Fock-Darwin States Research Articles

Spatial and magnetic confinement of massless Dirac fermions

The massless Dirac fermions and the ease to introduce spatial and magnetic confinement in graphene provide us unprecedented opportunity to explore confined relativistic matter in this condensed-matter system. Here we report the interplay between the confinement induced by external electric fields and magnetic fields of the massless Dirac fermions in graphene. When the magnetic length lB is larger than the characteristic length of the confined electric potential lV, the spatial confinement dominates and a relatively small critical magnetic field splits the spatial-confinement-induced atomic-like shell states by switching on a pi Berry phase of the quasiparticles. When the lB becomes smaller than the lV, the transition from spatial confinement to magnetic confinement occurs and the atomic-like shell states condense into Landau levels (LLs) of the Fock-Darwin states in graphene. Our experiment demonstrates that the spatial confinement dramatically changes the energy spacing between the LLs and generates large electron-hole asymmetry of the energy spacing between the LLs. These results shed light on puzzling observations in previous experiments, which hitherto remained unaddressed.

Rising and lowering operator approach to the problem of a charged particle in a uniform magnetic field

In this paper, we analyse the quantization of cyclotron orbits for a 2D non-relativistic and spinless charged particle under the influence of a static and uniform magnetic field, the Landau problem. We follow the algebraic approach of rising and lowering operators. The Hamiltonian is expressed in terms of the usual rising and lowering operators of an isotropic quantum oscillator. Through a Bogoliubov transformation, we show that the Hamiltonian may be reduced to the one of a quantum oscillator that accounts for the formation of infinitely degenerate Landau levels. The explicit form of the position-space wave functions of Fock–Darwin states for a vanishing parabolic confinement potential is worked out by applying the techniques of ladder operators and the induction method. To explore the internal dynamics of the electron states, we compute the Fock–Darwin states’ density currents expressing them in terms of the ladder operators. In the supplementary material, we provide the reader with Mathematica algorithm that can help perform some of the calculations exposed in the paper. The charged particle in a magnetic field is used to explore some of the fundamental concepts of quantum mechanics as commuting operators, simultaneous eigenstates, degeneracy, rising and lowering operators, etc. This approach can be used as supplementary material in undergraduate final-year courses like quantum mechanics or solid state physics.

The transition from double to single quantum dot induced by THz laser field

Abstract The combined influence of an intense laser field and a homogeneous magnetic field is considered on laterally coupled double quantum dot with isotropic and anisotropic parabolic confinement. It is shown that intense laser field can recover typical properties to a single quantum dot: it creates Fock–Darwin states in a double quantum dot with anisotropic confinement. Also, the laser and the magnetic field effects on the intraband absorption spectrum reveal blueshift if the laser field is increased and redshift with the gain of the magnetic field.

Generation of Coulomb matrix elements for the 2D quantum harmonic oscillator

The Fock-Darwin states are the natural basis functions for a system of interacting electrons trapped inside a 2D quantum dot. Interaction effects at the mean field level or more elaborate quantum many body descriptions rely on an accurate evaluation of Coulomb matrix elements. In this work, we derive a highly efficient recurrence scheme to compute these elements in Fock-Darwin basis. The algorithm is best implemented on symbolic calculus platforms, preventing the appearance of rounding-off errors. The high speed achieved allows us to calculate all matrix elements in a basis set of several hundred states in very reasonable times (a few hours on a standard computer). We also make use of symmetry to reduce the number of elements to be computed. Finally, we check the reliability of floating-point evaluation for existing analytical expressions. We confirm the stability of a two-folded nonalternating sum for elements involving only the lowest energy levels and put on display the daunting limitations of the expressions commonly used to compute the elements in the general case.

Spin and orbital structure of the first six holes in a silicon metal-oxide-semiconductor quantum dot

Valence band holes confined in silicon quantum dots are attracting significant attention for use as spin qubits. However, experimental studies of single-hole spins have been hindered by challenges in fabrication and stability of devices capable of confining a single hole. To fully utilize hole spins as qubits, it is crucial to have a detailed understanding of the spin and orbital states. Here we show a planar silicon metal-oxide-semiconductor-based quantum dot device and demonstrate operation down to the last hole. Magneto-spectroscopy studies show magic number shell filling consistent with the Fock–Darwin states of a circular two-dimensional quantum dot, with the spin filling sequence of the first six holes consistent with Hund’s rule. Next, we use pulse-bias spectroscopy to determine that the orbital spectrum is heavily influenced by the strong hole–hole interactions. These results provide a path towards scalable silicon hole-spin qubits.

