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

Abstract

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 (e_m) 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 e_ms and represents progress towards the goal of engineering anyon properties for fault-tolerant topological quantum gates.

1. Introduction

The integer (IQHE) [1] and fractional (FQHE) [2] quantum Hall effects, which reveal a skipping-type dissipationless edge transport in two-dimensional (2D) electron (e) gas semiconductor structures in a strong perpendicular magnetic field, are two of the most important discoveries in the physics of the last decades of the 20th century. The IQHE provides a precise method of measuring the fundamental fine-structure constant and gave the first example of edge/surface topological states that are robust to external perturbations [3,4]. The FQHE yields very rich physics involving fractionally charged anyon states [5] and Majorana zero-energy modes [6]. These open up the possibility of building topologically protected quantum gates (TQG) [6,7], which promise to dramatically improve the operation of quantum computers and accelerate the creation of their compact hardware—one of the challenges of science and technology in the 21st century.

The IQHE is well understood in terms of noninteracting electrons and the quantization of their Hall conductance by integer Landau-level fillings, ν [8]. The FQHE—representing quantization by fractional fillings ν = p/k [9], where k and p are the rational integers—is instead considered to be a property of strongly interacting electrons condensing into incompressible liquids of anyon quasiparticles (QPs) having charge 1/k, as was suggested by Laughlin [10]. This liquid state is described by Laughlin’s wavefunction with k zeros (k-WZs) localized at the positions of each electron keeping them separated, thus reducing the interaction energy. The FQHE can also be described using the concept of composite fermion (CFs) particles consisting of an electron with 2p magnetic-flux-quanta (MFQ) attached, as was shown further by Jain [11]. In this approach, the FQHE states are the IQHE states of weakly interacting CFs and the CF wave-function of k Landau level state has 2kp-WZs, i.e., ν = k/(2pk + 1). Thus, both approaches are equivalent. While these theoretical approaches have described a rich set of experimental data since the discovery of the FQHE nearly 40 years ago, a microscopic description thereof remains elusive [12]. This results from the fact that they do not clarify the connection between WZs and MFQ, i.e., the microscopic mechanism of MFQ attachment, and its dependence on the localization of the electrons. This creates uncertainty in the interpretation of FQHE magneto-conductivity experiments related to the realization of topological quantum gates (TQG) [13,14,15].

Theoretical proposals [16,17] and a methodology [18] for topological quantum computing are based on Laughlin’s QP or Jain’s CF description of the FQHE, and recommend using a set of localized QPs/CFs pulled from the appropriate host liquids. Thus, experiments demonstrating the possibility of the creation of single localized fractionally charged QPs/CFs are a necessary step for the realization of TQGs, and are critical for our understanding of the microscopic nature of the FQHE. So far, however, the existence of localized QP/CF puddles/molecules has only been observed using high-spatial-resolution electrometry of 2D quantum Hall structures [19,20] or near-field photoluminescence measurements of quantum dots (QDs) [21], while the localization of single fractionally charged QP/CF has not been reported.

Here, we demonstrate such localization and report the formation of fractionally charged states of single electrons, which gives insights into the mechanism of its coupling to MFQ. This clarifies the unresolved issues in the microscopic description of the FQHE and gives new perspectives for the realization of TQG. We use measurements of magneto-photoluminescence (magneto-PL) spectra of quasi-2D islands represented by self-organized semiconductor InP/GaInP₂ QDs for the detection of the charge of single-electron states for a Wigner-Seitz radius r_s~4 in a range of magnetic field 0–10 T. These QDs are large (~150 nm), providing a strong Wigner localization regime, together with in situ electron doping and a built-in magnetic field [22], and, thus, have unique properties compared to other QD material systems [23,24]. Using these, we measured a fractional charge in the range 1/3–1/9, which decreased linearly with an increase in the size of the QD and the angular momentum. This implies the formation of MFO vortexes and their fixed radius, which is limited by the Bohr radius. Thus, we revealed a critical role of localization in the coupling of the electron and MFO-vortexes, resulting in the formation of a magneto-electron (e_m) particle. Its formation can be explained by the nano-superconducting properties of the electrons occupying quantized states in the Wigner localization regime and the self-generation of vortexes. The fractional charge of such e_m appears to be due to vortex screening of the electric field. Our observation shows that Laughlin’s k-WZs are related to close-packed topological vortex structures of e_m, and provides an explanation for FQHE magneto-conductivity plateaus in terms of self-confinement of e_m’s in a quasi-ordered potential modulation, which, in turn, opens up new possibilities of deterministic engineering of anyon particle properties for use in TQGs.

