A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.

Three real-space discretization techniques in electronic structure calculations

We investigate the effects of impurities and changing ring geometry on the energetics of quantum rings under different magnetic field strengths. We show that as the magnetic field and/or the electron number are/is increased, both the quasiperiodic Aharonov-Bohm oscillations and various magnetic phases become insensitive to whether the ring is circular or square in shape. This is in qualitative agreement with experiments. However, we also find that the Aharonov-Bohm oscillation can be greatly phase shifted by only a few impurities and can be completely obliterated by a high level of impurity density. In the many-electron calculations we use a recently developed fourth-order imaginary time projection algorithm that can exactly compute the density matrix of a free electron in a uniform magnetic field.

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Effects of geometry and impurities on quantum rings in magnetic fields

We show that the method of splitting the operator ee(T+V) to fourth order with purely positive coefficients produces excellent algorithms for solving the time-dependent Schrodinger equation. These algorithms require knowing the potential and the gradient of the potential. One fourth order algorithm only requires four fast Fourier transformations per iteration. In a one dimensional scattering problem, the fourth order error coefficients of these new algorithms are roughly 500 times smaller than fourth order algorithms with negative coefficient, such as those based on the traditional Forest–Ruth symplectic integrator. These algorithms can produce converged results of conventional second or fourth order algorithms using time steps 5 to 10 times as large. Iterating these positive coefficient algorithms to sixth order also produced better converged algorithms than iterating the Forest–Ruth algorithm to sixth order or using Yoshida’s sixth order algorithm A directly.

Fourth Order Gradient Symplectic Integrator Methods for Solving the Time-Dependent Schrodinger Equation

The electronic states of quantum rings with centerlines of arbitrary shape and non-uniform width in a threading magnetic field are calculated. The solutions of the Schr\"odinger equation with Dirichlet boundary conditions are obtained by a variational separation of variables in curvilinear coordinates. We obtain a width profile that compensates for the main effects of the curvature variations in the centerline. Numerical results are shown for circular, elliptical, and lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on-shaped quantum rings. We also show that smooth and tiny variations in the width may strongly affect the Aharonov--Bohm oscillations.

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Quantum rings of arbitrary shape and non-uniform width in a threading magnetic field

Using the variational method and the effective mass and parabolic band approximations, the behaviour of the binding energy and photo-ionization cross section of a hydrogenic-like donor impurity in an InAs quantum ring, with Poschl-Teller confinement potential along the axial direction, has been studied. In the investigation, the combined effects of hydrostatic pressure and electric and magnetic fields applied in the direction of growth have been taken into account. Parallel polarization of the incident radiation and several values of the applied electric and magnetic fields, hydrostatic pressure, and parameters of the Poschl-Teller confinement potential were considered. The results obtained can be summarised as follows: (1) the influence of the applied electric and magnetic fields and the asymmetry degree of the Poschl-Teller confinement potential on the donor binding energy is strongly dependent on the impurity position along the growth and radial directions of the quantum ring, (2) the binding energy is an increasing function of hydrostatic pressure and (3) the decrease (increase) in the binding energy with the electric and magnetic fields and parameters of the confinement potential (hydrostatic pressure) leads to a red shift (blue shift) of the maximum of the photo-ionization cross section spectrum of the on-centre impurity.

Hydrostatic pressure, impurity position and electric and magnetic field effects on the binding energy and photo-ionization cross section of a hydrogenic donor impurity in an InAs Pöschl-Teller quantum ring

We have calculated the energetics of $^{4}\mathrm{He}$ and $^{3}\mathrm{He}$ atoms in one, two, and three carbon nanotubes. The binding energy and low-lying excited states are obtained for three characteristic sites: inside a tube in the presence of other tubes, within an interstitial channel, and in the outside grooves. Calculations have been performed using both an averaged helium-carbon potential that depends only on the distance to the nanotube axis, and a potential constructed as a sum of individual carbon-helium interactions. That way, we assess the effect of corrugation on the binding properties of helium atoms. We have also studied the energetics and structure of helium dimers, trimers, and tetramers in some of these geometries and found strong dependence of the binding energy on the geometry. From the single-particle spectrum, we obtain the thermodynamic properties of adsorbed helium atoms in carbon nanonotube bundles. As a consequence of corrugation a departure from the one-dimensional ideal gas behavior is observed; this deviation depends on the temperature, density, and bundle model.

He atoms in carbon nanotubes and nanotube bundles

We study the effects of external impurities on the shell structure in semiconductor quantum dots by using a fast response-function method for solving the Kohn-Sham equations. We perform statistics of the addition energies up to 20 interacting electrons. The results show that the shell structure is generally preserved even if there are clear signs of disorder in the energetics. The Coulomb interaction and the variation in ground-state spins have a strong effect on the addition-energy distributions, which in the noninteracting single-electron picture correspond to level statistics showing mixtures of Poisson and Wigner forms.

Stability of the shell structure in two-dimensional quantum dots

Statistics of closed quantum dots : Effects of disorder and interactions

We study computationally the ground-state properties of large quantum rings in the filling-factor ν>1 quantum Hall regime. We show that the arrangement of electrons into different Landau levels leads to clear signatures in the total energies as a function of the magnetic field. In this context, we discuss possible approximations for the filling factor ν in the system. We are able to characterize integer-ν states in quantum rings in an analogy with conventional quantum Hall droplets. We also find a partially spin-polarized state between ν = 2 and 3. Despite the specific topology of a quantum ring, this state is strikingly reminiscent of the recently found ν = 5/2 state in a quantum dot.

M. Aichinger

Papers

Effects of geometry and impurities on quantum rings in magnetic fields

He atoms in carbon nanotubes and nanotube bundles

Stability of the shell structure in two-dimensional quantum dots

Statistics of closed quantum dots : Effects of disorder and interactions

Large quantum rings in the ν>1 quantum Hall regime