On the validity of microscopic calculations of double-quantum-dot spin qubits based on Fock-Darwin states

We consider two typical approximations that are used in the microscopic calculations of double-quantum dot spin qubits, namely, the Heitler-London (HL) and the Hund-Mulliken (HM) approximations, which use linear combinations of Fock-Darwin states to approximate the two-electron states under the double-well confinement potential. We compared these results to a case in which the solution to a one-dimensional Schr\"odinger equation was exactly known and found that typical microscopic calculations based on Fock-Darwin states substantially underestimate the value of the exchange interaction, which is the key parameter that controls the quantum dot spin qubits. This underestimation originates from the lack of tunneling of Fock-Darwin states, which is accurate only in the case with a single potential well. Our results suggest that the accuracies of the current two-dimensional molecular-orbit-theoretical calculations based on Fock-Darwin states should be revisited since underestimation could only deteriorate in dimensions that are higher than one.

Long distance coupling of resonant exchange qubits

We investigate the effectiveness of a microwave cavity as a mediator of interactions between two resonant exchange (RX) qubits in semiconductor quantum dots (QDs) over long distances, limited only by the extension of the cavity. Our interaction model includes the orthonormalized Wannier orbitals constructed from Fock-Darwin states under the assumption of a harmonic QD confinement potential. We calculate the qubit-cavity coupling strength in a Jaynes Cummings Hamiltonian, and find that dipole transitions between two states with an asymmetric charge configuration constitute the relevant RX quoit-cavity coupling mechanism. The effective coupling between two RX qubits in a shared cavity yields a universal two-qubit iSWAP-gate with gate times on the order of nanoseconds over distances on the order of up to a millimeter.

Strong Rashba Spin–Orbit Interaction Intensity in Low-Potential-Barrier Quantum Dots

We study the spin splitting energies of different orbital states of quantum dots with a low-potential barrier. The experimental results show that the splitting energies are orbital state dependent. The theoretical analysis is done with a generalization of the Fock–Darwin states in the presence of spin–orbit interactions. The theoretical predictions match well with the experimental observations and exhibits that the Rashba interaction strength in vertical In0.05Ga0.95As/GaAs quantum dots is in the range 80≤λR≤120 meV Å. This enhanced Rashba spin–orbit interaction intensity can be understood from the high penetration of the electron wavefunction into the quantum well with a low-potential barrier.

Erratum: Strong enhancement of Rashba spin-orbit coupling with increasing anisotropy in the Fock-Darwin states of a quantum dot [Phys. Rev. B85, 153301 (2012)

Received 22 November 2012DOI:https://doi.org/10.1103/PhysRevB.86.239901©2012 American Physical Society

Stabilization of single-electron pumps by high magnetic fields

We study the effect of perpendicular magnetic fields on a single-electron system with a strongly time-dependent electrostatic potential. Continuous improvements to the current quantization in these electron pumps are revealed by high-resolution measurements. Simulations show that the sensitivity of tunnel rates to the barrier potential is enhanced, stabilizing particular charge states. Nonadiabatic excitations are also suppressed due to a reduced sensitivity of the Fock-Darwin states to electrostatic potential. The combination of these effects leads to significantly more accurate current quantization.

Comment on “Fock-Darwin States of Dirac Electrons in Graphene-Based Artificial Atoms”

Chen, Apalkov, and Chakraborty (Phys. Rev. Lett. 98, 186803 (2007)) have computed Fock-Darwin levels of a graphene dot by including only basis states with energies larger than or equal to zero. We show that their results violate the Hellman-Feynman theorem. A correct treatment must include both positive and negative energy basis states. Additional basis states lead to new energy levels in the optical spectrum and anticrossings between optical transition lines.

Strong enhancement of Rashba spin-orbit coupling with increasing anisotropy in the Fock-Darwin states of a quantum dot

We have investigated the electronic properties of elliptical quantum dots in a perpendicular external magnetic field, and in the presence of the Rashba spin-orbit interaction. Our work indicates that the Fock-Darwin spectra display strong signature of Rashba spin-orbit coupling even for a low magnetic field, as the anisotropy of the quantum dot is increased. An explanation of this pronounced effect with respect to the anisotropy is presented. The strong spin-orbit coupling effect manifests itself prominently in the corresponding dipole-allowed optical transitions, and hence is susceptible to direct experimental observation.

Enhancement of Rashba coupling in vertical In0.05Ga0.95As/GaAs quantum dots

We study the spin-splitting energies in low-potential-barrier quantum dots, finding splitting energies that are orbital state dependent. The theoretical analysis is done with a generalization of the Fock-Darwin states in the presence of spin-orbit interactions. We discuss experimental evidence indicating that the Rashba interaction strength in vertical In${}_{x}$Ga${}_{1\ensuremath{-}x}$As/GaAs quantum dots is in the range 80 meV \AA{} $\ensuremath{\leqslant}{\ensuremath{\lambda}}_{R}\ensuremath{\leqslant}$ 120 meV \AA{}. This enhanced spin-orbit interaction can be understood from the high penetration of the electron wave function into the quantum well with low-potential barrier.