2. Single-Electron InP/GaInP₂ QDs

The measured InP/GaInP₂ QDs (see Appendix A.1 for growth details) had a flat lens shape, a lateral size D~50–180 nm, up to 20 electrons and a Wigner-Seitz radius of up to r_s~5, as described in [22,25]. The dots revealed a small elongation and asymmetry, which, in most cases, could be described as a combination of a ~5% elliptical distortion (D||/D⊥ = 1.05) and a 10% change of R⊥. The dots had up to ~0.2 content of Ga (x_Ga) due to intermixing during growth, which resulted in an increase of the emission energy of the 00 transition E₀ (see below) on ~150 meV.

The InP QDs are surrounded by atomically ordered GaInP₂ 20–50 nm size domains (AODs) having a CuPt_B-type crystal structure, which is a [111]_B-oriented GaP/InP monolayer superlattice [22]. Thus, they represent core-shell QD-AOD composites. Relaxation of the bonds of the domain was observed for some dots and resulted in an 80 meV decrease of E₀. The domains generates a strong [111]_B-oriented piezo-electric field, which produces electron doping of QDs and induces a built-in magnetic field (B_bi) [21,26].

The structures studied had a relatively large dot density (~10⁹ cm⁻²) and we used an optical fiber probe cryo-magnetic near-field scanning optical microscope (NSOM) with a spatial resolution of 20-300 nm to measure the PL spectra of a single QD (see Appendix A.2 for details of NSOM technique). Since the QDs had a wide range of sizes, electron populations and B_bi, the measurements presented certain technical challenges (including luck), the main one being the tedious scanning process to find the appropriate dot for a specific study, and separating its spectrum from neighboring dots. In our experiments, QDs with different electron populations were selected using analysis of the number of spectral features in PL spectra, as described in detail in [25]. The spectra were measured at 10 K and an external magnetic field B_e = 0–10 T.

In our NSOM measurements of InP/GaInP₂ structures, we found nearly ten single-electron QDs from a set of ~200 dots. Here, we present the data for eight dots, denoted S00, S02, S03, S04, S09, S10, S11 and S12.

Table 1. Parameters of electronic states of single electron InP/GaInP₂ QDs measured in magneto- PL spectra.

No	Type	ħω0 (me)	D(a) (nm)	rs	E0 (eV)	xGa	dWM (nm)	Bbi T
S00	trion	4.0	60	~1	1.720	0.15	-	0
S02 [25]	S	2.5	85	2.5	1.724	0.15	35	0
S09 [25]	S	2.0	90	2.6	1.712	0.15	40	0
S10 [25]	S	0.7	140	4.1	1.798	0.2	75	0
S12	S*	1.25	110	3.4	1.705	0.1	65	10
S03 [25]	S*	0.8	130	3.8	1.765	0.15	70	10
S11	S*	0.8	130	3.8	1.695	0.1	70	10
S04 [27]	S*	0.6	150	4.3	1.743	0.15	80	15

^(a) Averaged value (between D_|| and D_⊥) of the size of the area containing 96% of calculated electron density.

contains a list of these dots, their type (see below) and parameters, such as, quantum confinement (ħω₀), D, r_s, E₀, x_Ga, the separation of the electrons in a photo-excited state d_WM, and B_bi (see [25]). The type and parameters were determined using magneto-PL spectra (see Appendix B.1 for spectra processing).

