Near-Field Scanning Magneto-Optical Spectroscopy of Wigner Molecules
Few-electrons moving in a weak confinement potential can be localized and spatially ordered (crystallized) forming, so-called, electronic or Wigner molecule (WM). The possibility of such crystallization in infinite systems was predicted by Wigner in 1934  and it occurs when the energy of Coulomb interaction between the electrons exceeds their kinetic energy. Theory predicts that for finite, quantum confined systems a transition from droplet to molecular phase is gradual with increase of the Coulomb to kinetic energy ratio, which is described by Wigner-Seitz radius rs=1/[a*B(p·n)0.5], where n is electron density and a*B is Bohr radius. Theory also predicts, that spatial ordering starts develop at rs>2, and full localization in two-dimensional parabolic potential is realized at rs>100, when charged particles can be considered as point classical charges . Most of the experiments were performed for two electron (2e) WMs structures in epitaxial InSb nanowires , ultra-clean carbon nanotubes  and nano-fabricated GaAs/AlGaAs quantum dots (QDs) defined laterally  or vertically [6,7]. These systems were measured mainly using single-electron tunneling spectroscopy and provide parabolic confinement energy ħw0=7.8-1.6 meV and rs ~2 near the onset of the WM formation.
Here we study the formation of 2e-WMs in emission spectra of self-organized InP/GaInP QDs providing rs up to 5. We used measurements of magnetic-field-induce shifts to distinguish emission of 2e-WM formed in photo-excited state from singly charged exciton (trion). We also observed magnetic-field-induced molecular-droplet transition for two electron dot, having doubly charge exiton (tetron) in photoexcited state at zero magnetic field.
The samples we grown by metal-organic chemical vapor deposition (MOCVD) in a horizontal AIX200/4 reactor under 100 mbar pressure at temperature 725 oC. The sample structure was as follows: 500 µm (100) GaAs substrate misoriented by 2o towards the  direction, 50 nm thick GaAs buffer layer, 50 nm Ga0.52In0.48P latticed matched to the GaAs, seven monolayers of InP to form the quantum dots, and finally 60 nm GaInP cap layer.
The quantum dot size, size distribution and composition were determined using an FEI Titan 80-300 transmission electron microscope (TEM). The sample preparation was done using focused ion beams.
Electron accumulation was detected by negative degree of circular polarization of emission of QD ensemble . The measurements were carried out under non-resonant continuous-wave excitation using 632.8 nm line of He-Ne. Samples were placed in closed cycle helium cryostat with temperature~10 K. Helicity of light in the excitation channel was alternated in sign at a frequency of 26 kHz, using photo-elastic quartz modulator. Quarter-wave plate followed by linear polarizer in the detection channel were used as a circular-polarization analyzer. Luminescence was dispersed by double monochromator and detected with photomultiplier tube. The degree of circular polarization , where I++,I-+ is intensity of σ+ luminescence under σ+, σ- excitation.
Single dot magneto-photoluminescence was measure using cryo near-field scanning optical microscope operating at 10 K and magnetic fields up to 10 T. We used tapered fiber probes coated by Al, having aperture size ~100 nm in collection-illumination mode . The spectra were excited by the 514.5 nm Ar-laser line and measured using a CCD detector together with a 280 mm focal length monochromator. The excitation power was ~5 mW, which provided power density ~0.5 W/cm2. The spectral resolution of the system is 0.2–0.4 meV. For near-field (NF) imaging we scan the tip across sample surface with the step 100 nm, measured spectra at each step and then generated emission intensity maps for given energy.
In the weak confinement regime charged excitonic complexes can be approximated as point-like particles and their single particle spectrum is described by Fock-Darwin (FD) Hamiltonian with corresponding parabolic frequency. The emission energy of trion (TR) is Etr=E0+ ħ[wtr-we(2k+1)], where E0 is free 2D TR energy, wtr(we) are parabolic frequency of trion(electron), k is radial number of electron left in final state. Since wtr<we, Etr(B)-Etr(0)<0, i.e. main PL peak (<0|0>-transition) has a paramagnetic shift, and, since wtrnot equal we, weak shake-up, <0|k>-transitions are activated . Similar is valid for tetron (TE). For WM the electrons in photoexcited state are shifted from the position of electron in ground by half of molecular bond length d0 and thus <k|0>-transitions is activated similar to that of stretching vibrations in diatomic molecules . The intensities of <k|0>-transitions can thus be evaluated by Frank-Condon factors Fk=exp(-S)*Sk/k!, were S is Stokes shift S= mew02(d0/2)2/2ħw0 and me is electron effective mass. The magnetic field shift of <0|0>-transition is [E2e,r.m.(B)+EC,2e(B)]+Eh(B)+EC,2eh(B),where E2e,r.m.(B)/ Eh(B) is FD energy of relative electron/hole motion and two rest terms are Coulomb interaction energies. The term in squire brackets was analyzed in  and represent difference between interacting and non-interacting two electrons. We calculated this terms and used it to estimate EC,2eh from experimental data.
