Observation of a Biexciton Wigner Molecule by Fractional Optical

Nano Lett. , 2016, 16 (1), pp 27–33. DOI: 10.1021/acs.nanolett.5b02419. Publication Date (Web): December 9, 2015. Copyright © 2015 American Chemica...
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Observation of a Biexciton Wigner Molecule by Fractional Optical Aharonov-Bohm Oscillations in a Single Quantum Ring Hee Dae Kim,†,‡,§ Rin Okuyama,∥ Kwangseuk Kyhm,*,†,‡,⊥ Mikio Eto,∥ Robert A. Taylor,*,§ Aurelien L. Nicolet,# Marek Potemski,# Gilles Nogues,⊥ Le Si Dang,⊥ Ku-Chul Je,∇ Jongsu Kim,○ Ji-Hoon Kyhm,◆ Kyu Hyoek Yoen,◆ Eun Hye Lee,◆ Jun Young Kim,◆ Il Ki Han,◆ Wonjun Choi,◆ and Jindong Song◆ †

Department of Opto and Cognomechatronics, and ‡Department of Physics Education, Pusan National University, Busan 609-735, South Korea § Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom ∥ Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan ⊥ Department of NANOscience, Institut Néel, CNRS, rue des Martyrs 38054, Grenoble, France # Laboratoire National des Champs Magnetiques Intenses, CNRS-UJF-UPS-INSA, F-38042, Grenoble, France ∇ College of Liberal Arts and Sciences, Anyang University, Gyeonggi-do 430-714, South Korea ○ Department of Physics, Yeungnam University, Gyeonsan 712-749, South Korea ◆ Center for Optoelectronic Convergence Systems, KIST, Seoul, 136-791, South Korea S Supporting Information *

ABSTRACT: The Aharonov-Bohm effect in ring structures in the presence of electronic correlation and disorder is an open issue. We report novel oscillations of a strongly correlated exciton pair, similar to a Wigner molecule, in a single nanoquantum ring, where the emission energy changes abruptly at the transition magnetic field with a fractional oscillation period compared to that of the exciton, a so-called fractional optical Aharonov-Bohm oscillation. We have also observed modulated optical AharonovBohm oscillations of an electron−hole pair and an anticrossing of the photoluminescence spectrum at the transition magnetic field, which are associated with disorder effects such as localization, built-in electric field, and impurities. KEYWORDS: Optical Aharonov-Bohm effect, quantum ring, exciton, biexciton, Wigner molecule, disorder effect

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ecently, advanced droplet epitaxy1−3 has enabled an alternative nanoquantum ring (NQR) structure to be grown, where both electrical and optical Aharonov-Bohm (AB) effects are accessible at tens of Kelvin,4−6 while mesoscopic scale ring structures (≤1 μm) require extremely low temperatures (∼100 mK).7−10 Also, the AB effect in disordered systems is of fundamental importance; a persistent current (PC) in a normal metal is known to flow against the finite resistance produced by elastic scattering with impurities,10 where attractive electron−electron interactions might play a crucial role in the large PC and diamagnetic response.7,8,10 When a number of confined electrons are strongly correlated, N electrons in a ring structure behave as a composite single particle, called a Wigner molecule (WM).11−14 As a result, the © 2015 American Chemical Society

AB oscillation period of an N-electron WM with applied magnetic field (B) becomes fractional (ΔBNe = ΔBe/N) compared to that of a single electron (ΔBe).15−19 An optical analogy of fractional optical AB oscillations is possible if a multiexciton composite becomes a WM in a NQR under some particular conditions.18 However, this theoretical conjecture has remained unconfirmed experimentally until now. In this report we have observed fractional optical AB oscillations of biexcitons (XXs) and disorder effects associated with the modulated Received: June 18, 2015 Revised: October 30, 2015 Published: December 9, 2015 27