From this set, the dot S00 was the only one having weak Wigner localization (WL) (r_s~1) and the smallest size (60 nm), for which a singly charged exciton (trion) formed in the photo-excited state. The magneto-PL spectra of this dot are discussed in the Supplementary Materials (S1). The spectra were similar to those of the 2e excitonic (tetron) dot discussed in detail in [27], with a paramagnetic (negative) B-dispersion.

For the rest of the dots, which had stronger Wigner localization (r_s > 2), the trion becomes unstable decomposing into a 2e Wigner molecule (WM) weakly bound to the photo-excited hole. Two of the dots, S02 and S09 had r_s~2.5 (D < 100 nm), and were thus in the regime of the onset of WL; the rest had r_s~4 (D > 100 nm) and were thus in the regime of strong WL.

The change of D was from 85 nm (for S02) to 150 nm (for S04), which corresponded to a change in ħω₀ from 2.5 to 0.6 meV. The x_Ga varied from 0.1 for S11 and S12 to 0.2 for S10 with x_Ga~0.15 for the five remaining QDs. This corresponded to a change of emission energy E₀ from ~1.7 to 1.8 eV.

3. Magneto-PL Spectra: Experiment

Figure 1a compares the spectra of all seven WM QDs at B_e = 0 T plotted in Stokes shift energy units. As can be seen from Figure 1a, a zero-energy peak, denoted 00, and a few Stokes peaks, denoted 0N, which are the signature of 2e-WM emission [27], were observed for all dots. For dots S02, S09 and S10, only a single 01 Stokes component (SC) peak was clearly observed in the spectra, while N > 1 SCs had an order-of-magnitude smaller intensity. We denoted these single SC dots as S- type. For the rest QDs up to four SCs were observed; their intensities were larger or the same as that of 00 component. We denoted these multiple peak SC dots as S*-type. Note that the dots S11 and S03, which had the same ħω₀, had nearly the same intensity distribution of 0N components, in which 02 and 03 components were dominant. They differed only by the emission energy, i.e., x_Ga (see Table 1).

Figure 1b presents magneto-PL spectra of the S11 and S12 QDs measured in the B_e range 0–10 T. It shows that the S11 and S12 QDs were observed at the same NSOM spectra, i.e., probe position, and thus, neighbored one another, being separated by~200 nm. Their spectra partially overlapped (at 1.696 eV for B_e = 0 T) and were separated using Lorentzian decomposition. At B_e = 0 T, the spectrum of the S11 dot also partially overlapped at 1.688 eV with the spectrum of another QD (D01m) with eight electrons and ħω₀ = 3.5 meV. This dot will be discussed elsewhere.

The intensity of the spectra of the S11/S12 dots versus B_e strongly increased/decreased, most probably due to the lateral shift of the NSOM probe toward the S11 dot.

Thus, for B_e > 7 T, the spectral shape of S12 QD could hardly be analyzed, but the position of the center of gravity of the spectra could be clearly traced.

When B_e increased, the S*-type peaks of the S11 dot merged into two at 4 T and then into a single peak at 6 T. We denote these types of spectra as D and P, respectively. A further increase in B_e led to a splitting of the P-type spectrum and its transformation into the S-type spectrum at B_e = 10 T. For the S12 dot, a P-type single peak shape seemed to appear at B_e = 4 T and 8 T, revealing some intermediary fine structure. For the QD S04 (see [27] and Figure 5a below), the P- and D-type spectra seemed to appear at 1 T and 4 T, respectively, but no S-type spectrum was observed over the entire range of B_e 0–10 T.

