Majorana Anyon Composites in Magneto-Photoluminescence Spectra of Natural Quantum Hall Puddles

Abstract

In magneto-photoluminescence (magneto-PL) spectra of quasi two-dimensional islands (quantum dots) having seven electrons and Wigner–Seitz radius r_s~1.5, we revealed a suppression of magnetic field (B) dispersion, paramagnetic shifts, and jumps of the energy of the emission components for filling factors ν > 1 (B < 10 T). Additionally, we observed B-hysteresis of the jumps and a dependence of all these anomalous features on r_s. Using a theoretical description of the magneto-PL spectra and an analysis of the electronic structure of these dots based on the single-particle Fock–Darwin spectrum and many-particle configuration-interaction calculations, we show that these observations can be described by the r_s-dependent formation of the anyon (magneto-electron) composites (ACs) involving single-particle states having non-zero angular momentum and that the anyon states observed involve Majorana modes (MMs), including zero-B modes having an equal number of vortexes and anti-vortexes, which can be considered as Majorana anyons. We show that the paramagnetic shift corresponds to a destruction of the equilibrium self-formed ν~5/2 AC by the external magnetic field and that the jumps and their hysteresis can be described in terms of Majorana qubit states controlled by B and r_s. Our results show a critical role of quantum confinement in the formation of magneto-electrons and implies the liquid-crystal nature of fractional quantum Hall effect states, the Majorana anyon origin of the states having even ν, i.e., composite fermions, which provide new opportunities for topological quantum computing.

1. Introduction

Superconducting (dissipationless) transport corresponding to zero electrical resistance of the materials, discovered in 1913 [1], gave a quantum mechanical illustration of Newton’s first law of motion, which states that moving material objects will conserve a constant velocity if no external force acts on them, and which represents an obvious paradox for our imagination, since natural movements observed in everyday life are subject to a frictional force and are slowed down. Thus, in superconducting materials, the persistent currents loops (PCLs) can be thought to be perpetual and laboratory measurements show their life-time to be at least 100,000 years [2]. In an external magnetic field (B_e), the persistent vortex currents induce a perfect diamagnetism, i.e., the expulsion of B_e from the superconductor (SC), known as the Meissner effect [3]. PCLs in SCs provide the quantization of the magnetic flux [4] in multiples of ϕ₀^* = h/2e₀, where e₀ is the electron charge, which was documented in the measurements of the magnetization of micrometer size SC cylinders [5,6]. Single ϕ₀^* vortexes, also called Abrikosov vortexes or fluxons, arranged in regular lattices, are generated in major SCs above the critical field B_e1 [7,8]. The vortexes consist PCLs tens of nanometers in size, whose density is proportional to B_e, and they merge at a critical field B_e2 > B_e1, transferring the material into a resistive state. Thus, the existence of ϕ₀^* is directly related to the existence of systems having SC-type dissipationless transport. Such transport corresponds to a suppression of the scattering (“friction”) of charged carriers and exists in systems having a gap in the density of conducting states, which in conventional SCs arises from the formation of the bound electron pair state known as a Cooper pair [9].

While it is commonly accepted to associate superconductivity with Cooper-type electron pairing, a zero electrical resistance, i.e., SC-type dissipationless transport, is observed in two-dimensional (2D) electron (e) gas semiconductor structures in the quantum Hall effect, i.e., at high perpendicular B_e, at integer (IQHE) [10] and fractional (FQHE) [11] Landau level (LL) filling factors ν = nϕ₀/B_e, where ϕ₀ = 2ϕ₀^* and n is the electron density. This transport is provided by a skipping-type edge current [12], which for the FQHE involves composite quasi-particles, called composite fermions (CFs) [13], consisting of e with 1/ν ϕ₀ magnetic-flux-quanta vortexes (Vs) attached, resulting in a fractional charge equal to νe₀. The presence of the Vs and SC-type gapped state, however, implies PCLs generated by CF e. While the necessity of such intrinsic sub-e PCLs was not considered in the commonly used description of the FQHE states, which is based on Laughlin’s approach [14], their existence was revealed in our recent observation of FQHE-type states for single [15] and five [16] electrons confined in quasi-2D InP/GaInP₂ islands-quantum dots (QDs) having dimensionless Wigner–Seitz radius r_s~4 and ~2, respectively, where r_s = 1/[a_B^*(πn)^0.5] and a_B^* is a Bohr radius.

