No. 516

**Waves and Trajectories
in
Quantum Dots **

Tomas Lundberg

**Akademisk avhandling **

som för avläggande av teknologie doktorsexamen vid Linköpings
Universitet kommer att offentligt försvaras i hörsal Planck,
Fysikhuset, Linköpings Universitet, fredagen den 13 mars 1998, kl.10^{15}.
Opponent är Professor Joan Martorell, Departament d'Estructura i
Constituents de la Materia, Facultat de Física, Universitat de
Barcelona, Spanien.

**Abstract**

This thesis deals with electron transport in low-dimensional semiconductor nanostructures, and in particular, systems with quasi zero-dimensional properties, called quantum dots. These systems are so small that the electron wavelength is comparable to the size of systems, which means that the wave nature of the electrons is of fundamental importance. At the same time, however, they are almost large enough to be seen with the naked eye, and hence they are in the intermediate region between macroscopic (classical) physics and microscopic (quantum) physics. This region has been called mesoscopic physics.

Several different kinds of quantum dots has been examined. The first one is a crossbar structure, consisting of two perpendicular, intersecting channels, of which one connects to two semi-infinite regions which acts like source and drain. The structure is modelled with hard wall potentials, which is a rather accurate model of the true confining potential. To increase the correspondence with real devices, the corners of the hard wall potentials, which usually are treated as being sharp and right-angled, are here replaced with smooth rounded corners. We have studied the effect of this replacement on the conductance and charge and current distributions. We found that for low Fermi energies, i.e. long electron wavelengths, the differences between sharp and rounded corners are small, but as the energy increases, discrepancies appear. At some energies the change in conductance can even be rather drastic, changing a resonance peak to an anti-resonance dip.

Earlier in the literature, a simple tunnelling model has been introduced to explain oscillations in the conductance of large circular quantum dots as a function of both magnetic field and size. In this model, the overall level structure is essentially the same as for an isolated dot, and the only effect of the leads is to make the individual energy levels Lorentzian broadened. To check the validity of this simple model we have calculated the full quantum mechanical solution for a large circular dot with hard wall potential and two conical connecting leads. At zero temperature the fluctuations in the conductance as a function of Fermi energy predicted by this model bears little resemblance with the result from the simplified model, but as the temperature is increased, rapid oscillations are smoothened out, and we find an overall qualitative agreement between the two models. The conclusion here is that transport through this dot is effectively mediated by one eigenstate or a superposition of a few eigenstates of the corresponding closed dot. We find further evidence for this when we also study scarred states-states where the probability density forms patterns which often resembles classical trajectories. We have found that these scarred states also can be explained as superpositions of eigenstates of the closed dot, and can hardly be interpreted as the remnants of classical trajectories, which frequently is argued in the literature.

We have also modelled a quantum dot using a more realistic model in which two saddle-point potentials, offset laterally, are smoothly joined to create a double-saddle-point potential. The calculated conductance from this potential was compared with experiments and a rather good agreement was found. The simple double-saddle-point potential however completely neglects any electron-electron interaction, which are known to affect the confining potential. To investigate the effect of electron-electron interaction we used a simple but realistic model, where we truncated the bottom of the well in the centre of the double-saddle-point potential. Surprisingly enough, the conductance was very robust with regard to this truncation.

Since the nanostructures are in the intermediate region between the classical and quantum worlds, it is not unreasonable to suspect that semiclassical methods might be useful in the studies of nanostructures, and semiclassical methods have indeed been developed and used with success. Since most semiclassical methods treats the electrons particles following classical trajectories but with quantum phase factors added, it is of interest of studying their quantum mechanical counterparts. We have therefore used the de Broglie-Bohm interpretation, in which the concept of trajectories comes naturally, and studied the transport through some of the quantum dots considered earlier. In some of the calculation we included the effects of inelastic scattering, which usually are neglected, by introducing a complex potential.

In all these studies, computer visualization of charge and current distributions, as well as the quantal trajectories, are enormously helpful, and almost essential for understanding the complex behaviour of nanostructures.

Department of Physics and Measurement Technology

Linköping University, S-581 83 Linköping, Sweden

Linköping

ISBN 91-7219-160-0 ISSN 0345-75241