Fractional Charge States in the Magneto-Photoluminescence Spectra of Single-Electron InP/GaInP2 Quantum Dots

We used photoluminescence spectra of single electron quasi-two-dimensional InP/GaInP2 islands having Wigner-Seitz radius ~4 to measure the magnetic-field dispersion of the lowest s, p, and d single-particle states in the range 0–10 T. The measured dispersion revealed up to a nine-fold reduction of the cyclotron frequency, indicating the formation of nano-superconducting anyon or magneto-electron (em) states, in which the corresponding number of magnetic-flux-quanta vortexes and fractional charge were self-generated. We observed a linear increase in the number of vortexes versus the island size, which corresponded to a critical vortex radius equal to the Bohr radius and closed-packed topological vortex arrangements. Our observation explains the microscopic mechanism of vortex attachment in composite fermion theory of the fractional quantum Hall effect, allows its description in terms of self-localization of ems and represents progress towards the goal of engineering anyon properties for fault-tolerant topological quantum gates.

an exciton radius of ~7 nm, which is close to a Bohr radius aB * of InP QDs [4]. The paramagnetic dispersion follows well trion FD spectrum downwards Be~5 T, where the decomposition of the trion and formation of the 2e WM in PS is expected [2]. For larger fields the paramagnetic dispersion becomes weaker compared to FDL and reveals a small kink at Be=8 T. These two values of Be are close to effective Landau level filling factors ν~1 and 1/3, which allows relate them to the signatures indicating a formation of a magneto-electron.
[4] In effective atomic units Bohr radius a*B=ħ 2 (4πεε0)/m*e 2 , where ε and m* are dielectric constant of the material and effective mass of electron, respectively. For our dots it is 8.5 nm.

s3. Theoretical modeling of electronic structure in magnetic field
We model the quantum dots as a confined 2D electron gas in the effective-mass approximation, with a uniform magnetic field applied perpendicular to theplane of the electrons. The Hamiltonian for confined electrons is (in atomic units, where ext ( ) is the effective 2D confining potential in the -plane and the magnetic vector potential is given by ( , ) = ( /2)(− , , 0). We take the conduction-band effective mass of InP to be * = 0.077 and the dielectric constant to be = 12.61. The appropriate value of the effective -factor * for our quantum dots is unclear to us, so for illustrative purposes the spin Zeeman term is included using the value for bulk InP, * = +1.20.
The Hamiltonian (S1) is discretized on a 2D Cartesian grid using a high-order (11point) finite-difference method, adapted to take account of the rapid variation of the phase of the complex wave function between adjacent grid points in the presence of a strong magnetic field. Typical grid dimensions are 70 × 70 for < 5 T (and larger for higher ). The many-particle wave functions and energies of confined electrons are computed by a configuration-interaction (CI) approach, which is based on a generalization of earlier work [1][2][3] where no magnetic field was present. The first step is to generate a single-particle basis set including states up to a high energy cutoff, using either a spin-polarized Hartree-Fock (HF) potential or an analytical approximation to the HF (mean-field) potential. For = 2 confined electrons, we then find the many-body wave function and total energy by full CI (including up to double excitations) using the HF basis set. To evaluate the Coulomb integrals required, we solve Poisson's equation in the -plane by a complex version of the algorithm described in Ref.
[1], which employs fast Fourier transforms and a quintic interpolation of the density between grid points. A useful overall check on the manybody calculation is provided by the fact that we can obtain total energies for the 2 system that agree to six or more digits with a variety of different basis sets, for the ground state as well as the low-lying excited states.
We model the experimental dots with a simple 2D effective confining potential, The physical boundary of the dot (where the vertical height of the InP deposit goes to zero) is represented by a radius function ( ), which in general depends on the polar angle in the -plane, so that the dot has an angular deformation. The parameter 0 = 〈 ( )〉 is the average value of ( ) over the polar angle 0 ≤ ≤ 2 . For small radii ≪ ( ), the potential ext ( ) is approximately harmonic with frequency 0 , while at large radii ≫ ( ), the potential develops a hard wall (controlled by the parameter ), which forces the electron wave function to zero. The angular deformation of ( ) is chosen to be typical of the dots synthesized experimentally.
Our grid-based numerical algorithm makes it straightforward to incorporate the symmetry-breaking effect of a deformed dot by direct calculation with the confining potential. As a result of the angular deformation, the orbital angular momentum is not an exact quantum number and localized electron states can be observed directly in the one-body electron density (S3) Figure SM3 shows this phenomenon in the ground-state densities for two potentials with harmonic parameters ℏ 0 = 1.3 meV (top row) and 0.6 meV (bottom row). The potential contours indicate the deformed shape of the confining potential and its outer physical radius ( ); the average physical radius is 0 ≈ 85 nm. For dots of this size and shape, even in zero field = 0, the ground state is a clearly defined Wigner molecule with two well separated electron peaks. For sufficiently large (in these examples, for >~2 T), the peaks become sharper as increases further, reflecting the magnetic length scale = �ℏ/ . For the shallower potential ℏ 0 = 0.6 meV, the peaks are further apart and begin to push up against the hard wall of the potential, while for ℏ 0 = 1.3 meV the confinement is mainly by the central harmonic part of the potential.
In a strong magnetic field, the lowest-energy states with = 0 consist of nearly degenerate triplet ( = 1 ) and singlet ( = 0) states. The total energy and the splittings of these states as a function of are shown in Figs. SM4a and b for the dot from Fig. SM3 with ℏ 0 = 0.6 meV. As increases and the two electrons become more localized, the exchange interaction between them is reduced and the singlettriplet splitting of the = 0 components becomes increasingly small [see Fig. SM4b].
The triplet states with ≠ 0 are Zeeman-shifted by an energy * relative to those with = 0, so that (assuming * > 0) the ground state at high is the = −1 component of the triplet state. The low-lying excited states in a strong field (not shown in the figure) also have = −1 and lie between the curves for = −1 and 0 in Fig. SM4(b). These excited states originate from excited triplet states at = 0, the = −1 components of which are Zeeman-shifted to lower energies as increases. For this confining potential, the average orbital angular momentum 〈 〉 of the ground state displays broadened plateaus at values of 〈 〉 = 1 and 3 [see Fig. SM4c]. These plateaus are washed out for 〈 〉 = 7 and higher, after which 〈 〉 shows a nearlinear dependence on .  (b) relative total energy of these states, with the zero of energy defined as the energy of the = 1, = 0 state; (c) average orbital angular momentum 〈 〉 of the = 1, = −1 state. The quantum dot is that from Fig. S1 with ℏ 0 = 0.6 meV.