The FQHE-type states have been observed in these QDs at zero B_e using high-spatial-resolution magneto-photoluminescence (magneto-PL) spectroscopy in the measurements of the B_e-dispersion of PL spectral components and their spatial localization. The appearance of these states results in a built-in magnetic field B_ν~6–15 T and a dependence of the ground-state, equilibrium fractional charge ν₀e₀ on the dot size D, for which ν₀ decreases with D. These observations imply a self-generation of Vs in a quantum state $|i\rangle$ occupied by a single e, and a fixed size of the V (at zero external field B_e) equal to 2a_B^*. The e, thus, can move without dissipation within a distance r_sa_B* and for r_s > 1 can be considered as nano-superconducting puddle, having charge-density distribution $\rho \left(r\right)={\left|{\psi }_i\left(r\right)\right|}^2$, where ${\psi }_i$is the wave-function of the state $|i\rangle$. Thus, ${\psi }_i$can be formally considered to be a complex SC order parameter of the linearized Ginzburg–Landau equation [17] used for the description of mesoscopic SC structures [18,19]. This single-e-puddle can support ~r_s sub-e PCLs generating 1/ν₀ Vs. This reduces the Coulomb interaction energy of es and, thus, the state having 1/ν₀ sub-e PCLs and charge ν₀e₀ is self-formed [16]. The ν₀ and corresponding B_ν = n2ϕ₀ν₀ are directly determined by the dot size (D) and the number of es (N), i.e., r_s. We called such a state a magneto-electron (em) and denoted it e^ν, where the em composition ν = n/k, with n (k) the number of es (Vs), is the quantity that substitutes the LL filling factor of the FQHE for quantum confined es. We observed ems having ν < 1, n = 1–4, k = 1–9 and molecular structures$6e^{1/4}$, $5e^{1/4}$, $3e^{2/7},$ and $3e^{1/3}+e^{4/15}$, having e^ν size ~40 nm and bond length ~60 nm for N = 5 and 6. The last two structures are transformed into one another by photoexcitation involving the braiding and fusion of e^νs, which are the elementary topological quantum computing (TQC) operations [20].

We should point out that our observations of the fractional charge and the self-formation of Vs of a single e at zero magnetic field sound like a paradox (similar to the perpetual persistent current in SC) or an artifact, not taking into account the quantum nature of single e states, for which non-zero angular momentum states consist of “fractional charge” parts, and relying on the famous experiment of Millikan demonstrating a “fixed” e₀ value in charged oil droplets [21]. However, at the same time, in parallel experiments made by Ehrenhaft, much smaller charges, i.e., sub-es, have been revealed in metal (Au, Pt, …) particles (see description of Millikan-Ehrenhaft dispute on sub-e in Ref. [22]). Moreover, much later, a transfer of 1/3e₀ charge from tungsten to Nb balls has been detected in experiments on superconducting magnetic levitation [23]. We can suppose that these observations can be connected to our observation of em.

Nevertheless, since we observed e^ν ems in QDs having ν < 1, one can expect their formation in QDs having ν > 1, as for FQHE states [24]. These states correspond to partial (total) filling of the highest (lower) LLs and involve CFs, which are anyons [16], in the highest LL and IQHE es in the lower one.

A much investigated ν > 1 FQHE state is the ν = 5/2 state, which, according to the conventional description, corresponds to the half-filled second LL and fully filled lowest one. The CFs in the second LL show non-Abelian anyon properties [25], which implies that they have half-flux Vs with a zero-energy excited state, known as a Majorana zero mode or simply a Majorana mode (MM) [26,27,28,29], and thus are a perspective for the realization of TQC [30,31].

In em, the MM is a particle-antiparticle pair, consisting of Vs having an opposite, anti-parallel direction/orientation. It can be naturally generated in a state having non-zero angular momentum l_z, for which the e-wave-function has “anti-phase spatial splitting”. In such states, only half of the ${\psi }_i$ can generate V along the B_e direction, which thus is a half-V having a MM excited state. For the em the MM adds two more Vs resulting in a two-times smaller charge, the signature of which was observed for the 5/2 state [32]. Moreover, since a MM adds zero magnetic flux, it can possibly be generated at zero B_e.

A realization of self-formed ν₀ > 1 ems/anyons in general and ν₀ = 5/2 in particular requires a decrease of r_s. In our earlier studies [33], we have measured a quantum Hall-type InP/GaInP₂ QD having N~9, r_s~1.5 and B_ν~2 T, which approximately matches that expected for ν₀ = 5/2.

Here, we analyze a few such dots in detail using measurements of the B_e-dispersion of their PL spectra components. In the measurements, we have revealed several anomalies, which include suppression of the dispersion, paramagnetic shifts, jumps (up to ~2 meV), and their hysteresis. We found that the appearance of these anomalies depends on the dot size and the direction of change of B_e. We analyzed the experimental data using a theoretical description of PL spectra and the electronic structure of such dots based on single-particle Fock–Darwin (FD) states and a many-particle configuration-interaction (CI) approach, which allows one to explain the observed anomalies by the self-formation of em-MM-anyon composites corresponding to ν₀~5/2, their collapse and the emergence of the anti-em-MM composites induced by the magnetic field. Our analysis has shown that the formation of the ems corresponds to the generation of Vs by single and pair non-interacting es occupying quantum-confined states. This allows one to describe the FQHE in terms of liquid-crystal states involving Majorana anyons and opens new routes for the realization of topological quantum computing.

