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Excitonic Aharonov-Bohm effect in a two-dimensional quantum ring

We study theoretically the optical properties of an exciton in a two-dimensional ring threaded by a magnetic flux. We model the quantum ring by a confining potential that can be continuously tuned from strictly one-dimensional to truly two-dimensional with finite radius-to-width ratio. We present an analytic solution of the problem when the electron-hole interaction is short-ranged. The oscillatory dependence of the oscillator strength as a function of the magnetic flux is attributed to the Aharonov-Bohm effect. The amplitude of the oscillations changes upon increasing the width of the quantum ring. We find that the Aharonov-Bohm oscillations of the ground state of the exciton decrease with increasing the width, but remarkably the amplitude remains finite down to radius-to-width ratios less than unity. We attribute this resilience of the excitonic oscillations to the non-simply connectedness of our chosen confinement potential.


I. INTRODUCTION
Recent advances in nanofabrication of quantum rings and dots by self-assembling, 1-5 lithographic 6,7 or etching techniques 8 have opened an active area of research both theoretical and experimental. In such systems electrons and holes are confined in a small region and consequently the Coulomb interaction is enhanced. The existence of bound states of electron-hole pairs offers a unique opportunity to explore the Aharonov-Bohm (AB) effect [9][10][11][12] for excitons in quantum rings. 13,14 Despite the exciton being a neutral entity it has been predicted to be sensitive to a magnetic flux due to its finite size inside a quantum ring. 13,14 In experiments, this sensitivity would show as an oscillatory dependence of both the optical transition energy as well as the oscillator strength upon the magnetic flux. 4,7,8,[15][16][17] Theoretically, the excitonic AB effect has been studied by a variety of different approaches. A short-range interaction between the electron and the hole has been investigated for one-dimensional (1D) rings, 13,14,18,19 where also the effect of an external electric field can be included. 20 Intermediate models assume two-dimensional (2D) rings with narrow width under harmonic confinement and Coulomb-like interaction potentials between the electron and the hole, 21,22 or radial polarized excitons when electrons and holes move in different circles. [23][24][25] The excitonic AB effect in 2D rings has been studied in models with harmonic 26 and geometric 15,27,28 confining potential or using a 2D attractive annular Hubbard model. 29,30 In all cases, the excitonic AB effect for neutral excitons has been argued to be suppressed in 2D as the width of the ring is increased. 31 Recently, experimental results in molecular-beam epitaxy grown nanorings made by AsBr 3 8 etching and on selfassembled InAs/GaAs quantum dots 5 report oscillations in the binding energy of neutral excitons which may be accounted for by the excitonic AB effect.
In this paper, we consider the excitonic AB effect in a confining potential that can be continuously tuned from strictly 1D to truly 2D with finite radius-to-width ratio while preserving the central structure of a ring, namely, its non-simply connectedness due to an infinitely strong repulsion at the origin. 32 We present a simple analytic approach to the excitonic problem when the electron-hole attraction is short-ranged. 13,14 We then study how the amplitude of the AB oscillations in the oscillator strength changes upon increasing the width of the ring. We find that the AB oscillations of the exciton ground state energy decrease with increasing the width of the quantum ring, but nevertheless the effect remains noticeable down to regimes with radius-to-width ratios smaller than unity. This shows the robustness of the excitonic AB effect in 2D.