The key observation from the comparison of Figure 1a,b is that, in the spectrum of the QD S11 at B_e = 10 T and in the that of the QD10 at B_e = 0 T, the peak separation and their intensity distribution were nearly the same. The only difference was a slightly larger full-width-at-half-maximum (1.6 versus 1.2 meV), which may have been due to the larger value of x_Ga in S10 (see Table 1). These features, which indicate that these are nearly identical dots, are clearly seen in Figure 1c, in which we compare their spectra. In Figure 1d, we compare an NSOM image of the S10 dot (see [25]), d_WM estimated from the experimental spectrum (see below) and the calculated electron density for this dot for B_e = 0. The measured data and the calculation are in excellent agreement. This indicates that QD11 had zero internal magnetic field B at B_e = 10 T. This corresponded to B_bi directed opposite to B_e and equaled −10 T. Thus, we can conclude that the S*-type spectrum is a signature of B_bi and the fact that we did not observe S-type spectra for QD S04 corresponds to B_bi > 10 T.

The total internal field in the QD is B = B_e + B_bi, and a negative (positive) shift of the spectral components versus B_e corresponded to a positive (negative) shift versus B. This shift, outlined in Figure 1b by dashed lines connecting the peak maxima, revealed a weak positive B-dispersion in the S*-range and a negative B-dispersion in the S-, P- and D-ranges (see also the more detailed analysis in Figure 5a–c below). The average value of the dispersion of the P-spectra was a few times stronger than that of the S and S*, namely ~0.1 meV/T. For S04, a similar B-dispersion was observed [27]. The dispersion showed a dependence on the dot size, which may clearly be seen in Figure 1b by comparing the slope of the dashed lines of the P-type spectra of the S11 and S12 dots. The slope corresponds to a dispersion ~0.5 meV/T for S11 (D = 130 nm) and ~1 meV/T for S12 (D = 110 nm), thus increasing with a decrease in the dot size.

4. Magneto-PL Spectra: Phenomenology Description

4.1. Intensity Distribution of the Stokes Components

In Figure 2a, we show a cartoon of the confinement potential and energy levels, together with a cross-section of the corresponding wave-functions ψ_N(r,θ) and electron positions in the initial (IS) and photo-excited (PS) states of a single-electron QD having r_s = 3.8 (analog of S11). In the left and lower inserts, we show the calculated electron density distributions for the PS and the six lowest ISs (see S2). Note that in Figure 2a and elsewhere, for PL transitions (see vertical arrows in the right), we use the notations 0N, where two numbers are radial quantum numbers (see Appendix B.2) of the PS and IS states, respectively, and, for single-electron states having the same angular momentum, the notations N’s, N’p and N’d, where N’ = N + 1, to number the states. It can be seen from Figure 2a that in the dot, a 2e-WM with d_WM = 70 nm formed by photon absorption generating an electron-hole (e–h) pair. An e-h recombination, resulting in photon emission corresponding to a transition between the lowest PS-IS 1s-states, which have total angular momentum L_z = 0, left the residual 1s electron in a position shifted from that of the IS by a d_WM/2. This shift broke the selection rule ΔN = 0 for an optical transition, and thus generated 0N (N > 0) PL SCs with an energy separation 2ħω₀. The relative intensity of 0N PL transitions was equal to the ratio of the square of the overlap integrals $O{I}_N\left(d\right)={\displaystyle \int }{\psi }_0\left(r+d,\theta \right){\psi }_N\left(r,\theta \right)drd\theta$ for d= d_WM/2. In Figure 2b,c we present numerically calculated ratios ${[O{I}_N\left(d\right)/O{I}_0\left(d\right)]}^2$ for B = 0 and B~7 T (see below) and corresponding PL spectra for d = d^WM/2, respectively. These ratios are also shown as bars in Figure 1a. The comparison shows good agreement with the experiment. The enhanced intensity of higher N components in S*-dots was therefore due to B_bi and a corresponding increase of electron localization and L_z (see below).