2. Materials and Methods

2.1. Natural Quantum Hall Puddles

The details of the growth procedure and structural and emission properties of the InP/GaInP₂ QDs were described previously [34]. The dots have a flat lens shape (aspect ratio 10) and a lateral size D~50-180 nm. Their shape reveals a small elongation and asymmetry, which in most cases can be described as a combination of a ∼5% elliptical distortion (D_‖/D_⊥ = 1.05) and a 10% change of R_⊥. The specific structural property of this QD system [35] is an atomic ordering of the GaInP₂ matrix material, which results in composite core-shell structure consisting of InP QD having zinc blende crystal structure surrounded by a few atomically ordered GaInP₂ domains having rhombohedral crystal structure and size 10-50 nm. In this composite, the domains generate strong piezo-electric fields resulting in e doping (up to 20) and B_ν (up to 15 T) forming natural quantum Hall puddles.

Here, we studied three of such puddles denoted D01m, D07m and D09m. They have a number of electrons N = 6 and 7 and D~75 nm resulted in the electron density n~3 × 10¹¹ and r_s~1.3, which corresponds to a weak Wigner localization regime.

2.2. Single Dot Magneto-Photoluminescence Spectroscopy Measurements

We measured magneto-PL spectra of single puddles using a home-made near-field scanning optical microscope (NSOM) having spatial resolution up to 25 nm operating at 10 K and magnetic fields of up to 10 T. We used home-made tapered fiber probes coated with Al, having an aperture size of 50–300 nm in a collection-illumination mode. The spectra were excited by the 514.5 nm Ar-laser line (Edmundoptics, Cherry Hill, NJ, USA) and measured using a CCD (multi-channel) detector (Horiba, Piscataway, NJ, USA) together with a 280 mm focal length monochromator (Horiba, Piscataway, NJ, USA. The excitation power measured before fiber coupler was ~5 μW, which provided a power density of ~0.5 W/cm². The spectral resolution of the system is 0.2–0.4 meV. σ⁺ and σ⁻ circular polarization were measured using λ/4 plate (Newport, Irvine, CA, USA) and linear polarizer (Newport, Irvine, CA, USA). In the spectra, we measured the shifts of PL peaks related to occupied e-shells versus B_e, which were compared with the theoretical calculations and used to extract the charge of single particle states of the puddles.

2.3. Analysis of the Data

The shape of the PL spectra, i.e., the number of spectral components, their position and full-width-at-half-maxima, were analyzed using a multi-peak fitting procedure from graphic software OriginPro (version 20.0, Northampton, MA, USA)

The values of effective (screened) quantum confinement ħω₀^* (see Appendix A.3) and N were measured from the splitting and the number of anti-Stokes peaks in the PL spectra, respectively. For N, the measurements of the shifts of anti-Stokes components in magnetic field were used as well. The D values were estimated from the charge density distributions (CDD) calculated from the unscreened quantum confinement value ħω₀ = ħω₀^*/0.7 and N values using CI approach [16,34] and from NSOM scanning experiments. Using N and D values we calculate n and r_s. Note that these values are related to the photo-excited state having N* = N + 1 electrons.

2.4. Theory and Calculations

The theoretical description of the PL spectra of quantum Hall puddles in a magnetic field based on the Fock–Darwin (FD) spectrum is presented in Appendix A. The description involves a general model (Appendix A.1), an analysis of the single-particle states of InP/GaInP₂ QDs involved in PL transitions using the 8 band $\overline{k}·\overline{p}$ method, i.e., non-interacting electrons and holes (Appendix A.2), and the analysis of the effect of the Coulomb interaction on single-particle states using the Hartree–Fock (HF) method (Appendix A.3). The resulting formulas of the model, i.e., B_e–dispersion of PL peaks, are Equations (A4c) and (A5) in Appendix A.4.

The ground states (GSs) of the puddles in a magnetic field are analyzed in Appendix B. Total energy and total angular momentum values were calculated using the multi-particle CI method (Appendix B.1) and the single-particle FD spectrum, including the vortex contribution (Appendix B.2). Both methods revealed GSs having fractional ν, suggesting em-composite (em-C) formation.

The parameters of em-Cs, which are composition ν_n⁺(ν_n)_S, where n is number and S is a total spin, and charge e_N^* are specified in Appendix C.1 (see Equations (A6)–(A8)). There, a mismatch of the emission energy of em-Cs is analyzed and corresponding quantities, which are Δ_ν jumps and related PL shift (E_s^em-C), are described (Equations (A9) and (A10)).

The fitting of the Fock–Darwin spectrum for the measurements of ν and details of the construction of the V structure of em-Cs are described in Appendix C.2 and Appendix C.3, respectively.

2.5. Summary of Measured Parameters

In

Table 1. Properties of anyon(magneto-electron) composites of InP/GaInP₂ quantum Hall puddles.

