Theory

Spin Eigenfunctions

When measuring the component of an electron's spin angular momentum along an axis, the result is either or . Conventionally, we denote this axis as the -axis and work in units in which is one. The two spin eigenstates in which we are certain of the electron's spin projection are denoted and , which are the eigenfunctions of the projection operator. To find projected spin eigenfunctions for multiple electrons we can simply multiply single electron spin eigenstates (e.g., or for five electrons). We call these simple multiplications of 's and 's elementary spin eigenfunctions. Quantum mechanics also allows linear combinations of projected spin eigenfunctions (e.g., ). As long as each term in the spin function has the same number of 's and 's, we will still have an eigenfunction of , that is , where is the spin projection quantum number given by half the number of spin up electrons minus half the number of spin down electrons ( in this case).

The square of the total spin is also quantised:

where is a multi-electron spin eigenfunction (spin eigenfunction) and is the total spin quantum number. Elementary spin eigenfunctions with the same number of 's and 's will be eigenfunctions of , each with the same projected spin quantum number , but they are not in general eigenfunctions of . Thus will in general be a linear combination of elementary spin eigenfunctions.

One way to calculate the spin eigenfunction is to use the Dirac identity to write the operator in terms of permutations

where is the number of electrons confined in the quantum dot and is the permutation operator. To form eigenfunctions of , we can calculate the action of on each of the elementary spin eigenfunctions. We convert into a matrix with the vector of coefficients of the elementary spin eigenfunctions in the linear combination. We can then solve the matrix eigenvalue problem to calculate all the spin eigenfunctions. There may be more than one eigenfunction for a given total spin quantum number, and so we carry out an orthonormalisation procedure to calculate an orthonormal spin eigenspace.

In the SACI package, we define and implicitly using ElectronList, in which 1 represents a spin up electron and 0 represents a spin down electron.

is the length of the list and is defined by the number of ones and zeros in the list. In this case and . We then define the total spin quantum number .

must be compatible with the number of electrons and to get meaningful results. With , , and in place we can calculate the spin basis and then display it.

We can see that we have a 2D spin eigenspace as there are two basis vectors produced. Calculating the spin eigenfunctions is a prerequisite to the rest of the calculations. Henceforth, we label spin eigenvectors , where is the number of electrons, and are the spin quantum numbers, and is an index to identify vectors in a multi-dimensional basis.

Fock-Darwin Solutions

The solution for a single electron moving in a 2D harmonic well under the influence of a perpendicular magnetic field was solved independently by Fock [15] and Darwin [16]. With the Hamiltonian

we get the energy eigenvalues

and the wavefunctions

where is the effective mass, the harmonic well constant, the cyclotron frequency, the effective harmonic well constant, the magnetic length, , the magnetic field, and the generalised Laguerre polynomials.

To coincide with experiments done on a circularly symmetric quantum, we will choose solid-state parameters appropriate for GaAs (i.e., and , where is the electron mass and is the vacuum permittivity).

When we are working in effective atomic units, the unit of length is the effective Bohr radius. This is given by the function lengthscale with the result in nanometers.

Here is a plot of the probability density for in GaAs in a harmonic well in which the gap between the first and second energy levels is 10 meV and there is no magnetic field. Fixing the first energy gap is a convenient way to define the harmonic well constant as it can be determined experimentally. We also set the magnetic field to 1 Tesla and plot six effective Bohr radii from the centre of the well.

The base scale is in nanometers and indicates the scale of the quantum dot.

Multiplying one-electron functions together (e.g., ) allows us to form multi-electron spatial wavefunctions that can be combined with the spin eigenfunctions. However, the combinations of spin and spatial wavefunctions must obey Pauli's exclusion principle as described in the next section.