For larger L_z, N’p_y and N’d_y states appeared in the PL spectra. For these states, only zero Stokes energy components 1p_y and 1d_y were expected, owing to the spatial matching between their and the 1s² PS maxima of the electron density distributions (see inserts in Figure 2a). The 1p_x and 1d_x states were expected to have nearly zero intensity, owing to the opposite phase of the upper and lower maxima, leading to a zero overlap integral. As we will show below, a single-peak P-type spectrum can be attributed to transitions involving the 1d_y component of L_z = 1 PS and a second peak in the D-type spectra to the 1p_y component L_z = 3 of PS. The S*-type spectra can be attributed to transitions involving the 1s component of the L_z = 3 or 5 states.

4.2. Magic L_z Numbers of the Photo-Excited State

The increase of L_z in a few-electron 2D WMs versus B occurred via certain values known as “magic” numbers L_z^MN, which depend on the WM configuration [28]. L_z^MN appeared because of the necessity of matching the spatial structure of the electron wave-functions and the spatial symmetry of the electron arrangement, and resulted from the competition between the Coulomb interaction and quantum and magnetic field confinement [29]. For 2e-WM, L_z^MN are odd, i.e., L_z = 1, 3, and so on. They correspond to single electron PS configurations 1s + 1p_x, 1p_y + 1d_x, 1d_y + 1f_x, … for Lz = 1, 1p_x + 1d_x, 1s + 1f_x, … for L_z = 3 and so on.

In Figure 3, we present the results of our calculations (see S3 for details) of the average orbital angular momentum 〈L_z〉 = L_z of the 2e-WM for an InP/GaInP₂ QD having D = 150 nm (r_s = 4.3), corresponding to S04, versus B for B = 0-5 T. The inserts show the electron density distribution for B = 0, 1.0, 2.25, 3.0, 4.25 and 5 T. Results for smaller r_s are presented in S3.

Broadened plateaus at values of L_z^MN =1 and 3 centered at B1~0.5 T and B3~1.25 T are seen in Figure 3. These plateaus were washed out for L_z^MN = 7 and higher, after which L_z showed a near-linear dependence on B. The plateaus shifted to higher B as D (r_s) decreased, and for D = 110 nm (r_s =3.4), they were centered at fields that were two times larger. Assigning the P-, D- and S*-type spectra in Figure 1b to L_z = 1, 3 and 5, the experimental B_Lz fields (1.5, 4 and 8 T) seemed to be two times larger than the calculated ones (~0.7, 2 and 3 T). This may have indicated some mechanism for reducing the effective field acting on an electron, related to vortex formation (see below).

The electron density distributions in Figure 3 show a strong decrease of the size of single-electron maxima D_e versus B, which was nearly by a factor of two in the range from 0 to 5 T (~50 nm versus 30 nm). At the same time, d_WM did not change. Thus, the enhancement of higher-order SCs in the PL spectrum in S*-dots was related to a decrease in D_e. Such a decrease can be formally described as an increase of the effective mass, since D_e~ 1/m*². In Figure 2b, we present calculated OI_N² ratios using a nine-fold increase of m* corresponding to B~7 T. The calculations yielded an S*-type order-of-magnitude increase of SCs having N > 1, confirming that these dots had strong B_bi (see also calculated spectra in Figure 2c).

4.3. Magnetic Field Shifts

For the analysis of the B-dispersion of PL spectral components, we used the following expression for the emission energy of the ground state of a 2e-WM weakly bound to a hole (2eWM-h) in a parabolic potential [27]:

E^2eWM-h(B) = E₀^2eWM-h(0) + E_C,2e(B) + E_r.m.,2e(B) +ħω_0h(B)