QD	N*	ħω0*/ħω0 (meV)	D (nm)	rs	Bec/Bν (T)	ν0+(ν0)SeN*	νn+(νn)SeN*
D01m	7	5.5/9.2	65	1.2	3/3.6	−5/2(−21/8)3/23.75	23/9(∞)1/24.91	27/11(5/2)3/23.91	-
D07m	8	4.5/7.5	75	1.3	2.5/3.2	−21/8(−8/3)03.05	8/3(∞)15.66	5/2(5/2)15.41	10/7(7/4)24.25
D09m	8	3.5/5.8	85	1.5	2/3	−9/4(−5/2)02.08	32/13(∞)14.91	23/10(3)23.88	9/7(4/3)23.71

, we summarize the initial zero-field parameters of the puddles, which are the number of the electrons in photo-excited state N* = N + 1, D, r_s, ħω₀^*/ħω₀, and the final parameters of em-Cs formed in these puddles obtained from the measurements and the analysis of the data, which are two intrinsic magnetic fields B_ec and B_ν (see below), and ν₀(ν₀⁺)_S and ν_n(ν_n⁺)_S values of the equilibrium and B_e-induced em-Cs, respectively. In the notation of the compositions in Table 1, we include reduced charge value e_N^* as a superscript.

3. Experimental Results

3.1. Shell Structure and Built-In Magnetic Field of Quantum Hall Puddles

Figure 1a compares the PL spectra of the dots/puddles studied at zero internal field B (see below), plotted in Stokes energy units. The inserts show CI CDDs of the dots and a contour plot of spatially resolved PL spectra near the center of D07m dot. Figure 1b shows circularly polarized spectra of the D07m dot measured at B_e = 0, 1, 2, …, 10 T.

CDDs show a decrease of the size related to a decrease/increase of D/ħω₀ (see Table 1). The CDDs are elongated along x and their landscape has the same topology consisting from two minima of ~D/6 size separated along x by ~D/3, which indicates molecular structure (see Appendix B.1). The left insert shows the measured size of the emission area of the D07m dot of ~90 × 60 nm (average size of 75 nm), in agreement with the corre sponding CDD size.

In Figure 1a, it is seen that the spectra of the dots studied consist of the main peak denoted by s and two weaker anti-Stokes ones denoted by p and d. These three peaks are related to three occupied e-shells of the QD in the photo-excited state, as described in Appendix A.1. The peaks have a full-width-at-half-maximum ~3 meV and varying splitting corresponding to ħω₀^* (see Table 1), related to the size variation. The D09m dot also has a few times larger relative intensity of p- and d-components. There is an increase of the ratio of the s-p to p-d splitting related to the size increase.

In Figure 1b, the PL spectra of the D07m dot at zero B_e reveals nearly a two-times stronger intensity of the σ⁻- component, i.e., they are σ⁻-polarized, which indicates B_ν. At B_ec = 3 T, the σ⁻-polarization disappears and the spectra acquire a σ⁺-polarization at larger B_e. This indicates B_ν~−B_ec (see Table 1), where B_ec is a compensating field and B_ν has a direction opposite to B_e. Thus, in the range from 0 T to B_ec, the internal field B = B_e + B_ν is negative and decreases from B_ν to zero. In this range, which we denote B⁻↓ or B_e^a, B_e < |B_ν| and the PL spectra have an anomalous B_e-shift (see below). Such an anomalous shift is a direct signature of B_ν, which allows one to detect it independently from polarization measurements. For larger fields, i.e., B_e > |B_ν|, B becomes positive and increases versus B_e. In this range, which we denote by B⁺↑ or B_eⁿ, normal B_e-shifts are observed, but B is lower B_e on B_ec.

There is an anomalous B_e^a-range, indicating B_ν in D01m and D09m dots too.

3.2. Shell Structure in Magnetic Field

In Figure 2a–c, we present unpolarized spectra of the dots, measured at B_e = 0, 1, 2, … 10 T. The spectra were measured under an increase of the field from 0 to 10 T, denoted as B_e↑ and shown in the lower part of the graphs, and a decrease from 10 to 0 T, denoted as B_e↓ and shown in the upper parts of the graphs. For this range, the B_eⁿ and B_e^a ranges are the B⁺↓ and B⁻↑–ranges, respectively.

The spectra having B = 0 T and dividing the B_e^a- and B_eⁿ-ranges (B_ec = 3 and 2 T was found in D01m and D09m dots, respectively) are shown by thick solid lines. In Figure 2a–c, the peak maxima are connected by straight lines, which allows one to quali tatively trace their shift.

At all B_e values, the s-peak is dominant, and changes of the spectral shape are caused by the changes of the intensity and the position of the p- and d-peaks at B_e > 4 T. These are: the peak merging at B_e = 7 T in the D01m dot; the few-times intensity increase of the d-peak at B_e = 9 T and the appearance of the additional x-peak between the s- and p-peaks at B_e = 7 T in D07m dot; a few-times intensity decrease of the p- and d-peak at B_e = 8 T in the D09m dot.