The Antisymmetry and Representation Matrices

The Pauli exclusion principle states that because electrons are indistinguishable fermions (half-integer spin particles), any wavefunction that describes the motion of electrons must change sign if both the spin and spatial coordinates of any pair of electrons are interchanged. Consequently in the -electron case, if we apply any permutation to the electrons, the following holds:

where is the multi-electron wavefunction and is the parity of the permutation. To antisymmetrise a wavefunction we apply the antisymmetrisation operator

where is the symmetric group (the set of all permutations of electrons under composition). A wavefunction must be an eigenfunction of the antisymmetrisation operator if it is to fully describe the physical properties of the electrons.

Since the permutation and spin operators commute, if we apply a permutation to a spin eigenfunction, the result is another spin eigenfunction. This new spin eigenfunction is not, in general, one of the functions already calculated but a linear combination of them. In fact, the coefficients of the linear combination form matrices which together form a representation of the symmetric group. Explicitly

where is the spin eigenfunction for electrons with total spin quantum numbers and in an eigenspace of dimension , and is the element of the representation matrix . It can be shown that for any two permutations and

where is the composite permutation derived by acting with first then . Equation (9) satisfies the condition for a representation of the symmetric group. These representation matrices and their properties are useful in simplifying the calculation of the Hamiltonian matrix.

We can see that our spin eigenfunctions are eigenfunctions of by the representation matrix of The permutation is given by a reordering of the list. For example, in a three-electron system, represents the transposition of electrons 1 and 2 and the transposition of electrons 2 and 3.

This shows spin eigenfunction two is unchanged by the permutation and spin eigenfunction one changes sign under the permutation. In general the action of a permutation can lead to a linear combination of the spin eigenfunctions.

This shows that under the permutation the spin eigenfunctions transform via a linear combination. The following demonstrates the relation given by equation (9).

Spin-Adapted Basis

To get an -electron spin-spatial basis function we can multiply single-electron wavefunctions, given by equation (5), combine them with a spin eigenfunction of and , and then antisymmetrise the result. The antisymmetrisation process restricts which spatial and spin combinations give nonzero results as discussed in detail in reference [13].

We use a five-electron quantum dot as an example, where the system wavefunction and there are two doubly occupied spatial orbitals. Two possible spin eigenfunctions for are

Both and are antisymmetric under permutation , but is antisymmetric under permutation , while is symmetric under permutation . This means vanishes but is nonzero.

The CalculateSpinBasis function applies a procedure to ensure the spin basis functions are also eigenfunctions of the permutations , ... . This property is required for the Hamiltonian matrix element formulae, described in the Matrix Elements Rules section, to be valid. A basis whose spin eigenfunctions are simultaneously eigenfunctions of , ... are guaranteed to exist as the permutation operators together with and pairwise commute.

Diagonalisation

The multi-electron Hamiltonian is given as the following:

This is simply the sum of the single-electron energies defined by equation (3) plus the pairwise Coulomb interactions. The fact that we are working in a solid state media is modelled by using the relative effective mass and the effective permittivity .

We wish to solve the following equation:

for the energy levels and wavefunctions . The standard diagonalisation method is to approximate as a sum of a finite sum of basis functions with the convergence improving as we increase the size of the basis. This converts the problem into a matrix eigenvalue problem. However, to get a good approximation to the solution we need to choose the basis wisely as there is a limit to the size of the matrix that is computationally feasible. We also need to be able to efficiently calculate the Hamiltonian matrix elements.

Basis Choice

The SACI method uses normalised, antisymmetrised products of spin eigenfunctions and spatial wavefunctions given by products of Fock-Darwin solutions as basis elements. The elements of our Hamiltonian matrix are then for various spatial wavefunctions and , spin eigenfunctions and , and normalisation constants and .

The SACI method reduces the number of basis elements required by ensuring the wavefunction satisfies the Pauli exclusion principle from the outset and restricting the calculation to specific spin quantum numbers and .

Although each of the basis elements is a sum of terms, the properties of the antisymmetriser and the spin eigenfunctions allow enormous simplification of the calculation similar to the Slater-Condon rules for matrix elements between Slater determinates. In effect, only the product spatial wavefunction and spin eigenfunction need to be specified for each basis element with the normalisation constant and antisymmetrisation built into the rules for calculating the Hamiltonian elements.