(1)

where the first term is the “free” WM energy, the second term is the energy of the Coulomb interaction between the two electrons, the third term is the energy of the relative motion of the two electrons and the fourth term is the cyclotron energy of the hole (see Appendix B.2). In expression (1), we take advantage of the fact that cyclotron energy of center-of-mass c.m. motion does not depend on the number of electrons [30], i.e., E_c.m.,2e(B) = ħω_0e(B) + const. We also use the fact that m_h* > >m_e*, and we neglect the energy of the relative motion of electron and hole. We further assume, for simplicity, that the energy of the electron-hole interaction does not depend on B. The second term in (1) represents the difference between the energy of the 2e state of interacting and noninteracting electrons and it is approximately given by E_C,2e(B) = 0.1$\frac{{\omega }_c}{\sqrt{m^{*}}}/\left(\sqrt[4]{{\omega }_c^2+ 4{\omega }_0^2}\right)$) (see [31]). The third term gives L_z^MN transitions discussed above and makes much smaller contribution (see S3). Thus, the B-dispersion of the 00 transition (i.e., 1s state) is

E₀₀(B) = E^2eWM-h(B)-E₀^2eWM-h(0)≈ħω_0h(B),

(2)

accounting for the fact that ħω₀h(B) ≈ 2E_C,2e(B). For Ns, p_x,y and d_x,y transitions, we find from the Fock-Darwin spectrum (see Appendix B.2)

E_0N(B) ≈ ħω_0h(B) − (2N + 1)ħω_0e(B),
E_px,py(B) ≈ ħω0_h(B) − 2ħω_0e(B) ± 1/2ħω_ce (B),
E_dx,_dy(B) ≈ ħω_0h(B) − 3ħω_0e(B) ± 1ħω_ce(B),

(3)

In Figure 4 we present a calculation of the energies given by Equation (3) for an electron charge e* = 1 and 1/7, together with the experimental points for S* N’s peaks of S11 dot at B = 8 T, at which positive dispersion begins. The energies show a strong negative B-dispersion (~1 meV/T) for e* = 1 resulting, from the small value of m*_e and the fact that m*_e<<m*_h, i.e., a large ω_ce and ω_ce >>ω_ch. Such large dispersion also have 1p and 1d states presented in Figure 4 too. For e* = 1/7, a negative dispersion decreases down to −0.1 meV/T, owing to the decrease of ω_ce to values close to ω_ch.

The positive dispersion of the S*-region indicates ω_ce < ω_ch and for e* = 1/7 results in a more than three-fold decrease of m*_h (see example in Figure 4 for B> 8 T), which can indicate heavy-light hole mixing. A qualitative comparison of B-dispersions presented in Figure 4 with Figure 1b for B > 7 T shows that matching the peaks of the S- and S*-type spectra is achieved only for e* = 1/7, thus indicating the observation of the fractional charge of single-electron states. The appearance of the S*-type spectra implies a recombination involving e-h states having l_z = L_z^MN = 3 or 5, leaving the residual electron in the 1s state. Thus, positive dispersion is a signature for the mixing of the holes of the states having large angular momentum.

5. Magneto-Electrons

5.1. Size-Dependence of Fractional Charge

A quantitative comparison of B-dispersions and a fit using Equation (3) are shown for dots S04, S11 and S12 in Figure 5a–c, respectively. This gives fractional charge values 1/k, where k is the number of vortexes, versus D (see

Table 2. Measured fractional charge of single electron states.

Quantum Dot	Dot Size D (nm)	Fractional Charge 1/k
		N’s	1p	1d
S04	110	1/9	1/7	1/5
S11	130	1/7	1/5	1/4
S12	150	1/5	-	1/3

). In the figure, the PL spectra are shown as contour plots, on which peak positions (circles) and fitted curves were superimposed. Note that the fit includes B_bi, which is -10 T for the S11 and S12 and −15 T for the S04 QDs.

The fit makes it possible to consistently match all observed peaks to single-electron ISs N’s, 1p_y and 1d_y in these dots. We found that the fitting procedure was robust (in terms of self-consistency) against variations of level sequence, 1/k and B_bi.