The peak shift, further denoted as s, p- and d-shifts, in the B_eⁿ-range is mostly positive, i.e., increases with field increase, and at B_e = 10 T reaches average values of ~2, ~5, and ~3 meV, respectively. In the B_e^a-range, the shifts are negative (paramagnetic), i.e., “anomalous”, and have values from 0 to 2 meV. In both ranges, the shifts reveal a few bends (see spectra near B_e~3 T and 6 T).

3.3. Anomalous Shifts and Jumps Induced by Magneto-Electrons

In Figure 3a–c, we show the measured data points of the peak positions versus B and calculated B-dispersion of the peaks overlaid on the contour plots of the spectra. Calculated shifts are presented in the B_eⁿ-range, i.e., for B ≥ 0 T. The peaks are denoted by the quantum numbers of the Fock–Darwin (FD) spectrum kl (see Appendix A.2).

The s-shift data show that the bends seen in the spectra in Figure 2a–c are jumps having amplitude 1–2 meV over ΔB_e~1 T (see dashed ovals in Figure 2a–c). Their appearance is different in different dots and B/B_e ranges, revealing size dependence and B_e↑–B_e↓ asymmetry (hysteresis). In the D01m dot, one step is observed in both B⁻↓– and B⁻↑–ranges at B = −1 T, which is thus symmetric. In the D07m dot, two asymmetric steps are observed, one in the B⁻↓– and the other in the B⁺↓–ranges at B = −2 and 5 T, respectively, and no mirror steps appear in the ranges of reversed direction of field change. In the D09m dot, four asymmetric steps are observed in the B⁻↓– and B⁺↑–ranges at B = −2, 3, 5 and 7.

The calculated s-shift shows a weak, nearly linear dispersion with slope ~0.25 meV/T and in the regions where the jumps are absent the experimental slope is the same as the calculated one.

The p-shift follows a dispersion of 10-state up to B_c~5 T and at larger field the shift is saturated (for D01m dot) or becomes negative (for D07m and D09m) approximately reaching the 02–04 states, i.e., shows bending related to level crossing (see Figure A2c). The p-shift has a ~30% smaller slope than the 01-state (~1 meV/T).

The d-shift follows the 11-state for B_e < B_c for all dots and the 03-, 04- and 05-states for B > B_c and the D01m, D07m and D09m dots, respectively, revealing a level crossing similar to the p-shift. For the D01m dot, the d-shift is strongly suppressed compared to the 11-state.

In the B_e^a range, the experimental data points do not coincide with mirror counterparts in the B_eⁿ-range, i.e., the shifts have a B_e^a/B_eⁿ asymmetry. This is most clearly seen for the d-peak in the D01m dot.

The appearance of the B_e^a-range is direct evidence for the existence of the ems in this range. For the B_eⁿ-range, the existence of the ems follows from the observation of the jumps of the s-peak and the suppression of the B_e-dispersion of the p- and d-peaks for B_e < B_c. The former are Δ_ν steps and the latter is direct evidence for charge reduction of the e in the single-particle state (see Appendix C.2).

Since the B has opposite directions in the B_e^a and B_eⁿ ranges, the Vs and ems of the former are anti-Vs (aVs) and anti-ems (aems) in the latter (see Appendix C.1). The ems at zero B_e are self-organized and can be considered to be equilibrium ems, which we will denote as S-ems. Thus, the shifts of the peaks in the B^a_e-range are related to the destruction of S-ems by the external field, which implies that the internal field generated by S-ems can be B_ν ≠ B_ec and the B decrease versus B_e increase in this case is non-linear in the B_e^a-range, as observed. The ems in the B_eⁿ range are induced by the B_e and can be considered as stimulated, further denoted as B-ems.

3.4. Majorana Anyon Composites in Fock-Darwin Spectrum

In Figure 4a–c, we present the results of the e-em-FD spectrum fit to the experimental data. The data and fit are presented in the reduced field ν_B^-1 and energy ω/ω₀ units (see Figure A2c) and the e-em-FD-states (e-em-FDSs) are labeled as e^ν_l, where l is angular momentum of the state (see Appendix C.1). In the figures, we also show total charge values e_N^* (upper plots) and constructed V-structures (inserts). Total charge values are shown for photo-exited and initial states and were used to calculate the Δ_ν and E_S^em-C. The experimental data for B_e↓ and B_e↑ measurements are plotted in the same graph and they reveal a B_e↑–B_e↓ asymmetry of the s-peak jumps as discussed above, and the same for p-shift in D07m and D09m for ν_B⁻¹~0 and 0.4–0.9, respectively.

