Two Electrons in an Anisotropic System (Nanorod)

Since the shape of nanocrystals can be controlled effectively, we can tune the electronic structure of a nanocrystal by controlling its shape rather than applying a magnetic field. As shown in FIG. 3.1, the shell structure is changed due to the anisotropy, and thus the energy spectrum is also changed. Similar to our method in the previous section, we calculated the eigenenergy numerically by varying the aspect ratio.

The energy spectra of all the configurations are shown in FIG. 4.3.1 and FIG. 4.3.2 where the interaction terms are ignored/included. Note that the elongation of the system (from oblate to

prelate) makes the state energies decay very rapidly such that the energy difference between singlet and triplet states in FIG. 4.3.2 is not very obvious.

0.5 0.75 1.25 1.5 1.75 2 lzêl0

FIG. 4.3.1 Energy spectrum versus aspect ratio where all the possible configurations are considered in the 2-shell approximation when the interaction terms are ignored.

0.5 0.75 1.25 1.5 1.75 2 lzêl0

FIG. 4.3.2 Energy spectrum versus aspect ratio, where all the possible configurations are considered in the 2-shell approximation when the interaction terms are included. The state energies decay so rapidly such that the energy difference between singlet and triplet states is not obvious.

0.5 0.8 1.2 2 l_zêl⁰

FIG. 4.3.3 Illustration of the energy evolution of each state versus aspect ratio. The evolution can be separated into 4 stages by three critical values of the aspect ratio. At these critical aspect ratios, some exited states cross due to the varying shell structure caused by the anisotropy. Thus the order of the states from low to high energy in each stage is different.

−

Energy (in arbitrary unit)

aspect ratio (a=l lz/ )0

Energy (in arbitrary unit)

aspect ratio (a=l lz/ )0

FIG.4.3.4 The evolution of all the states with the aspect ratio. It can be well mapped to FIG.4.3.3, but the energy information is absent. Each transition line is labeled by a different color.

However, the absolute value scale of this difference is the same as that in the nanocrystal when the magnetic field is applied.

The energy evolution of each state with the aspect ratio is a slightly complex. In FIG. 4.3.3 we show the energy evolution of each state versus the aspect ratio. The evolution can be separated into 4 stages by three critical values of the aspect ratio determined by a single particle spectrum. The critical values are approximately 1

2 , 1, and 2, and at these critical values some exited states cross because of the varying shell structure caused by the anisotropy.

Thus the order of the states from low to high energy in each stage is different and can be tuned by controlling the aspect ratio. In FIG. 4.3.4 we show the evolution of all the states with the aspect ratio. It contains the complementary information of FIG. 4.3.3 and the transition lines are labeled with different colors. Let us check that does the singlet-triplet transition occur in the ground state. When the system is elongated, the energy of orbital p₀ lowers and the electrons in s probably jump to p₀. Thus the triplet states we should consider are

| 01t > , | 02t > , and | 03t > . The energy spectrum of the ground state and the the first excited state versus the aspect ratio as shown in FIG. 4.3.2 indicates that no transition occurs. We cannot check if the transition occurs with a high aspect ratio because the 2-shell approximation fails in a high anisotropy system. To verify the highly anisotropic conditions, we have to consider more configurations and perform additional calculations. The shell structure of lower states with high aspect ratio is just the 1D SHO states, and thus we should consider the configurations | 0,0,0> to | 0,0, q > . Besides, when the aspect ratio is small, we should take the Fock-Darwin configurations in the calculation. After all, we have demonstrated the 2-electron system in a simplified condition and shown that the electronic structure could be tuned by an external magnetic field and/or the anisotropy.

Chapter 5: Summary

In this thesis we develop a CI theory for interacting electrons in nanostructures with 3D confinement, i.e., nanocrystals and nanorods, based on the 3D parabolic model. We mainly focus on three types of systems: (1) isotropic nanocrystals without magnetic field, (2) isotropic nanocrystals in a magnetic field, and (3) anisotropic nanocrystals (nanorods).

As a first step, we demonstrated a two-electron system in the two-shell approximation. We show that one can tailor the electronic structure of NC/NRs and the particle-particle interaction by means of shape-control and applying an external magnetic field; the latter affects the electronic structure slightly while the former results in drastic change of electronic structure and many-body physics.

To obtain numerical results with high accuracy, we should take the number of configurations as many as possible. In practice, we can truncate the Hilbert space spanned by the considered configurations with some cut off energy determined by convergence study. Since the kinetic and interaction terms we derived are universal, the theory we built up could also be applied to describe the behavior of the holes in valence band. Therefore we could study the excitonic problems and explore the optical properties of NC/NR systems. The theory may be applied in other systems such as pillar quantum dots which shapes are similar to NC/NR’s such as pillar quantum dots.

In the future, the following subject can be studied as the extended work of this thesis:

1. We may explore the possibility of observing S/T transition by both magnetic field and the breaking of symmetry of NC’s.

2. The simple model in this thesis may be compared with the results by atomistic tight-binding theory.

3. The two-electron study may be compared with numerical results.

4. We may explore more-electrons systems by exact diagonalization. The theory is extendable to large scale exact diagonalization calculation.