The most straightforward fitting was obtained for S11 (see Figure 5b), which had a complete data set, in terms of the measured B-range, peak resolution and intensity. For this dot, the two low-field oscillations of the 1s-peak, which had an amplitude/period 0.3 meV/2T, could be attributed to switching between 1s and 1p_x/1p_y states; formation of P-type spectra could be attributed to the contribution of the 1d_y state and its crossing with the 2s state at B = 3 T; the formation of the D-type spectra could be attributed to the contribution of the 1p_y at B > 5 T; and the formation of S*-type spectra can be attributed to a positive dispersion at B > 7 T.

Almost all these features were resolved in the S04 dot (see Figure 5a), accounting for the limited B-range due to the large B_bi. For the S04 dot, however, the 2s state was not resolved, probably because of an overlap with the 1d_x state, and S*-type peaks were smeared out/crossed because of the smaller energy separation.

For the S12 dot, which had a limited set of identified peaks due to low PL intensity, it was possible to resolve (see Figure 5c) the following: the crossing between the 2s and 1d_y states, manifesting the onset of P-type spectra at B = 3 T; the transition between D- and S*-type spectra at B = 6 T; and the positive B-dispersion of the 3s and 4s peaks in the S*-range.

The measured fractional charge (see Table 2) was 1/9, 1/7 and 1/5 for the N’s states, 1/7 and 1/5 for 1p states and 1/5, ¼ and 1/3 for 1d states. This decreased with D and angular momentum. The number of vortexes versus D is shown in Figure 6 and reveals a linear dependence in the measured range. Also, for D = 150 nm, it shows a linear decrease versus l_z.

Note that the fit included a matching between N’s states in S- and S*-type spectra and revealed a suppression of the 1s (00) component in the S*-type spectra, which means that the peaks denoted 0N at B_e = 0 T in Figure 1b were actually 0(N + 1) peaks, as shown in Figure 5a,b. This corresponded to a much smaller “effective” D_e (see lower plot in Figure 1b) due to vortexes and related WZs (see inserts in Figure 6 below).

5.2. Magneto-Electrons and Fractional Quantum Hall Effect

Our results presented in Figure 5a–c and Figure 6 show self-generation of MFQ vortexes by a single electron in the strong Wigner-localization regime and imply a nano-superconducting characteristic of its quantum states. We denote such states as magneto-electron states or simply as magneto-electrons – e_mks. Note that for l_z > 0, vortexes of opposite directions were generated for the positive and negative parts of wave-functions, which was thus related to Majorana zero energy modes.

The k(D) dependence implies a critical size D₀ for which fractional charge can be generated, corresponding to k = 2. This, in turn, implies hard-wall vortexes, i.e., the existence of a fixed vortex size d_v and close-packed spatial structures. Assuming conservation of linear k(D) dependence for D < 100 nm and an effective size of the electron density generating vortexes equal to D_v = D/2, we could estimate d_v ≈ D₀/4 = 20 nm or a radius r_v~10 nm for the s-state. This value was close to the Bohr radius a_B* of InP/GaInP₂ QDs (~8 nm see Appendix B.1 and S1). Possible close-packed structures of a_B*-size-vortexes for different D and single-electron states are shown in the inserts in Figure 6. The observed decrease of k versus l_z may have been due to the reduction of D_v due the spatial “splitting” of electron density.

The vortexes of e_ms represented current loops and could be considered as a “bound” charge part of the electron wave function. Thus, a fractional charge 1/k appeared, owing to the vortex screening the electric field. In InP/GaInP₂ QDs, self-formation of e_ms generating vortexes and B_bi may have been due to the dissipationless electron motion in a lateral electric field produced by the AOD GaInP₂ shell. Thus, one can assume that for the S-type S10 dot, which had nearly the same D as the S*-type S11 and S03 dots, the lateral field was absent and B_bi was not generated. This may have been due to the different GaInP₂-shell structure, in which AODs were weakly ordered because of large x_Ga.