II. SINGLE PARTICLE STATES IN THE QUANTUM RING
In the absence of Coulomb interaction, the Hamiltonian of a single particle (electron or hole) subjected to a magnetic flux in a 2D quantum ring is given by where m i , p i and A are the effective mass, the momentum in the plane and magnetic vector potential, respectively. Here the subscript i = e, h refers to the electron and the hole, respectively. Electric charges are q e = −e and q h = e. The quantum ring is modeled by an anharmonic, axially symmetric potential with a centrifugal core 32,33 The confining potential has the minimum at |r i | = R (see Fig. 1) and, for this reason, R will be used as a convenient measure of the effective ring radius. Close to the minimum the potential reduces to the well-known displaced parabola, used in other theoretical studies of 2D quantum rings. 26 As r i → 0, we see from (2) that the centrifugal core assures the survival of the essential feature of a ring: its repulsive barrier in the center. The effective width W of the quantum ring can be estimated from the singleparticle ground state in the harmonic potential, namely W = (ǫ 0 /2V 0 ) 1/4 R, where ǫ 0 = 2 /2mR 2 is the ring-size quantization energy. 32 Notice that we assume that W is the same for electrons and holes, namely m e = m h ≡ m.
For the purpose of this work, all energies will be measured in units of ǫ 0 and we parametrize the strength of the confining potential by the radius-to-width ratio γ ≡ R/W = (2V 0 /ǫ 0 ) 1/4 . When γ → ∞, we approach the limit of a 1D ring, whereas γ → 0 corresponds to an anti-dot geometry. 32 Figure 1 shows the radial confining potential for different values of γ.
In order to study the AB effect in the quantum ring, we choose A ≡ (A r , A θ ) = (0, Φ h/e2πr), corresponding to an infinitely thin magnetic flux piercing the plane of the ring perpendicularly. Here Φ is the dimensionless flux through the ring and h/e the universal flux quantum. 34 We note that due to the axial symmetry aroud the ring axis, all our results for energies have to be periodic in Φ with period 1 and we hence restrict ourselves to the sector Φ ∈ [0, 1]. Then the Schrödinger equation for the electron in polar coordinates r e =(r e ,θ e ) is written in dimensionless form where M e = (n e , ℓ e ) represents the set of quantum numbers for the electron, which are n e = 0, 1, 2, . . . and ℓ e = 0, ±1 ±2, . . . For brevity we define the dimensionless energy λ Me = E Me /ǫ 0 and radial coordinate ρ e = r e /R. The Schrödinger equation for the hole is the same aside from a change in sign in the linear term on Φ, and with a set of quantum numbers M h = (n h , ℓ h ). The normalized eigenfunctions of Eq. (3) are given by 32 where k e = f 2 e + γ 4 /4 with f e = ℓ e − Φ defining an effective angular quantum number due to the confinement and the magnetic flux. L k n stands for the generalized Laguerre polynomials. The corresponding dimensionless energies are λ Me = γ 2 (2n e + 1 + k e ) − γ 4 /2. The eigenfunctions and energies for the hole are the same as for the electron, with a effective angular quantum number f h = ℓ h + Φ. The dimensionless zero point energy (n e = ℓ e = Φ = 0) for the electron is λ 0 e = γ 2 . Figure 2 shows the electron energy as a function of the parameter γ. We note that levels at higher values of the quantum number n e become increasingly uncoupled for γ > 5 and hence we expect to see nearly 1D behavior for γ values beyond this regime. 14 From the inset is clearly observed that in this 2D confinement regime, in the absence of interaction, the ground state energy for the electron (once sustracted the zero point energy) describes an oscillation with the magnetic flux. For γ > 5, the 2D oscillation is indistinguishable from the 1D case.