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Nano Letters exciton (X) AB effect in a single NQR, such as anisotropic structure,6 localization,6,20−22 built-in electric field,23 and impurity scattering.24,25 As excitons are charge neutral, the exciton AB effect in a ring structure requires a difference between the phases (Δϕ) acquired by the electron and the hole when a magnetic field threads the ring. This means that their orbits should be unequal, whereby a difference between the magnetic fluxes (ΔΦ) penetrating their paths results in the phase of a radially polarized exciton (Δϕ = eΔΦ/ℏ). Therefore, Δϕ is proportional to the radial dipole moment and can be measured through the photon emitted by the electron−hole pair, that is, the center-of-mass exciton energy is given by EX = ℏ2(L + Δϕ/ 2π)2/(2MR2X), where L, M, and RX are the angular momentum, the effective mass, and the orbit radius, respectively. Therefore, when an exciton is confined to a ring structure, an oscillation in the emission energy of the exciton as a function of magnetic flux can be observed. While the AB effect is generally restricted to ideal isotropic structures, an AFM image taken of an uncapped GaAs NQR is volcano-like, that is, the rim height is not constant around the azimuthal angle. In this anisotropic structure, the wave function is likely to be localized and have a crescent-like shape,6,20,22 and the coherence around the whole circumference is inhibited. However, when B is strong enough to overcome the potential barrier between the two separate localized states in a NQR, AB oscillations can emerge in the anisotropic NQR beyond a characteristic magnetic field (Bc).6 Therefore, Bc for a NQR can be a criterion for the onset of AB oscillations in the presence of structural disorder. For a selected single NQR, the magneto-PL spectrum of the Xs for an excitation intensity of Pex = 0.7 kWcm−2 is plotted in (Figure 1b) from Bc ∼ 2 T, below which the Zeeman splitting is not significant enough to be resolved. When the excitation becomes stronger by an order of magnitude (10Pex), XXs also emerge (Figure 1c and Supporting Information). All magnetoPL spectra (with ∼30 μeV spectral resolution) were carefully fitted by using Gaussian functions with a fitting accuracy of ∼2.5 μeV in order to obtain the central emission energy of X and XX as a function of B (Figure 2a). In the absence of B, we also measured an energy difference (ΔX ∼ 0.13 meV) in the X photoluminescence (PL) for perpendicular polarizations to evaluate the anisotropy of the localized states in a NQR (Figure 1d).22 With an excitation of 10Pex, a blueshift is observed as well as a decrease in ΔX (from ∼0.13 to ∼0.09 meV) (Figure 1e). These results are possibly associated with screening of the built-in lateral electric field (Elat).23 Because a GaAs/AlGaAs NQR is a lattice-matched system, Elat is possibly generated by defects at the interface between the GaAs and AlGaAs such as Ga-antisite and As-vacancies, where trapped carriers may cause Elat, and a large number of optically injected carriers in the AlGaAs can suppress Elat by screening. In addition, the energy difference of the XX emission energy for perpendicular polarizations (ΔXX ∼ 0.13 meV) is different from that of the excitons (ΔX ∼ 0.09 meV) (Figure 1e). In the case of an elliptical quantum dot (QD), an asymmetric electron−hole exchange interaction causes a splitting (ΔX) of the spin-degenerate exciton states into two singlet states, where two linearly orthogonally polarized dipoles are defined along the symmetric axes of the QD. The selective transition from the polarized XX and X states in an elliptical QD also gives rise to an identical splitting in the XX PL spectrum (ΔXX = ΔX). Therefore, the differences seen in a NQR (ΔXX ≠ ΔX) can be

Figure 1. Magneto-PL spectrum of a single quantum ring. (a) AFM image of a nanoquantum ring. Magneto-PL spectrum with an excitation intensity of Pex = 0.7 kWcm−2 (b) and 10Pex (c), where the energy difference for perpendicular polarizations was also measured separately without B for both Xs (ΔX) and XXs (ΔXX), respectively (d,e).

attributed to a selection rule change in the anisotropic structure.21,22 Given the inherent structural anisotropy and Elat in a NQR, the AB oscillations can be modulated by morphological control, excitation intensity, and an external electric field. In this case, disorder effects in NQRs are of great potential for quantum coherence control. As shown in Figure 2a, optical AB oscillations of the X energy with B are significant for both weak (Pex) and strong (10Pex) excitation intensity, and both these cases have the same period (ΔBX ∼ 1.8 T). In order to predict the period of X AB oscillations theoretically, we used a quasi-one-dimensional potential Ve,h(r, Re,h) for the electron and hole of a NQR, which is anharmonic and axially symmetric with a centrifugal core (Supporting Information). Here the quasi-one-dimensional model implies the width (We,h) of Ve,h(r) is still smaller than the orbit size (Re,h). It is also known that the electron and hole are likely to rotate along different orbits (Re ≠ Rh) due to their different effective masses, deformation potential, and strain,5,23,26−28 whereby the phase 28