A comparison of the plots in Figure 4a–c shows different ν_B⁻¹ ranges, acquired using the same B_e range, in the dots studied, which is due to the different dot size giving an increase of the ν = 1 field B₀ as D decreases (see Appendix B.1). The final ν_B⁻¹value is 0.6 for the smallest D01m dot and 1.1 for the largest D09m dot. At the same time, the ν_B⁻¹ starting value is the same ~−0.29, i.e., ν~−7/2.

In addition, a comparison of the plots shows that the ν_B⁻¹-dispersion of the FDSs is smoothed out as D increases (see the shallowing of 2e₁ level “depth” at ν_B⁻¹~0.5), which shows a suppression of the quantum confinement effect.

The data in Figure 4a–c show that the ems are observed in the p- and d-single particle states forming B-em-Cs over the entire B_eⁿ-range measured. The composites correspond to a set of n GSs |ν_nS1⁺ > ^N_1/νB given in Table 1. In all B-em-Cs, only the 2e s-state is em-free and has ν = 2 (e²₀-state) for n = 1 and 2 and ν = 1 (2e¹₀-state) for the n=3 composites. The rest are B-em states, which have ν = 1 and 2/3 for 2e p_y/p_x-states and ν = 1/2, 1/3, 14 and 1/8 for 1e 10(p_x)-, 02(d_y)-, 20(d_x)- and 03(f_y)–states, depending on B, N^* and D.

The n = 1 structures in the inserts in Figure 4a–c show that in the D01m dot B-em^M-C has two single-state Majorana ems (em^M) (see Appendix C.1) e¹_-1(0) and e^1/4₋₂(0) and one two-state em^M, formed by s-e and e^2/3₁ states, in which the V of the former is compensated by the aV of the latter. Note, however, that this compensation is not complete since two s-es in the ν = 2 state generate two-times smaller flux. In the D07m dot, the n = 1 em^M-C has the same number of Vs N_V since the extra e generates e^1/2₂(0) em^M in a d_x-state taking two Vs from the d_y-state. In the D09m dot, MMs e^1/2₋₂(2) and e^1/2₂(2) are generated in d_x- and d_y-states increasing the N_V on four Vs/aVs.

For n = 2 and n = 3, em^M-C are formed by the addition of Vs and elimination of aVs, resulting in an increase/decrease of N_V/ν_nS⁺. Thus, we observed the expected N_V increase, with an increase of D and B, and the minimum N_V value of 9 has n = 1 em^M-C in D01m, while the maximum of 21 has n = 3 em-C in D09m.

The charge values obtained are in the range e_N = (3−6)e₀, which is thus up to two times smaller than that without ems, i.e., (7–8)e₀. The reduced charge for n = 1 corresponds to a zero field B_ν generated by the composite, i.e., ν_1S⁺ = ∞, and thus can be considered as Majorana anyon/em-C (em^M-C). This em^M-C has e_N^* = 4.91, 5.66, and 4.91 in the dots D01m, D07m and D09m, respectively, which shows that the increase of e_N of D07m relative to D01m due to an extra e is compensated in D09m by its decrease due to the increase of the N_V.

3.5. Charge Hysteresis and Majorana Modes

The Δ_ν values calculated show an absence of the steps in the D01m and D07m dots and their presence in the D09m dot (see the corresponding curves in Figure 4a–c). Zero Δ_ν (no steps) occurs when the photo-excited (topmost) em states and the initial states of the neighboring em-Cs have the same ν and N_V, respectively (see Equation (A9)). This is the case for the smallest D01m dot for both B_e↑ and B_e↓ measurements.

For the intermediate size D07m dot, this took place only in the B_e↑ data, and in the B_e↓ data a negative jump occurs at ν_B⁻¹~0.27 (ν~7/2). After the jump, the low-energy position of the s-peak is maintained under a further field decrease down to zero. This shows that two em^M-Cs exist in the initial state in the D07m dot, which are 3(∞) and 19/7(∞) em^M-Cs (see V-structure in the lower right insert in Figure 4b), among which the latter is a MM of the former appearing because of the generation of an e¹₋₁(0) em^M. The B_e↓-jump is thus related to a charge increase in the initial state due to N_V reduction in the 3(∞)-em^M-C and is reproduced well by E^em-C_s(B). Thus, B_e↑–B_e↓ measurements reveal charge hysteresis (CH). CH is accompanied by B_e↑–B_e↓ p-peak splitting at ν_B⁻¹, B = 0, which can indicate generation of additional Vs in the p-state.

In the largest dot D09m the ν_l values of the four topmost em states are 1/4₂, 1/8₋₂, 1/2₁, and 1/3₋₃, which are different and thus give three Δ_ν steps at ν = 7/2, 2 and 5/4, the first and the third of which are negative, as is seen in Figure 4c. The number and position of these steps match well to that of the B_e↑ experiment, but the sign of the first and third jumps is positive instead. This implies a MM in the corresponding initial state, substituting Vs of the photo-excited em. Analysis has shown that this is the e¹₋₁(0) em^M (the same as in the D07m dot) forming the e^2/5₋₁(1) em from the e^2/3₋₁(1) one. The calculated E_s^em-C accounting for this MM shows good agreement. The B_e↓ data show two CH starting at ν~5/4 (CH₁) and 2 (CH₂), respectively. In the CH₂ em-C, there is an increase of p-peak energy, indicating the generation of additional Vs.