For calculations of energy transitions of InP/GaInP QDs we used COMSOL Multiphysics 5.1 program and material parameters from Ref. 
Structural data versus near-field imaging and spectroscopy
The dot lateral size, D, measured using TEM is varied from 80 to 220 nm. Using energy dispersive X-ray analysis of cross section TEM specimens we reveal intermixing resulted in incorporation of ~20% Ga in QDs. The dots are lens shaped having heights h=5 and 20 nm. These heights were also observed by us using atomic force microscopy of uncapped dots and was previously reported in Refs [14-17]. In low-temperature ensemble PL spectra the dots having h=20 nm give contribution to the main peak centered at 1.72 eV, while dots having h=5 nm form high-energy tail centered at 1.79 eV. The difference between the emission energy of these dots (~70 meV) is reproduced well in calculations of optical transitions of the lens shape InP/GaInP QDs. The intermixing results in a higher band-gap of QD material, which is Ga0.2In0.8P, and shifts ensemble PL peak on ~60 meV. Negative degree of circular polarization was observed in the range 1.68-1.74 eV, which indicated electron accumulation of the dots having h=20 nm.
The spatially resolved near-field spectra we observed population of the dot by up to 20 electrons, as was reported by us previously . This follows from the observation of a multi peak structure consisting of s-, p- and d-peaks and related to electron occupation of corresponding electronic shells, and a low energy tail of s-peak, which is shake-up (Stokes) component, related to the modes of relative and center-of-mass motion of electrons. The s-p splitting of the dots was 1-6 meV and it is approximately equal to the energy spacing of the levels of parabolic potential ħw0. The some dots we see only Stokes component, which indicates that it contains a single electron. For these dots we observed a several <k|0>-transitions, nw0 which we assign to higher energy quantum confined levels.
The NF image of the emission intensity main peaks has lateral size at the base ~ 300 nm. We used these images to estimate the size of emission area of QD, which is ~50-150 nm and agree well with the calculations.
Wigner molecule versus charged exciton emission
We found that for single electron dots the number of shake-up <0|k>-transitions and distribution of their intensity dramatically depends on ħw0: only two with <0|0> dominant were observed for ħw0=4 meV and up to six with <0|2> dominant were observed for ħw0=1.2 meV. The relative intensity of the transition are described very well by Fk of vibronic model giving S values 0.5, 1.5 and 2.5 and d0=40, 110 and 140 nm for ħw0=4, 1.8 and 1.2 meV, respectively. The d0–values obtained corresponds to a dot size, which is unexpectedly, since calculations  give nearly two times smaller values.
We observed that in magnetic field a single elctron dot having ħw0=1.2 meV the energy of the s-peak increases with increasing of magnetic field, indicating that a stable 2e-WM is formed in photoexcited state of this dot.
For one dot we observed a single s-peak at zero field. For this dot s-peak shifts to lower energy up to 8T and at higher fields it starts shifted to higher energy. For this dot a Stokes component having energy 1.3 meV appears at 4 T and p-shell component having energy 2meV appears at 10 T. The measured low energy shift for B< 8 T is well reproduced by FD shift of doubly charged exciton (tetron). For higher field appearance of Stokes component and p-peaks indicated formation of Wigner molecule.
We also observed a mixed trion-WM behavior in which dot has two emission peaks one has negative, trion shif in magnetic field and the other positive, WM shiftt.
We study the emission spectra of single self-organized InP/GaInP QDs (size 100-220 nm) using high-spatial-resolution low-temperature (5K) near-field scanning optical microscope (NSOM) operating at magnetic field strength B=0-10T. The dots contain up to twenty electrons and represent natural Wigner molecules (WM). We observed vibronic-type shake-up structure in single electron QDs describing well by vibronic Frank-Condon factors and manifesting formation of two electron (2e) WM in photo-excited state. We used measurements of magnetic field shifts to distinguish emission of 2e-WM from singly charged exciton (trion). We also observed magnetic-field-induced molecular-droplet transition for two electron dot, having doubly charge exiton (tetron) in photoexcited state at zero field.
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