Magnetoluminescence from trion and biexciton in type-II quantum dot
© Okuyama et al; licensee Springer. 2011
Received: 15 August 2010
Accepted: 20 April 2011
Published: 20 April 2011
We theoretically investigate optical Aharonov-Bohm (AB) effects on trion and biexciton in the type-II semiconductor quantum dots, in which holes are localized near the center of the dot, and electrons are confined in a ring structure formed around the dot. Many-particle states are calculated numerically by the exact diagonalization method. Two electrons in trion and biexciton are strongly correlated to each other, forming a Wigner molecule. Since the relative motion of electrons are frozen, the Wigner molecule behaves as a composite particle whose mass and charges are twice those of an electron. As a result, the period of AB oscillation for trion and biexciton becomes h/2e as a function of magnetic flux penetrating the ring. We find that the magnetoluminescence spectra from trion and biexciton change discontinuously as the magnetic flux increases by h/2e.
PACS: 71.35.Ji, 73.21.-b, 73.21.La, 78.67.Hc
Rapid advance in nanotechnology has allowed us to fabricate ring structures whose circumference is shorter than the phase coherent length. In these systems, the persistent current induced by the Aharonov-Bohm (AB) effect was predicted theoretically , and observed both for metallic rings in the diffusive regime and semiconductor rings in the ballistic regime [2, 3]. In the semiconductor rings, the theory well explains the experimental results. In the metallic rings, however, the observed current was much larger than the theoretical prediction. This should be ascribed to the electron-electron interaction in the rings, which has not been fully understood.
In this study, we theoretically investigate the correlation effect when more than one electron is put in a type-II quantum dot. First, we calculate the many-electron states in the quasi-one-dimensional ring and find the formation of Wigner molecules . Since the relative motion of electrons is frozen due to the strong correlation, an N-electron molecule behaves as a composite particle whose charge and mass are N times of those of an electron. In consequence, the energy oscillates with Φ by the period of h/Ne. This is known as a fractional AB effect . Next, we examine the magnetoluminescence from trion and biexciton in the type-II quantum dot. We show that the peak position and intensity of the luminescence change discontinuously as Φ increases by h/2e. This indicates the possible observation of Wigner molecules by the optical experiment.
Model and calculation method
We consider a type-II semiconductor quantum dot formed in a plane. A ring-like potential is imposed on electrons, while a harmonic potential on holes. Here, m e and m h are the effective masses of electrons and holes, respectively. A magnetic field is applied perpendicularly to the quantum dot.
Parameters ω e, ω h, V 0, and α are chosen so that R, at which V e(r) has the minimum, is eight times larger than the size of hole confinement . The expectation value of the electron radius is approximately R in our model.
where ψ e(r) [ψ h(r)] is an envelope function for an electron [hole] state, and u c(r) [u v(r)] is the Bloch function of the conduction [valence] band edge. u c(r) and u v(r) mainly consist of s- and p-waves, respectively. Since they oscillate in space by the period of the lattice constant, a, the exchange interaction between electron and hole is smaller by the order of (a/R)2 than other terms, e.g., the exchange interaction between two electrons.
The strength of the magnetic field is measured by Φ = πR 2 B, the flux penetrating the ring of radius R. The strength of the Coulomb potential against the kinetic energy increases with R/a B, where a B = 4πϵħ 2/m e e 2 is the effective Bohr radius. R/a B ≳ 1 in the experimental situations [4, 5].
The exact diagonalization method is used to take full account of the Coulomb interaction. We calculate the luminescence spectra by the dipole approximation, using obtained energies and wavefunctions of many-body states.
Results and discussion
Few electrons without hole
First, we calculate the electronic states in the absence of holes. Figure 1 shows Φ dependence of low-lying energies for (a) one, (b) two, and (c) three electrons confined in V e(r) with R/a B = 1. The total angular momentum L is indicated in the figure. For one electron, the angular momentum increases by one in the ground state Φ as increases by about h/e, and the energy oscillates quasi-periodically with Φ. by the period of h/e. This suggests that the electronic confinement V e(r) realizes a quasi-one-dimensional electron ring. In contrast to the perfect one-dimensional ring, on the other hand, a diamagnetic shift is seen in our model. As a whole, the energy increases with Φ. This is because the electron radius is shrunk by the magnetic field. For two and three electrons, the angular momentum increases, and the energy oscillates quasi-periodically with Φ in the ground state. The diamagnetic shift is also present. However, the period of AB oscillation becomes about h/Ne for N electrons.
Similarly, three electrons are localized at apices of an equilateral triangle inscribed in the ring to form a Wigner molecule. The period of AB oscillation in the ground-state energy becomes about h/3e for R/a B ≳ 1.
The total spin S of the ground state changes with L, as shown in Figures 1 and 2. For two electrons, S = 1 (S = 0) when L is even (odd). In the case of three electrons, S = 3/2 if L is a multiple of 3, S = 1/2 otherwise. This is explained by the N-fold rotational symmetry of the electron configuration in the Wigner molecule .
Electron-hole complex and optical spectrum
Next, we investigate electron-hole complexes: exciton, trion, and biexciton. We fix R/a B = 1. Since the hole motion is almost frozen due to the strong confinement in the quantum dot, the Φ dependence of the ground state of exciton is qualitatively the same as that of an electron confined in V e(r). In the same manner, the Φ dependence of the ground state of trion and biexciton mimics that of two electrons . In particular, two electrons in trion or biexciton form a Wigner molecule, and the period of AB oscillation in the ground-state energy becomes about h/2e as a function of Φ for trion and biexciton .
We have examined the optical AB effect on trion and biexciton in the type-II semiconductor quantum dots. We have found that two electrons in trion and biexciton form a Wigner molecule. As a result, the ground-state energy oscillates as a function of the magnetic flux by the period of about h/2e. We have shown that the luminescence spectra from them change discontinuously as the magnetic flux increases by about h/2e. This indicates the possible observation of Wigner molecules by the optical experiment.
We note that the discontinuous change in the luminescence peaks and intensity stems from the selection rule, which is broken in the presence of disorder. By the selection rule, excitons with the angular momentum L ≠ 0 should be dark. However, transitions from excitons with finite L were observed by experiments in both ZnSeTe and SiGe [4, 5]. Possibly, the sudden change of the luminescence spectra would be smeared in such systems. However, the fractional period of h/2e is a ground-state property and hence, it is expected to be observed even in dirty samples.
This work was partly supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science. R. O. was funded by Institutional Program for Young Researcher Oversea Visits from the Japan Society for the Promotion of Science.
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