Abstract
We theoretically investigate optical AharonovBohm (AB) effects on trion and biexciton in the typeII 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. Manyparticle 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
Introduction
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 AharonovBohm (AB) effect was predicted theoretically [1], 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 electronelectron interaction in the rings, which has not been fully understood.
In typeII semiconductor quantum dots, such as ZnSeTe and SiGe, holes are localized inside the quantum dots while electrons move in a ring structure formed around the dots (inset in Figure 1a). In a perpendicular magnetic field B, the electrons acquire the AB phase. For the sake of simplicity, suppose that an electron moves in a perfect onedimensional ring of radius R, the Hamiltonian is written as
where is the angular momentum operator, m_{e }is the effective mass of electron, and Φ= π R^{2}B is the magnetic flux penetrating the ring. As a result, the angular momentum increases with in the ground state, and the energy oscillates as a function of Φ by the period of h/e [1]. This AB effect was observed experimentally as the B dependence of peak position of luminescence from excitons [4,5], in which the hole motion is almost frozen due to the strong confinement [6]. This is called an optical AB effect.
Figure 1. Lowlying energies for (a) one, (b) two, and (c) three electrons in the typeII quantum dot, as a function of the magnetic flux Φ. The dot radius R equals to the effective Bohr radius a_{B }= 4πϵħ^{2}/m_{e}e^{2}. Solid and dash lines indicate spinsinglet and triplet, respectively, in (b), whereas they indicate quartet and doublet in (c). The period of AB oscillation becomes h/Ne for N electrons.
In this study, we theoretically investigate the correlation effect when more than one electron is put in a typeII quantum dot. First, we calculate the manyelectron states in the quasionedimensional ring and find the formation of Wigner molecules [7]. Since the relative motion of electrons is frozen due to the strong correlation, an Nelectron 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 [8]. Next, we examine the magnetoluminescence from trion and biexciton in the typeII 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 typeII semiconductor quantum dot formed in a plane. A ringlike 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.
Using the field operator for electron, , and for hole, , the effective mass Hamiltonian is written as
where q_{e }=  e, q_{h }= e, and A(r) is the vector potential; ∇ × A =  Be_{z}. Note that the exchange interaction between electron and hole is omitted here for the following reason. An electron [hole] wave function Ψ_{e}(r) [Ψ_{h}(r)] is written as
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 pwaves, 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 manybody 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 lowlying 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 quasiperiodically with Φ. by the period of h/e. This suggests that the electronic confinement V_{e}(r) realizes a quasionedimensional electron ring. In contrast to the perfect onedimensional 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 quasiperiodically 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.
In order to elucidate the relation between the electronelectron interaction and the fractional period of AB oscillation, we examine manybody states for two electrons with changing R/a_{B}. Figure 2 shows lowlying energies with (a) R = a_{B }= 0.01, (b) 0.1, (c) 1, and (d) 10. Without the Coulomb interaction, two electrons occupy the lowest orbital shown in Figure 1a in the ground state. Consequently, the total angular momentum is always even, and the total spin is a singlet. As the strength of the Coulomb interaction increases with R/a_{B}, the exchange interaction lowers energies for spintriplet states compared to singlet states. For R/a_{B }≳ 1, singlet and triplet states alternatively appear as Φ increases by about h/2e. Hence, the groundstate energy oscillates with Φ by the period of h/2e. Note that the period of AB oscillation in the case of R/a_{B }= 10 is slightly shorter than that of R/a_{B }= 1. This is because the Coulomb repulsion between electrons tends to increase the expectation value of the electron radius.
Figure 2. Lowlying energies for two electrons in the typeII quantum dot, as a function of the magnetic flux Φ. Solid and dash lines indicate spinsinglet and triplet, respectively. The ratio of the dot radius R to the effective Bohr radius a_{B }= 4πϵħ^{2}/m_{e}e^{2 }is (a) 0.01, (b) 0.1, (c) 1, and (d) 10. The groundstate energy oscillates by the period of h/2e for R/a_{B }≳ 1.