Magnetotunneling into Fock–Darwin‐like quantum dot states: Lateral matrix elements and the role of selection rules

AbstractWe study theoretically the magnetotunneling transport through quantum dots formed by thermal diffusion of charged manganese interstitials in the vicinity of a GaAs quantum well [Phys. Rev. Lett. 101, 226807 (2008)]. In particular, we examine the lateral matrix elements between Landau subbands in the contact and Fock–Darwin‐like states of an individual dot at high magnetic fields. We explicitly demonstrate the effect of spatial deformation of the dot on the wave function's overlap. The comparison with measured data suggests a selection rule similar to angular momentum conservation for tunneling into perfect Fock–Darwin states. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fabrication and Characterization of an Enhancement-Mode Planar Resonant Tunneling Transistor

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We report the fabrication and characterization of a planar resonant tunneling transistor (PRTT) using a GaAs/AlGaAs heterostructure wafer. Our PRTT operates as an enhancement-mode device, and shows the negative differential resistance (NDR) modulated with the gate bias in an extremely wide range (from the fully depleted dot to the dot in a single-electron transistor regime). This NDR spectrum represents a full, gate-bias-modulated evolution of the 0-D states in a quantum dot. Our data are also consistent with the ground (<formula formulatype="inline"><tex Notation="TeX"> $E_{1}$</tex></formula>) and the first excited state (<formula formulatype="inline"><tex Notation="TeX">$E_{2}$</tex></formula>) of the Fock–Darwin states. </para>

Variational calculation of the depolarization of the maximum density droplet in two-dimensional quantum dots

We show that, if a variational parameter in the Fock-Darwin states is optimized, the lowest Landau level (LLL) approximation agrees well with configuration interaction results using several Landau levels for a wide range of confining and Coulomb interaction strengths. Within the optimized LLL approximation, we study several phase transitions beyond the maximum density droplet for four to nine electrons and find similar patterns in the phase-space diagrams for angular momenta up to $N(N+1)∕2$. Calculations for larger angular momentum reveal unpolarized phases for filling factors up to $\ensuremath{\nu}=1∕3$.

Fock-Darwin States of Dirac Electrons in Graphene-Based Artificial Atoms

We investigate the Fock-Darwin states of the massless chiral fermions confined in a graphitic parabolic quantum dot. In light of Klein tunneling, we analyze the condition for confinement of the Dirac fermions in a cylindrically symmetric potential. New features of the energy levels of the Dirac electrons as compared to the conventional electronic systems are discussed. We also evaluate the dipole-allowed transitions in the energy levels of the dots. We propose that in the high magnetic field limit, the band parameters can be accurately determined from the dipole-allowed transitions.

Exchange energy in coupled quantum dots

In this work, the exchange energy J for a system of two laterally-coupled quantum dots, each one with an electron, is calculated analytically and in a detailed form, considering them as hydrogen-like atoms, under the Heitler-London approach. The atomic orbitals, associated to each quantum dot, are obtained from translation relations, as functions of the Fock-Darwin states. Our results agree with the reported ones by Burkard, Loss and DiVincenzo in their model of quantum gates based on quantum dots, as well as with some recent experimental reports.

Three Electrons in a Two-Dimensional Parabolic Trap: The Relative Motion Solution

In agreement with the Kohn theorem the relative motion (rel) of three electrons in a two-dimensional parabolic trap separates from the centre-of-mass (CM) motion. By introducing new coordinates the Hamiltonian for relative motion in the approximation of non-interacting electrons can be taken to the normal form. The eigenstates of the normalized Hamiltonian are products of the Fock-Darwin states for normal modes. The energy levels for relative motion are obtained by diagonalizing the exact Hamiltonian in the eigenbasis for the non-interacting case. In this basis the interaction matrix elements can be obtained in the analytical form. Since the rank of the Hamiltonian matrix is significantly reduced, the calculations are faster and more accurate than those for the full (CM + rel) motion. This advantage is especially important for the calculations of excited states and the analysis of energy spectra.

Configuration interaction method for Fock–Darwin states

We present a configuration interaction method optimized for Fock–Darwin states of two-dimensional quantum dots with an axially symmetric, parabolic confinement potential subject to a perpendicular magnetic field. The optimization explicitly accounts for geometrical and dynamical symmetries of the Fock–Darwin single-particle states and for many-particle symmetries associated with the center-of-mass motion and with the total spin. This results in a basis set of reduced size and improved accuracy. The numerical results compare well with the quantum Monte Carlo and stochastic-variational methods. The method is illustrated by the evolution of a strongly correlated few-electron droplet in a magnetic field in the regime of the fractional quantum Hall effect.