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Figure 2. Optical AB oscillations of the exciton in a single NQR. (a) Central energy of the X emission for weak (Pex = 0.7 kWcm−2) and strong (10Pex) excitation intensity is plotted from 2 to 11 T. When B ≤ 2 T, the spectral splitting is too small to distinguish between the peaks X (σ+) and X (σ−) . (b) Theoretical X energy change (ELX(B) − ELX(B = 0)) of the various orbital angular momenta (L = 0,−1,−2,−3,...) with B for a quasi-onedimensional isotropic quantum ring model, where the many-body correlation is fully taken into account for the Coulomb interaction. (c) Theory and experiment are compared after removing the quadratic fitting term, where the period of exciton AB oscillations (ΔBX ∼ 1.8 T) is the same but a magnetic field difference in the oscillation extremum (δBX ∼ 0.7 T for Pex and δBX ∼ 0.4 T for 10Pex) is observed. (d) When the strong Coulomb interaction between the electron and hole is considered, the X can also be approximated as a single particle (θe ≈ θh) with its orbit at RX.

After removing the fitted quadratic functions in Figure 2a,b, which are determined by the exciton g-factor (gX ∼ −1.3) and the diamagnetic coefficient (γX ∼ 10 μeV/T2) of the exciton, the experimental data are compared with theory (Figure 2c). The oscillation maxima and minima for weak (Pex) and strong (10Pex) excitation occur at different magnetic fields although all show the same ΔBX ∼ 1.8 T. This result can be associated with Bc, that is, as the AB effect emerges at finite Bc, the first transition from L = 0 to L = 1 also occurs at a relatively large magnetic field compared to the case of ideal theory. For a weak excitation Pex, a magnetic field difference for the oscillation extremum between theory and experiment (δBX ∼ 0.7 T) is obtained (Figure 2c). On the other hand, when the excitation is 10Pex, a small δBX ∼ 0.4 T is obtained. Recently, changes in the sequence of maxima and minima of the X PL intensity oscillations with B were claimed to be due to Elat and thermal effects.23 Also, the presence of Bc was shown experimentally in a volcano-like anisotropic NQR.6 Therefore, the so-called modulated X AB oscillations in Figure 2c possibly result from a combined effect of Bc, Elat, and thermal phonons in an anisotropic NQR. Even though the wave function is delocalized around the anisotropic rim, the rotational symmetry of a NQR most likely breaks down due to the scattering potential arising from anisotropy and localized impurities. In this case, the orbital angular momentum L is not a good quantum number, and the coupling between different L states are involved. As a result, an

difference (Δϕ) between the electron and hole as a function of B controls the X AB oscillations.5,24,25 When the Coulomb interaction between the confined electron and hole in Ve,h(r, Re,h, We,h) was considered, we obtained the X eigenstates of different orbital angular momentum (L) as a function of B by using an exact diagonalization method (Supporting Information), where the many-body states are expanded by a large number of the singleparticle states. We also found that the exciton can be approximated as a single composite particle (θe ≈ θh) with its orbit at RX and angular momentum L = le + lh, where le and lh are constituent orbital angular momenta for the electron and the hole (Figure 2d and Supporting Information). Recently, it was found that the ratio of exciton orbit-to-confinement width (RX/We,h) is crucial in determining the amplitude of the excitonic AB oscillations,29,30 whereby the emergence of the X AB oscillations occurs under limited conditions (1 ≤ RX/We,h ≤ 10). We found optimum values for the different orbits (Re = 32 nm, Rh = 15 nm) and the rim width (We = 5 nm, Wh = 5 nm) for the electron and the hole empirically to obtain the measured AB oscillation period (ΔBX ∼ 1.8 T), which is also the case for the limiting condition (RX/We,h ∼ 4.7) of the AB oscillation. Figure 2a shows an average theoretical energy between the two X states for spins parallel (X(σ+)) and antiparallel (X(σ−)) to B, but it should be noted that we have ignored disorder effects in the calculation. 29

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Figure 3. The anticrossing splitting in the presence of disorder. Provided that the rotational symmetry of a NQR breaks down due to disorder, the anticrossing splitting, that is, the dip in the middle of the PL peak, is significant when L changes but eventually disappears when B is strong enough to overcome the disorder effect for weak (Pex = 0.7 kWcm−2) (a) and strong (10Pex) (b) excitation intensity.