III. SOLUTION OF THE EXCITONIC CASE
Within the effective-mass approximation, the Hamiltonian of the interacting electron-hole pair is given by where H e−h is the interaction term. We model the excitonic interaction between the electron and the hole as a short-range potential of the form This contact interaction is the same used in Refs. 14 and 20 extended to a 2D case, where the area of the ring is 2πRW . In this definition we have carefully chosen the prefactors such that in the 1D limit, γ → ∞, the values of v 0 become identical to the corresponding 1D parameter and facilitate comparison with the results of Refs. 14 and 20. Thus, we express v 0 as −α/π 2 where α denotes the ratio of 1D excitonic Bohr radius to ring circumference. 14 Before continuing with the detailed study of the model, let us discuss some of the assumptions made and the limitations which we will encounter. Let us first emphasise that the restriction to equal electron and hole masses is simply a presentational convenience; all calculations shown here can easily be generalised to the case of unequal masses 20 but with a certain loss of clarity in the mathematical expressions. Nevertheless, we shall present some results for unequal masses later. The assumption of an infinitely thin current-carrying solenoid generating the magnetic flux Φ is a theoretical construct. The experiments cited in the introduction all use a magnetic field B to generate the required Φ. This results in an additional, diamagnetic term proportional to B 2 , which we ignore here similarly to the experimental papers. 4,7,8,[15][16][17] Certainly the most drastic assumption seems to be the δ-function potential for the two-particle interaction. Its use is of course motivated by our resulting ability to reduce the computational difficulties as we will show below. Nevertheless, we wish to emphasise that there are also certain conceptual advantages associated with it: (i) in 1D, the δ-function interacting many-particle problem has been solved exactly and hence the expression for the exciton binding energy on a line is known in terms of v 0 . 35 (ii) In Ref. 14, it was shown how the Bohr radius of the exciton similarly depends on v 0 . Both these parameters will of course vary when another form of interaction is considered. However, as also shown in Ref. 14, it is the ratio α introduced above which governs the strength of the AB oscillations. The effect of other two-particle interaction potentials along the ring, when expressed in terms of α, will lead to similar AB oscillations and we expect at least qualitative agreement. Even for a long-range potential such as the Coulomb interaction, we expect this to hold as long as the overlap of wave packets on opposite sides of the ring, i.e. across the origin at r = 0, can be neglected. For the confining potential considered here, with its strong centrifugal core, this should be a rather good approximation.
We construct the exciton eigenfunction as a linear com-bination of the electron and hole single-particle eigenfunctions The Schrödinger equation for the electron-hole pair may now be cast in equivalent form where ∆ is the excitonic energy in units of ǫ 0 . Following an analogous procedure as in Ref. 20 the coefficients A MeM h are obtained multiplying Eq. (6) by ψ † Me (r e )ψ † M h (r h ) and integrating over the coordinates where we have defined Setting r e = r h = r in the expansion of (5), multiplying by ψ † (r) and integrating over the coordinates we finally obtain with To proceed we define the total angular momentum of the electron-hole pair in units of as L = ℓ e +ℓ h . Because the system is axially symmetric, only states with L = L ′ can contribute to the excitonic system. This condition is even more restrictive under the dipole approximation, i.e. only excitons with total angular momentum L = 0 can absorb light polarized perpendicular to the ring. Therefore Eq. (10) reduces to We note that in the limit γ → ∞, the integrals in (11) reduce to 1/ √ 2πRW for n e = n h = n ′ e = n ′ h = 0 and l e +l h = l ′ e +l ′ h . For other combinations of n e , n h , n ′ e , and n ′ h , the corresponding P MeM h M ′ e M ′ h are less important due to the energy denominator in (11).
Similarly as for 1D quantum rings, 14,20 there is no analytical solution of Eq. (9) for finite values of v 0 . In order to find approximate solutions, we hence need to cut off the sums at some maximally allowed values for M e and M h . Figure 2 shows that for smaller values of γ, i.e. increasing ring width W , the level separation between the quantum states of the single particles is decreased. Therefore we use different ℓ max and n max values depending on our choice of γ. We have tested that our results do not change appreciably for the range of Φ and v 0 considered here. As in Ref. 20, Eq. (9) is reformulated as a standard left-eigenvalue equation G K ′ = K G K P KK ′ (∆) after mapping the quantum numbers M e , M h → K and M ′ e , M ′ h → K ′ according to K = (ℓ + ℓ max )(n max + 1) 2 + n e (n max + 1) + n h + 1 such that K, K ′ = 1, 2, . . . , (1 + 2ℓ max )(1 + n max ) 2 . The excitonic energies are obtained numerically by determining the values of ∆ which result in the matrix P KK ′ having an eigenvalue equal to 1. For a given ∆, all eigenstates can be found using (9), (7) and (5). An advantage of our approach is that it allows us to target the ground state directly by choosing a suitable starting value for ∆.