3.6. Collapse of S-Magneto-Electron-Composites

The structure of S-em-Cs and the ν₀⁺(B_ν) values can be derived from the crossing of the positions of the p- and d-peaks at B_e = 0 and em-FDSs of the B_n-range. Thus, a position of the topmost d-peak corresponds to a d_x-state at ν ~5/2 and S = 3/2 in the D01m dot and to d_y-state at ν~5/2 and S = 0 in the D07m and D09m dots. Note that while for the D01m dot the total spin of the mirror aem S- and B-composites is the same (3/2), for the D07m and D09m dots it is different (0 versus 1 and 2). Note also that the ν₀⁺ value of all dots (see Table 1) are nearly the same, i.e., ~5/2 (for dots D01m and D07m ν₀⁺ = 5/2 + 1/8 and 5/2 + 1/6, respectively), and the corresponding B_ν values are larger than B_ec on ~1 T.

The preference of the self-formation of the S-em-Cs follows from their smaller charge compared to the B-em-Cs; it is 3.75e₀, 3.05e₀ and 2.08e₀ for the S-em-Cs versus 3.91e₀, 5.41e₀ and 3.85e₀ for the B-em-Cs for the D01m, D07m, and D09m dots, respectively. Charge reduction is provided by the extra Vs in the p-states (see the corresponding V-structures in the left inserts in Figure 4a–c) and thus there can be some mechanism for selecting the V direction in these states. These preferential Vs create e^*1/2₋₁(2) and e^*1/6₁(4) S-ems (we add a superscript * to distinguish S-ems from B-ems).

In the D01m dot, a symmetric positive s-jump at ν_B⁻¹~0.1 related to S-em-C destruction indicates the generation of aVs/Vs decreasing/increasing B_ν needed for formation of n = 1 B-em-C/S-em-C in B_e↑/B_e↓ measurements. A transition to S-em-C for the B_e↓ data is accompanied by a red shift of the p-peak, indicating emission of the p_y-component, which is not seen for B-em-Cs. Its appearance implies a redistribution of the hole density (see (Appendix A.3) under V formation and the instability of B-em-C.

In the D07m and D09m dots, negative B_e↑ s-jumps correspond to the annihilation of MMs under a decrease of B and their absence in the B_e↓ data indicate a suppression of the MM under the collapse of n = 1 B-em-Cs. The signatures of S-em-C instability in these dots is a B_e↑–B_e↓ splitting of the p-peak, which is stronger in D07m dot.

4. Discussion

4.1. Majorana Anyons

A subset of the GSs measured is different from that obtained in the exact quantum mechanical CI calculations in Appendix B.1. For D09m dot, they are |∞₁, 3₂, 4/3₂ > ⁸_1.1 (see Table 1) and |8₀, 7/2₁, 7/3₀, 2₁ > ⁸_0.6 (see Figure A2a,b), respectively, showing a reduction of the number of B-em-C GSs compared to e-GSs due to a suppression of the level crossing (see Appendix C.2). The CI set includes fractional ν GSs, which, however, do not imply a generation of Vs by single electrons, i.e., fractional charge, and reflect only a matching of the number of Vs to the total angular momentum value [36,37]. Moreover, the observed B-em-Cs have a variety of MM states, including d-type MMs, em^M and em^M-Cs, which is unexpected and very important accounting for the limited experimental data on MMs (mainly for the p-type-states) in FQHE [25], one-dimensional hybrid superconductor topological structures [38,39] and in topological superconductor [40]. Thus, our observations give high impact to the physics of anyon/Majorana states and to the perspectives of their use for TQC. Note that from five em-Cs observed there is only one MM-free, which is a ν_2,1⁺ = 5/2 composite in the D07m dot, and, thus, MMs seen are intrinsic features of the confined electrons for ν > 1, in contrast to ν < 1 molecular states [16]. In the context of em^M and em^M-Cs, a SC analog of such states has been reported in the compound EuFe₂(As_1−xP_x)₂ [41].

The Δ_ν steps observed demonstrate detection of fractional charge variations related to MMs using PL spectra. Moreover, the observed CHs are related to the states with and without MM, i.e., to a MM qubit (see Appendix D), and for the D07m and D09m dots these are zero-field qubits. This two-level qubit can be used in conventional schemes of quantum computing and/or quantized information bit of classical Boolean logics [42]. The data show that the appearance of the Δ_ν steps depends on D, N and B, or more generally on N_V and r_s, and is suppressed with r_s decrease. This gives the opportunity for engineering of MM qubits. The CH is related to the suppression of aV generation under a decrease of the field and evidently reflects some fundamental interaction between em Vs and a sign of the magnetic field variation, which needs further investigation.

