview
- Compressed Matrix Storage Mode
- Storage by Indices
- Storage by Columns or Rows
- Compre s sed Diagonal M ode
- Supercell Application
The densest parts are the M x M blocks of the form uvut which have no non-zero elements (cf. the sparsity pattern or arrangement of the non-zero elements in the matrix can vary from well-ordered (such as when all non-zero elements are concentrated along a few diagonals) to random.
Interior
Supercell Orbital Orbiting
Plane 1
- Benchmarks
- Concurrent Considerations
- Overview
- Amdahl's Law
- Topologies
- Load Balancing
Of the methods discussed, we have implemented four to solve (2.19): LU decomposition, the conjugate gradient method, the quasi-minimum residual method with preconditioning, and the quasi-minimum residual method without preconditioning. Amdahl's law states that the ratio between the execution time on a single processor and on N processors is.
Amdahl's Law
Implementations
The package provides a library of routines to perform inter-node communications, processor synchronization, input/output services, and processor allocation and control. More complicated algorithms present a greater coding challenge, and parallel programming is becoming an area of vigorous research.
Application to Supercell Calculations
This example raises some tricky algorithm design issues, such as load balancing. Needless to say, solving our problem on a parallel machine in this case would be a challenging and fascinating research problem.
Chapter 4
Neutral Impurities in Tunneling Structures
Introduction
- Background
- Outline of Chapter
Double-barrier structures with a diluted impurity concentration in the borehole were also considered [12]. We then consider level splitting and transfer effects in the case of two closely spaced impurities.
Simulation and Results
- Isolated Impurity
The level splitting manifests itself in the transmission differently for different orientations of the impurity separation direction relative to the incident plane wave. This is due to interference caused by repetition of the supercells (and thus impurity sites) in the growth planes.
Isolated Attractive Impurity
Locally Favorable Current Path
Resonance Shape
We find that the resonance width decreases as the impurity is moved towards the center of the barrier, another sign of increasing isolation from the electrodes. The maximum transmission increases to unity as the impurity approaches the middle layer of the barrier.
Two Impurities
We plot the transmission coefficient versus energy for different relative orientations of the impurity separation direction and the incident plane wave direction. Thus, the relative orientation of the incident direction and the impurity separation direction can play a significant role in the transmission properties of a tunnel structure.
Wavefunction Magnitude Along Line of Centers
Multiple Impurities
The upper panel of Figure 4.10 shows the transmission coefficient curves for different concentrations of attractive impurities in the well. The lower panel of Figure 4.10 shows the transmission in a double barrier structure with attractive impurities in the barriers.
Current-Voltage Calculation
To calculate the current at a particular bias, we need to integrate the transmission coefficient over the Fermi distribution in the emitter and over the in-plane momenta, k11, as described in Section 2.2.4. To confirm this, we plotted the transmission coefficient versus energy near resonance for plane waves incident with three different k11, all of the same magnitude, in the top panel of Figure 4.11.
Comparison with Experiment
In the bottom panel of Figure 4-11, we plot the transmission coefficients near resonance as a function of Ez, the energy corresponding to kz in the transmitter, for different lk1il. The resonances have slightly different widths and substantially different energy positions. We present here the results of our supercell calculation of the OK current-voltage characteristic of a single barrier with an attractive impurity in the middle layer.
In-plane Momentum Dependence
Summary
We have studied the interaction of two closely spaced impurities and found that the manifestation of level splitting in the transmission depends on the relationship between the direction of the incident electron and the direction of separation of the impurities. Depending on the impurity concentration and the area over which a structure is being investigated, the impurities can shift and broaden the resonances in a double barrier and increase the overall transmission in a single barrier or give rise to new resonances.
Chapter 5
Interface Roughness and Alloy Disorder
Introduction
- Background
- Outline of Chapter
In this thesis, we examine three-dimensional quantum transport in double barrier structures with interface roughness and alloy disorder. We first calculate transmission coefficient curves for a series of double barrier structures with interface roughness characterized by different island sizes.