IV. RESULTS
In Fig. 3 we plot the ground state energy ∆ defined by Eq. (9) with ℓ max = 40 for γ > 0.5 and n max = 5 (K max = 2916) for γ < 5 or n max = 2 (K max = 729) for γ > 5 and as a function of v 0 for different values of γ. For γ = 0.5 we have used ℓ max = 30 and n max = 6 (K max = 2989). Here and in all following figures, when plotting the excitonic energies ∆, we have subtracted the zero-point energy 2γ 2 of the non-interacting electron-hole system.
We see that for all γ and Φ values, the increase of the interaction strength v 0 leads to the formation of a state with decreasing energy values below the onset of the freeparticle continuum. We also compare in Fig. 3 the 2D exciton results with the 1D ring studied in Refs. 14 and 20. When the radius of the ring is 10 times its width (γ = 10) the 2D excitonic behavior is essentially indistinguishable from the 1D results in the range of v 0 values studied. In particular, the differences between energies at different flux values at large γ decrease. Nevertheless, for small γ 3 different magnetic flux values lead to quite distinct ∆ values -even in a 2D quantum ring the exciton is sensitive to the magnetic flux. It is also interesting to note that for large γ, the bound state energies are more negative for larger values of v 0 [14] whereas for γ 1 we find evidence that smaller γ values lead to smaller differences between different values of Φ.
The above results have been obtained assuming the same effective mass for the electron and the hole. To addrees the question of the robustness of the excitonic AB effect in a more realistic situation with different effectives electron and hole masses, we have calculated the exciton energy as a function of the ratio m e /m h . Let us define the amplitude of the excitonic AB oscillations as the difference of the exciton energy at Φ = 1/2 and Φ = 0, namely ∆(1/2) − ∆(0). The inset of Fig. 3 shows this amplitude as a function of the ratio m e /m h for different values of the radius-to-width ratio γ at interaction strength v 0 = −2/π 2 . In 2D rings (γ = 0.5) the energy difference is almost constant and the assumption of equal masses is well justified. Upon approaching the 1D limit, i.e. increasing γ, the amplitude of excitonic AB oscillations is reduced but the effect is still revealed. As an example, in common III-V compound semiconductors the ratio of the electron and ligh hole masses typically ranges from 0.6 to 0.9, and it can be seen in the inset of Fig. 3 that the reduction of the amplitude is small. Figure 4 shows the AB oscillations of the exciton energy as a function of the magnetic flux Φ within one flux period at different values of γ and v 0 . In agreement with Fig. 3, we find that the AB oscillations are retained for radius-to-width ratios ranging from γ = 0.5 to 10. This shows that the excitonic AB effect remains robust even in a ring of finite width. Upon increasing the γ values for different Φ values, we find mostly a moderate increase of the exciton energy, except in the vicinity of Φ = 0.5 where even the reverse tendency can be observed.
In Fig. 5 we plot the amplitude of the excitonic AB oscillations for different interaction strength v 0 = −1/π 2 , −2/π 2 , −3/π 2 as γ is varied. We see that upon decreasing γ from the nearly 1D behaviour at γ = 10 towards γ ≈ 1.5, there is only a slight decrease in the amplitude of the AB oscillations. Upon further decreasing γ, the oscillations weaken more rapidly, but even at In Fig. 6 we plot the exciton probablity density |Ψ(r e , r h )| 2 for different values of γ. We integrate |Ψ| 2 over the radial coordinates ρ e , ρ h and hence retain the angular dependence in Figs. 6a and 6c, whereas in Figs. 6b and 6d we integrate out the angular degrees of freedom and retain the ρ e , ρ h dependence. From these figures we conclude that the exciton fills the available width of the ring. Figures 6a and 6c show that the exciton is indeed bound, i.e. the majority of the weight of |Ψ| 2 resides along the diagonal θ e = θ h . Analogous results are obtained for different Φ and v 0 . This is similar to the 1D behavior described in Refs. 14 and 20.
The oscillator strength, defined as is plotted in Fig. 7. Large values of F corresponds to a large transition matrix element from the exciton ground state into the vacuum. We find from Fig. 7 that the results for large γ are in good agreement with the 1D results. 14 And when decreasing the radius-to-width ratio γ, the value of F does not suddenly drop to zero, again emphasizing the robustness of the excitonic AB effect in ring of finite width.

V. CONCLUSIONS
Our results suggest that the excitonic AB effect originally predicted for a 1D model 13,14 remains essentially unchanged when allowing for rings of finite widths as given by (2). We find that when we enlarge the ring width by one order of magnitude from 1/γ = 1/10 to 1, the magnitude of the AB oscillations drops by about 15% only. In addition, we show that the qualitative behavior of the oscillations both for the spectral position as well as the oscillator strengths of the exciton luminescence lines are again governed by the relative strength of attractive Coulomb interaction to ring radius. Our results are in good agreement with recent experimental observations where the magnitude of the excitonic AB oscillations was observed to be about 0.5 meV at binding energies of 4.35 mV for rings of about 11-22 nm radius and γ ≈ 1. 5 We also note that our confining potential (2) has been chosen to retain its non-simply connectedness due to the infinitely repulsive centrifugal core at the centre. Hence even for very wide rings, there is an essential difference with respect to the previously considered 2D confining potentials. 15,21,22,[26][27][28] This demonstrates that it is not so much the width or the exact shape of the confining potential, but rather the avoidance of the ring centre which is the important ingredient needed for the experimental observation of the excitonic AB effect.
Last, we expect that the effects of external electric fields 20,31,36 and the formation of charged excitons remain similarly robust in 2D, whereas disorder effects 24,37,38 should be less important than in the 1D case.

ACKNOWLEDGMENTS
We thank Andrea Fischer for valuable discussions and a critical reading of the manuscript. CGS is grateful to the Centre for Scientific Computing for hospitality and to Ministerio de Educación, Comunidad de Madrid and the European Social Fund for funding the research stays at Warwick during which much of this work was done.