is rigidly fixed due to correlation effects. In order to explain the fractional period, we have calculated the theoretical eigenenergy levels of the XXs (ELXX) and Xs (ELX) for various Ls in a quasione-dimensional NQR as a function of B (Figure 4c,d), where the Coulomb interactions between the two electrons and two holes are fully taken into account (Supporting Information). Note that the XX PL results from a transition from ELXX and ELX, but the energy oscillation periods of ELXX and ELX are different. For example, at the (i)-transition marked in Figure 4c,d, both the XX and X have the same orbital angular momentum L = 0. On the other hand, at the (iii)-transition, the XX has already changed its orbital angular momentum from L = 0 to L = −1 to minimize its energy. However, emission occurs between EL=−1 XX and EL=−1 due to a selection rule. In other words, emission X L=0 L=−1 and EL=0 . between EL=−1 XX X is not allowed although EX < EX Consequently, while B increases, an abrupt change of the PL energy can be seen near the (ii)-transition. More specifically, two PL peaks are expected at the (ii) transition, that is, EL=0 XX − L=−1 L=−1 , and the energy separation between the EL=0 X and EXX − EX two peaks is also expected to increase with B. Additionally, when considering disorder an anticrossing splitting is also expected for the XX. Interestingly, no splitting was found in the XX PL spectrum at any B. If a splitting of the XX PL spectrum were present, a fast intrarelaxation between the two states could be one explanation, whereby only the low emission energy

anticrossing splitting of the PL spectrum appears at the transition B-field of the AB oscillations, where the exciton changes L to minimize its energy.24,25 As evidence of the anticrossing splitting, a dip in the middle of the Gaussian PL peak was observed up to the transition from L = −2 to L = −3 for both weak and strong excitation (Figure 3a,b). For the case of impurity disorder, a δ-function scattering potential (Uimp e,h δ(θe−θ0i )) can be used, where θ0i is a fixed angle of the impurity localized at the electron/hole orbit.24 In comparison to the measured anticrossing (60−120 μeV), we estimate ∼110 μeV 24 for the splitting with Uimp e,h ∼ 50 μeV by using recent theory. However, a dip is not seen at the transition from L = −3 to L = −4 for both weak and strong excitation. This result suggests that the rotational symmetry recovers with a good quantum number L when a strong magnetic field overcomes disorder effects. As the electronic correlation effect is enhanced in a quasione-dimensional NQR, an electronic WM can be formed. For optical experiments, an XX in a NQR is a good candidate for an excitonic WM. In Figure 4a, the central emission energies of the X and XX with B are compared, where both the X and XX show an oscillatory behavior, although the AB period of the XX (ΔBXX ∼ 0.5 T) is shorter than that of the X (ΔBX ∼ 1.8 T). We claim that the fractional AB oscillation arises from the XX WM in a NQR, where the relative position of the two excitons 30

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Figure 4. The AB oscillation period of XXs in a single NQR is fractional compared to that of Xs. (a) Central emission energy of the X and XX for strong (10Pex) excitation intensity is plotted from 2 to 11 T. XX in a NQR can be considered as a pair of the interacting X dipoles, where the dipole− dipole interaction becomes minimized when the both are located at opposite positions as shown schematically. (b) Sawlike AB oscillations arising from the emission between ELXX(B) and ELX(B) in experiment and theory are compared after removing the quadratic fitted functions. (c,d) Theoretical energy change of the XX (ELXX(B) − ELXX(B = 0)) and X (ELX(B) − ELX(B = 0)) for the various orbital angular momenta (L = 0,−1,−2,−3,...) with increasing B.

Therefore, the description of a XX in a NQR can be simplified as a pair of the interacting X dipoles as shown schematically in Figure 4a, where the two Xs should be maximally separated to minimize the dipole−dipole interaction. Note also that ΔBXX ∼ 0.5 T is not half of ΔBX ∼ 1.8 T. In order to obtain ΔBXX ∼ 0.5 T, we found Re,h and We,h of Ve,h(r) for a XX should be 1.4 times those for an X, that is, both the orbit and width of the confinement potential for XX increase. The larger orbit for XX is due to the electron−electron and hole−hole repulsion in a NQR with redistribution of the surrounding electrons and holes, whereby the confinement potential of XX becomes different from Ve,h(r) determined by the geometric structure. Therefore, a modified Ve,h(r) for XX can be an effective correction, although the quantum Monte Carlo method would be the most suitable technique for accurate calculation. Partial Wigner localization can also be a candidate13 but the exact reason remains an open question. In conclusion, we found disorder effects are associated with the optical AB effect of a NQR. While the AB period of the X PL spectrum is determined by a difference between the magnetic fluxes penetrating the separate orbits of the electron and hole, the AB oscillations become modulated by structure anisotropy and excitation intensity, where the degree of localization and screening of the built-in electric field are determined. As the rotational symmetry breaks down in an NQR, we also observed an anticrossing splitting of the X PL spectrum at the transition magnetic field, where different Lstates cross over. Furthermore, we observed the AB oscillation period of the XX PL spectrum becomes fractional compared to