We need to be careful in the selection of basis elements, however, as we need an orthonormal basis for the diagonalisation expansion to be valid. Remember, some spin and spatial combinations vanish and thus cannot be included in the basis. We must also make sure the set of basis elements is mutually orthogonal and normalised (i.e., if and , and 0 otherwise). To do this, we select only spatial wavefunctions that have doubly occupied orbitals in the first and second position, the third and fourth position, ... to match the spin eigenfunctions symmetry or antisymmetry under the operators ,  , ... . The properties of the antisymmetriser then guarantee the production of a complete orthonormal basis.

The SACI package automates the process of the orthonormal basis generation. As we can select only a finite number of basis elements out of an infinite set, we need to have some method of selecting appropriate basis elements. As we are usually interested in the ground and first few excited states of a quantum dot, we have chosen to use basis elements in increasing order of the sum of their component one-electron energies.

Elaborating, with no magnetic field the energy levels of the Fock-Darwin wavefunctions are given by . If there were no interaction between the electrons, the energy of a multi-electron state would simply be the sum of the one-electron energies, for example, the energy of would be . We select a basis consisting of all those elements with a noninteracting energy less than a certain cutoff.

The variable cutoffenergy is set to a cutoff energy of . We can then create a basis.

BasisCreate produces a list of basis elements with noninteracting energy less the specified cutoff. The spatial component of each basis element is given by a list of pairs of integers denoting the quantum numbers and for each orbital. The number at the end of the list denotes the index of the spin eigenfunction (its position when ShowSpinStatus runs) with which it is combined.

We notice that spin eigenfunction "1" is antisymmetric with respect to the permutation , so it can be combined with the doubly occupied orbitals, but spin eigenfunction "2" is symmetric with respect to , so it can only be combined with nondoubly occupied orbitals.

Matrix Elements Rules

As mentioned previously, there are general rules for calculating Hamiltonian matrix elements. These rules are coded in the function InteractionHamiltonianMatrixElement, and their derivation can be found in [13]. In brief, when the spatial wavefunctions of the two basis elements differ by more than two orbitals (i.e., after being sorted so the orbitals are in maximum correspondence), then the matrix element is zero; when the spatial orbitals differ by two or less orbitals, a different rule applies for each of the three cases (i.e., zero, one, and two orbitals differ). These formulas are quite general and can be used for any interaction Hamiltonian that acts pairwise.

In the Hamiltonian element formulas we must calculate the two-electron interaction integral. For the case of electrons in a 2D harmonic well interaction through a Coulomb potential under the influence of a perpendicular magnetic field, the integral is in the following form:

where

We need to be able to do this integral for many combinations of the quantum numbers , and . We notice that when and , we have a singularity as the Coulomb interaction goes to infinity as the electrons move closer together.

However by using Mathematica and a little ingenuity, we can do this integral exactly in all cases needed in computation. The detailed derivation (breaking down and transforming the integral into a sum of components) can be found in [13]. For example, we set the quantum numbers , and in order.

We also note that if (i.e., the sum or the orbital angular momentum quantum numbers are not equal), the interaction integral is zero.

These integrals are done with the effective harmonic well constant set to one. Values of the interaction integral for the effective harmonic well constant can easily be transformed to any value of . Thus when we do an interaction integral, it is done for and transformed to the correct value of afterwards. This allows us to store interaction integral values for various combinations of , and as function definitions and save them to use later. This process is done dynamically so that values are saved as definitions only if they are used.

Here are some examples of interaction Hamiltonian matrix elements indicating that they can be done exactly.

It can also be proved that the interaction element is zero if the sum of the orbital angular momentum quantum numbers are not the same, hence, the last calculation leads to zero.

The preceding calculations show that the spin eigenfunction with which a spatial wavefunction is combined makes a nontrivial contribution to the Hamiltonian matrix element calculation.