4.2. Magneto-Electrons and Fractional Quantum Hall Effect States

Our experimental results presented here and in the previous studies [15,16] show that the fractionally charged em-anyons emerge in the quantum confined (gapped) e-states, which allows superconducting-type sub-e PCLs/Vs generated by a single or a pair of es, creating ems. This allows consider a description of the FQHE in terms of a liquid crystal (LC), rather than as an incompressible liquid, in which specific ν_nS⁺-B-em-C is a unit cell of the crystal. The nucleation of the ν_nS⁺-B-em-C LC is provided by a quasi-ordering of potential fluctuations (PF) [43] of the corresponding size as was suggested by us in [15]. The experimental evidence of such quasi-ordering is the observation of the Wigner-crystal-type network patterns of the localized electron states for ν = 1, observed in a scanning electrometer probe experiment in quantum Hall 2D-e structures [44].

Within the LC description, the conductivity plateaus are quantum-confined-type L^S plateaus (see Figure A2b,d) rather than a set of states above the mobility edge of the disorder-broadened LLs. The L^S plateaus strongly overlap, as follows from the E_L^S curves in Figure A2e, which is consistent with the models describing the temperature-dependence of the width of IQHE and FQHE plateaus [45], revealing a Lorentzian broadening of LLs. Moreover, LC description does not involve e-e interaction as shown in Appendix B.

We can suggest that the em^M Majorana anyons revealed are the CFs of the quasi-particle description of the FQHE and they naturally explain the zero internal field of CFs, leading to a termination of the conductance plateaus at even fractional filling factors. The description of CFs using LC implies B-em-C consisting of e^1/2₀ and e^1/2₋₁(0) ems, in which the latter contribute to the conductivity. The S-em-C of such type can explain the ½ and ¼ fractional quantization of the conductivity of the holes and electrons in Ge [46] and GaAs [47] nanowires, respectively, in a zero magnetic field and the “0.25 anomaly” of the conductivity of a point quantum contact [48,49].

The signatures of em^Ms in these experiments using confined geometries, i.e., probing of submicron areas, imply that the corresponding PFs are naturally formed in 2D-e structures and that their short- versus long-range order arrangement determine the formation the corresponding LCs and FQHE conductivity plateaus. Thus, in the extremely pure, high mobility structures in which FQHE plateaus emerge, the imperfection density is in a dilute regime (~1%), which provides a wide spectrum and long-range order of the PFs [43], induced by S- and B-em-Cs. The PFs depth in 2D-e-structures is ~10 meV [50], which is an order of magnitude smaller than in InP/GaInP₂ QDs, and we can expect that the FQHE LC states will include em-Cs having size ~50 nm, adopting only s- and p-ems, and assume extended imperfections having a size of at least a few nanometers.

4.3. Majorana Anyons and Topological Quantum Computing

The proposal of the topologically protected qubit involving the ν = 5/2 state [31] is based on Laughlin’s quasi-particle description of FQHE states [14], in which this state is assigned to a non-Abelian topological phase characterized by a Pfaffian wave function [51,52]. This state is considered as a set of degenerate p-wave paired fermions having charge e₀/4. The qubit is formed by a zero-energy Majorana excitation of a fused fermion having charge e₀/2. We can suggest that the em analog of the e₀/4 fermion is e²₀ + e^1/8₋₁(2) B-em-C, the e₀/2 state is e²₀ + e^1/8₋₁(2) + e^1/8₁(2) B-em-C, formed by addition of two es, and MM is e²₀ + e^1/8₋₁(2) + e^1/8₁(0) B-em-C. It follows from our results that it may be possible to realize and measure such B-em-Cs in InP/GaInP₂ using appropriate N and D, and to check the formation of the corresponding LC states in 2D-e structures using high-spatial-resolution optical [16] and electrometrical [53] measurements. Moreover, it may be possible to realize similar states using S-ems, thus creating magnetic-field-free TQCs.

5. Conclusions

We have used the magneto-PL spectra of InP/GaInP₂ QDs having seven electrons and a Wigner–Seitz radius ~1.3 to demonstrate the self-formation of fractionally charged anion(magneto-electron) composites having ν~5/2 and Majorana modes/anyons. We observed the destruction of these composites and the appearance of anti-em composites induced by a magnetic field. Using a theoretical description of magneto-PL spectra and an analysis of the electronic structure of these dots based on single-particle Fock–Darwin spectrum and many-particle configuration interaction calculations, we show the formation of fractionally charged anion states based on the mechanism in which non-interacting quantum confined electrons generate sub-electron persistent current loops (vortexes). This implies the liquid-crystal nature of the fractional quantum Hall effect states, in which Majorana anyons are spontaneously formed, and opens new perspectives for the realization of topological quantum computing.