We should point out that different vortex-type single-electron eigenstates were considered theoretically in billiard models of square and related shape geometries in magnetic fields [32]. In this case, these states are excited states and vortex solutions arise because of the matching of the spatial geometry of the eigenstates to the shape of the confining potential. These vortex eigenstates appeared to be similar to the vortexes in mesoscopic superconductors of similar shape [33], which link these systems to e_m and give rise to the notion of the superconducting 2D “droplet” formed by a single-electron quantum state.

For r_s~4, the FQHE regime begins at B~0.2 T, corresponding to ν = 1, where ν = 4h/(πD²)/e₀/B is effective Landau level filling factor, assuming uniform charge distribution within QD. The formation of e_mk has thresholds B~0.2k T corresponding to ν = 1/k and D_k~ka_B*. This makes it possible to describe the FQHE, i.e., the appearance of corresponding conductance plateaus in quantum Hall (QH) bar structures, using a localization of e_mks in the disorder-induced potential fluctuations. The key effect in such a description is a quasi-ordering of the fluctuations [34], which is analogous to spontaneous symmetry breaking induced by quantum monitoring [35]. This results in a spatial modulation having a half-period equal to the spatial size ka_B*. This implies a self-confinement of e_mks at the corresponding ν. Therefore, localized/delocalized e_mks exist in the range between 1/k and 1/(k +1). In this range, delocalized e_mks provide edge hopping conductivity. For IQHE, Wigner-crystal-type network patterns of the localized electron states, observed in a scanning electrometer probe experiment for ν = 1 [36], naturally emerge from such self-confinement.

Within this picture, the alternation of ¼ and ½ charge versus size of the resonator, i.e., the number of electrons, observed in Aharonov-Bohm phase interferometry measurements of quantum Hall (QH) bars for ν = 5/2 [14], can be explained by alternation of the size of e_m by the Majorana modes.

5.3. Topological Quantum Computing

While in our experiment, the effective density of electrons was ~10¹⁰ cm⁻², the density used in QH bars for experiments aiming to realize TQGs [16] was nearly twenty times larger [14]. At such high density, the anyon e_mks are hard to control locally, making it difficult to measure their statistics [17] and perform elementary topological quantum computing operations, i.e., anyon interchange (braiding) and pairing (fusion) [16]. The same control seems more manageable at low densities; we have already demonstrated optically induced pairing and the interchange of e_ms in the same InP/GaInP₂ QDs, but with six e_mks [21]. Moreover, owing to B_bi, these were done at zero magnetic field and at relatively high temperature (10 K), which was more practical. Also, in InP/GaInP₂ QDs, the properties of e_mk anyons could be controlled by Ne, D, QD shape and B_bi. For InP/GaInP₂ QDs, TQG could be realized using adjacent nano-electrodes attached to single electron transistors [20]. Such a chip made it possible to move e_mks around each other and to control the local charge to measure the initialization and manipulation of q-bits.

6. Conclusions

In conclusion, using measurements of the magneto-PL spectra of single-electron InP/GaInP₂ quantum dots having Wigner-Seitz radius ~4, we demonstrated the existence of a magneto-electron state having a fractional charge. This is the anyon state, in which a few magnetic flux quanta vortexes are self-generated by a single electron in Wigner localization regime, providing nano-superconducting properties of quantum states. The emergence of the magneto-electron corresponds to the three- to nine-fold reduction of the cyclotron frequency of single electron Fork-Darwin states and its dependence from quantum dot size measured. These imply a close-packed arrangements of the magnetic-flux vortexes and make it possible to describe the fractional quantum Hall effect in terms of magneto-electron self-confinement in the disorder-induced potential fluctuations. Such self-confinement gives rise to the possibility of engineering of anyon properties (statistics) of magneto-electrons, and could be used to build anyon-based fault-tolerant topological quantum gates.