Calculations

In this section we will demonstrate how the package can be used to do various calculations such as energy levels, wavefunctions, convergence plots, and energy level evolutions with magnetic fields.

Energy Levels

To calculate energy levels, we need to choose a basis. For larger basis sizes the calculation will take longer and require more memory (the size of the matrix is the size of the basis squared) but the convergence will be better.

We choose a basis with a single total orbital angular momentum quantum number . The properties of the circularly symmetric quantum dot allow us to separate calculations for different values of , thus reducing the size of the basis needed for each calculation allowing for greater accuracy. We then calculate the interaction Hamiltonian using the function InteractionHamiltonian.

The calculation of this matrix can be time-consuming, so it is worthwhile to save the result, as the interaction Hamiltonian calculated for a particular number of electrons and spin quantum numbers and can then be used for any value of the harmonic well constant, magnetic field, relative effective mass, and relative permittivity. We have also loaded previously calculated integral definitions using the function LoadIntegralandHamiltonianData to speed up our calculation and saved the data afterward using SaveIntegralandHamiltonianData to speed up future calculations run in other sessions.

Next we set the magnetic field strength to 0 Tesla remembering that firstenergygap was set to 10 meV previously and relativeeffectivemass and relativepermittivity are set to the parameters for GaAs.

We then can calculate the Hamiltonian for GaAs with these particular values of the harmonic well constant and magnetic field and plot the energy levels.

The plot shows the first 10 energy levels for a GaAs quantum dot system of three electrons with spin quantum numbers , and total angular momentum quantum number in a zero magnetic field. The harmonic well constant is implicitly defined by a first energy gap of 10 meV.

Convergence Plots

As mentioned, the accuracy of our result as well as the computational time required increases with the size of the basis. We can test the convergence with the following series of functions.

As the interaction Hamiltonian for lower cutoff energies is a submatrix of interaction Hamiltonians with higher cutoff energies, we can do convergence tests using a single interaction Hamiltonian. We will use the interaction Hamiltonian from the previous section.

The Convergence function takes submatrices of the Hamiltonian using different basis sizes as specified by the input sizelist and calculates the energy of the levels specified by wantedlevels. The ground state is specified by 1, the first excited state by 2, and so on. This allows the convergence properties of the solution to be ascertained.

The list basis size can be specified manually, but a convenient method is to choose all the basis sizes that correspond to the different cutoff energies.

In the preceding code, we have specified wantedlevels to examine the ground state and first excited state. We set cutoffmin to be the first cutoff energy that will produce the highest energy level requested, as we can only approximate up to levels when we have basis elements.

As there may be large gaps in the basis sizes, we can also add in points at intermediate intervals. Here, we add a basis size of 85.

We can then calculate the energies of the ground state and first excited state as we increase the basis size. The energies are in meV.

The following plot shows convergence of energies of the two lowest states.

We can see that the convergence is quite good for relatively small basis sizes in this case.

Electron Density Plots

We can extract electron density plots for various energy levels by calculating the normalised energy eigenvectors of the full Hamiltonian and using the function ElectronDensity. We can then plot the electron density and use lengthscale to give the length dimensions in nanometers.

First we calculate the energy eigenvectors for the first six energy levels.

Next we use ElectronDensityFunction to convert the normalised energy eigenvector into a probability density function by combining it with the basis basisLzero. This is then simplified to cancel out the complex components. As the density function is a measurable quantity, we are guaranteed that the complex components will cancel. Chop is used to remove any complex components that remain due to rounding.

We can see that we end up with a numerical approximation to the wavefunction. Here is an example.

The following plot shows the electron probability density functions for the first six energy levels with , , and for the three-electron quantum dot system that has been the subject of our examples.

Energy Levels versus Magnetic Fields

The SACI package also allows the calculation of the evolution of the ground state with a magnetic field using a single interaction Hamiltonian.

We can use LevelVsField to calculate a list of energy levels for a range of magnetic fields defined by magmin, magmax, and step given in Tesla, and for the levels defined by wantedlevels. Here we choose the first six energy levels and plot for the magnetic field range of 0 to 4 Tesla at intervals of every 0.5 Tesla.

Calculating the energy levels for one value at a time has the benefit of producing more accurate results for a given matrix size and allows for easier separation of different energy states from the table of eigenvalues. The following plot shows the results.

Note that although we are plotting six levels, two pairs of these remain degenerate as the magnetic field changes, so we only see four lines in the plot. Running the program several times, it is possible to plot curves with different spin quantum numbers and total orbital angular momentum quantum numbers on the same graph. From this, it is possible to see the graphs for different quantum numbers intersecting as the magnetic field increases. This corresponds to magnetic transitions in the energy levels. Magnetic transitions in the ground state of quantum dots have been measured experimentally.