Simulation and Results
- Interface Roughness
Different degrees of disorder and clustering of the barriers have different consequences for transmission. As the cluster size increases, the barriers become less restrictive, causing the resonances to broaden and shift to a lower energy.
Double Barrier With Interface Roughness
We observe satellite peaks above the n = 1 resonance in all cases except the virtual crystal calculation. For very large supercell size (as would be necessary to represent a macroscopic sample), the satellite peaks would merge into a broad collision above the n = 1 resonance.
Double Barrier With Large Interface Islands
Alloy Disorder
We draw a transmission coefficient curve for such a structure in Figure 5.5, where the alloy barriers are composed of an uncorrelated random distribution of AlAs and GaAs.
Double Barrier with Interface Roughness Probability Density lsosurface
Double Barrier With Alloy Barriers
For comparison, we include the results of a virtual crystal approximation calculation, where the alloy barriers are replaced by the same fictitious material as for the approximate interfaces in the previous section. For island sizes larger than about 5 - 6 nm, a substantial new structure develops in the transmission coefficient curves.
Double Barrier With Alloy Clustering in Barriers
Summary
We investigated the effects of interface roughness and alloy disorder on the transmission properties of resonant double-barrier tunnel structures. We found that interface roughness can cause in-plane momentum scattering, resulting in an additional resonance structure above the n = 1 peak, and a localization of the wave function that shifts down and broadens the n = 1 resonance by island sizes on the order of the electron deBroglie wavelength .
Chapter 6
Introduction
- Background
- Outline of Chapter
Interface roughness at the wire boundaries in the range of a few monolayers is currently unavoidable. In Section 6.2.1, we give some examples of supercell calculations of imperfection effects such as interface roughness, impurities, and structural variations in a quantum dot.
Simulation and Results
- D e vice Imp e rfections
The dark shaded circles represent electrode material, the solid circles represent barrier material, the open circles represent the well material in the cavity, and the light shaded circle represents an impurity.
Quantum Dot Waveguide
Interface Roughness Fluctuations
We see that there is about 10% - 20% variation in the resonance width and about 5% variation in the maximum transmission. This means that the total number of barrier sites in the shell can vary, giving different effective levels of confinement.
Quantum Dot Waveguides With Rough Interfaces
Neutral Impurities
We calculated a series of transmission coefficient curves for a point with interface roughness and an impurity in the middle. The rough wall spot is that of sample 1 in the previous section, and the impurity strength is varied from jj.U jt = -4.9 (strongly attractive) to 2.2 (repulsive).
Neutral Impurity
The n = 1 cavity mode has an antinode in the center, at the location of the impurity. Here, the n = 1 resonance is lowered less for an impurity in the center than for an impurity near the ends of the dot.
Weakly Attractive Impurity
Strongly Attractive Impurity
Here, the standard deviation of the n = 1 resonance position for the ten samples is 0.0007eV compared to 0.008eV without the impurity. In the presence of the impurity, the standard deviation of the widths for the ten samples is 6.3%, and that for the maximum transmission coefficients is 0.5%; without the impurity, the standard deviation is 6.8% for the widths and 2.2% for the maximum transmission coefficients.
Quantum Dot Waveguides With Rough Walls
Summary
Bottom panel: fluctuations in the resonance width and maximum transmission for quantum dots with ten different rough-walled configurations, each with a moderately attractive (6.U jt) impurity in the center. Values for the resonance width and maximum transmission coefficient of 3.5 nm x 3.5 nm x 4.5 nm and 2.5 nm x 2.5 nm x 3.5 nm smooth-walled dots with an impurity in the center are also shown for reference.
Quantum Dot With Rough Walls
Nevertheless, the presence of more than one impurity in the dot can lead to a complex resonance structure dependent on the impurity configuration, especially at high concentrations. Based on the concept demonstration of fluctuation reduction by an isolated impurity near the center of a dot, there is hope that problems with variation at the atomic scale can be overcome.