We calculate the twobody density
to examine the electric correlation. Figure 3 shows the twobody density for the twoelectron ground state at zero magnetic field with (a) R/a_{B }= 0.01, (b) 0.1, (c) 1, and (d) 10. r_{0 }is fixed at (R, 0), which is indicated by a circle in the plots. For R/a_{B }≳ 1, electrons maximize their distance to be localized at the other side in the ring, that is, a Wigner molecule is formed. Since the relative motion of electrons is frozen, the Wigner molecule behaves as a composite particle whose mass and charge are twice those of an electron. In consequence the groundstate energy oscillates with Φ by the period of about h/2e.
Figure 3. Gray scale plots of the twobody density for the twoelectron ground state at zero magnetic field with (a) R/a_{B }= 0.01, (b) 0.1, (c) 1, and (d) 10. One electron is fixed at the point indicated by a circle.
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 groundstate 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 Nfold rotational symmetry of the electron configuration in the Wigner molecule [9].
Electronhole complex and optical spectrum
Next, we investigate electronhole 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 [10]. In particular, two electrons in trion or biexciton form a Wigner molecule, and the period of AB oscillation in the groundstate energy becomes about h/2e as a function of Φ for trion and biexciton [10].
We examine recombination phenomena. Figure 4 shows the Φ dependence of the luminescence peak from (a) exciton and (b) trion. The behavior of the biexciton peak is qualitatively the same as in Figure 4b. The exciton peak oscillates by the period of about h/e. On the other hand, the trion peak increases with an increase in Φ and suddenly drops by the period of about h/2e. The fractional period of h/2e comes from the period of AB oscillation in the ground state of trion. The discontinuous change is explained by a selection rule for the recombination: The optical transition conserves the orbital angular momentum in twodimensional systems [11]. A trion with the angular momentum L has to decay into an electron with the same angular momentum. As a result, both of the initial and final states of the recombination change at the transition of the trion state. In the case of exciton recombination, the final state is always the vacuum state of the quantum dot, and the peak position is continuous as a function of Φ, as seen in Figure 4a (The recombination of exciton with the angular momentum L ≠ 0 is forbidden by a selection rule. After the first transition of the electronic state at Φ ≄ h/2e, excitons get dark in our model. However the forbidden transitions were observed in experiments. This should be ascribed to the disorder of samples which breaks the selection rule.)
Figure 4. The luminescence peaks from (a) exciton and (b) trion in the typeII semiconductor quantum dot, as a function of the magnetic flux Φ. The trion peak suddenly drops as Φ increases by h/2e.
Figure 5 shows the intensity of the trion peak as a function of Φ. The intensity of the biexciton peak is approximately the same. The intensity decreases discontinuously at the transition of the electronic state, and approximately takes a constant value until the next transition occurs. Roughly speaking, the height of the intensity plateaus indicates a ratio of 4:3:1:0. The intensity reflects properties of the twoelectron wavefunction. This is in good agreement with our theory based on the HeitlerLondon approximation, in which the correlation effect between electrons is taken into account by a linear combination of two Slater determinants [10].
Figure 5. The intensity of the trion luminescence peak as a function of the magnetic flux Φ. Reflecting the twoelectron wave function, the intensity decreases discontinuously as Φ increases by h/2e.
Conclusions
We have examined the optical AB effect on trion and biexciton in the typeII semiconductor quantum dots. We have found that two electrons in trion and biexciton form a Wigner molecule. As a result, the groundstate 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 groundstate property and hence, it is expected to be observed even in dirty samples.
Abbreviations
AB: AharonovBohm.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
RO developed the numerical model, ran the simulation and acquired data. The interpretation of data has been carried out together with RO and ME. ME and HH conceived of the study and participated in its design and coordination.
Acknowledgements
This work was partly supported by a GrantinAid 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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