would be dominant at the (ii)-transition. Alternatively, the XX WM is possibly less susceptible to anisotropic disorder when it consists of interacting Xs located at opposite positions in the NQR (Figure 4a). Nevertheless, the exact reason is not clear at the moment. Ultrafast spectroscopy at the transition magnetic field could be employed to clarify this issue. In Figure 4b, ΔBXX ∼ 0.5 T was reproduced using the quasione-dimensional NQR model and compared to the experimental data after removing the fitted quadratic terms from Figure 4a,c,d, whereby the diamagnetic coefficient of the XX (γXX ∼ 12.4 μeV/T2) was also obtained. Because of the abrupt change of the XX PL energy near the transition magnetic field, sawlike AB oscillations are seen. Also, the magnetic-field difference of the XX oscillation extremum between theory and experiment is the same as that of the modulated X AB oscillations (δBXX ≈ δBX ∼ 0.4 T) due to structural anisotropy. When a XX becomes a WM in a NQR, two cases are possible; either two Xs form a WM or electrons and holes form a WM independently. In order to better understand this issue, correlation effects associated with the formation of a WM can be visualized in terms of a two-body charge density, which is the probability to find either an electron (ρee) or a hole (ρhe) at the position of a fixed electron (Supporting Information). Figure 5a shows that ρhe for X is localized due to the strong Coulomb interaction. This result verifies that the exciton can be approximated as a single particle (Figure 2d). On the other hand, the localized distribution of ρee and ρhe for XX (Figure 5b,c) suggests that a XX WM in a NQR is the former case, where the pair of Xs are strongly localized opposite each other. 31

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Aharonov-Bohm effect, and approximation into a simple one-dimensional model for Xs and XXs. (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Fax: +82-(0)51-583-1623. *E-mail: [email protected]. Fax: +44-(0)1865-272400. Author Contributions

H.D.K. and R.O. contributed equally. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the European Commission from the seventh framework programme “Transnational Access”, contract no. 228043-EuroMagNet II-Integrated Activities, Korean Grant (NRF-2014R1A1A2058789, GRL program), and French-Korean LIA.

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Figure 5. Two-body charge density of a NQR for B = 0. The probability distribution of a hole with respect to the electron fixed at the marked position for an X (a) and the probability distribution of another electron (b) and two holes (c) to a fixed electron for XXs localized with Wigner molecule character.

that of the X spectrum as a consequence of the XX WM in a NQR, where a pair of the interacting X dipoles are rigidly localized at opposite positions.

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REFERENCES

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b02419. Experimental method, identification of charged Xs and XXs in a single NQR, theoretical model with quasi-onedimensional confinement potential for the optical 32

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Nano Letters (21) Kim, H. D.; Kyhm, K.; Taylor, R. A.; Nogues, G.; Je, K. C.; Lee, E. H.; Song, J. D. Appl. Phys. Lett. 2013, 102, 033112. (22) Kim, H. D.; Kyhm, K.; Taylor, R. A.; Nicolet, A. A. L.; Potemski, M.; Nogues, G.; Je, K. C.; Lee, E. H.; Song, J. D. Appl. Phys. Lett. 2013, 103, 173106. (23) Teodoro, M. D.; Campo, V. L.; Lopez-Richard, J. V.; Marega, E.; Marques, J. G. E.; Gobato, Y. G.; Iikawa, F.; Brasil, M. J. S. P.; AbuWaar, Z. Y.; Dorogan, V. G.; Mazur, Y. I.; Benamara, M.; Salamo, G. J. Phys. Rev. Lett. 2010, 104, 086401. (24) Dias da Silva, L. G. G. V.; Ulloa, S. E.; Govorov, A. O. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 70, 155318. (25) Dias da Silva, L. G. G. V.; Ulloa, S. E.; Shahbazyan, T. V. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 72, 125327. (26) Maslov, A. V.; Citrin, D. S. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 67, 121304(R). (27) Barker, J. A.; Warburton, R. J.; O’Reilly, E. P. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 69, 035327. (28) Arsoski, V. V.; Tadic, M. Z.; Peeters, F. M. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 085314. (29) Gonzalez-Santander, C.; Dominguez-Adame, F.; Romer, R. A. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 235103. (30) Tadic, M.; Cukaric, N.; Arsoski, V.; Peeters, F. M. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 125307.

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