# Magneto-Optical Quantum Switching in a System of Spinor Excitons

## Abstract

In this work we investigate magneto-optical properties of two-dimensional semiconductor quantum-ring excitons with Rashba and Dresselhaus spin-orbit interactions threaded by a magnetic flux perpendicular to the plane of the ring. By calculating the excitonic Aharonov-Bohm spectrum, we study the Coulomb and spin-orbit effects on the Aharonov-Bohm features. From the light-matter interactions of the excitons, we find that for scalar excitons, there are open channels for spontaneous recombination resulting in a bright photoluminescence spectrum, whereas the forbidden recombination of dipolar excitons results in a dark photoluminescence spectrum. We investigate the generation of persistent charge and spin currents. The exploration of spin orientations manifests that by adjusting the strength of the spin-orbit interactions, the exciton can be constructed as a squeezed complex with specific spin polarization. Moreover, a coherently moving dipolar exciton acquires a nontrivial dual Aharonov-Casher phase, creating the possibility to generate persistent dipole currents and spin dipole currents. Our study reveals that a manipulation of the spin-orbit interactions provides a potential application for quantum-ring spinor excitons to be utilized in nano-scaled magneto-optical switches.

## I Introduction

While the relativistic nature of a spin-orbit interaction (SOI) in atomic systems
is given to be in the order of eV,
a large SOI field can be obtained either in the presence of a large electric field or in materials where the mass gap is reduced.
Therefore, in recent decades, studies of spin-dependent effects in semiconductor nanostructuresManuel ValÃn-RodrÃguez *et al.* (2002); Meijer *et al.* (2002); Tsitsishvili *et al.* (2004); Kuan *et al.* (2004) and nano-devicesDatta and Das (1990); Wang *et al.* (2002); Koga *et al.* (2002) have been attracted to both the experimental and the theoretical aspects and have opened up the field of
spintronics.Wolf *et al.* (2001); Å½utiÄ *et al.* (2004) In narrow-gap heterostructures
such as InAlAs/InGaAs, a spontaneous spin-splitting can be realized by structural inversion asymmetry (SIA).Rashba (1960) The kind of asymmetry, corresponding to an inhomogeneous built-in electric field due to band-bending and discontinuity which gives rise to the 2DEG confining potential at the interface, leads to the splitting termed the Rashba effect.Bychkov and Rashba (1984) On the other hand, in wide-gap zinc-blend structures, bulk inversion asymmetry (BIA) may induce a coupling of electronic states which is cubic in the electronic momentum . The spin splitting of electron and hole states at nonzero , even at zero magnetic field, is known as the Dresselhaus effect.Dresselhaus (1955)
Both spin splittings can be tuned continuously by means of external gates.

Achievements in the state-of-the-art material engineering and nanofabrication
techniques have successfully led to the realization of advanced semiconductor devices and made
possible the investigation of quantum phenomena in these systems.Fuhrer *et al.* (2001); Mano *et al.* (2005) In ring geometries, systems that undergo a slow, cyclic evolution were predicted
to induce a Berry phase for the electronic states.Berry (1984)
The existence of the geometric quantum phase reveals the
significance of electromagnetic potentials in the quantum theory. The periodic interference effect identified by Aharonov and Bohm (AB)Aharonov and Bohm (1959) was measured via spectroscopy of nanoscopic semiconductor ringsLorke *et al.* (2000) and AB oscillations of the spin components were manifested in the resistance in a GaAs 2D hole system with a strong SOI.Yau *et al.* (2002) Many electronic properties, like optical absorption and transmission,Murdin *et al.* (1999) cyclotron-resonance,Krahne *et al.* (2001) and Rabi oscillations,Kamada *et al.* (2001); Stievater *et al.* (2001); Htoon *et al.* (2002); Borri *et al.* (2002); Stufler *et al.* (2005) have been examined in combination with the optical AB effect, with or without SOIs. However, the AB effect for another simple many-body complex, the exciton, has not caught the attention of researchers,
nor has its existence even been debated.

Viewed as a neutral particle, the exciton was not expected to demonstrate
an AB effect. But it was pointed out that the optical AB effect could be manifested for polarized excitons in ideal nano rings,Govorov *et al.* (2002) and for Wigner molecules in Type-II quantum dots.Okuyama *et al.* (2011) Whenever there is a net magnetic flux traversing the particles, the electron-hole pair would be dragged along together, and their coherence would be revealed in an oscillatory energy spectrum. A robust AB oscillation
observed in columnar ZnTe/ZnSe quantum dots has provided the evidence to support
the formation of coherently rotating states of neutral excitons.Sellers *et al.* (2008); Roy *et al.* (2012)

Based on previous experience, we proceed for more detailed and realistic
simulations by adopting a double-ring confining potential with a finite width. The geometry provides the degree of freedom to adjust the separation and the barrier between an electron and a hole, hence establishing an opportunity to determine the density correlations displaying the coherence properties of the system.
Meanwhile, the inclusion of the SOIs is aimed at testifying the AB stiffness under external perturbations. The model describes electrons in the -6 band under the influence of the Rashba effect and heavy holes in the -8 band experiencing a Dresselhaus effect from the -linear interaction.Tsitsishvili *et al.* (2004) Next, we explore the light-matter interactions of the spinor exciton via photoluminescence spectroscopy (PL).
As the time periods between absorption and emission of photons may range from short femtoseconds up to milliseconds, the microscopic description of the PL has to be accomplished by a semiconductor luminescence equations.Koch *et al.* (2006); Kira *et al.* (1998); Koch *et al.* (2006) Rather than tracing the dynamics of the optical transitions, finding the conservation laws supporting open channels for bright excitons in the electron-hole recombination is the first priority in the present work. An analysis of the magneto-PL spectrum of the spinor exciton can lead to an understanding
of how the presence of the SOIs influences the diamagnetic responseAmbegaokar and Eckern (1990) observed
in spin degenerate excitons.
Furthermore, a thermodynamic persistent, charge current (CC), , has been theoretically suggestedBÃ¼ttiker *et al.* (1983); Byers and Yang (1961); Chakraborty and PietilÃ¤inen (1994) and was observed both in metal and semiconductor loopsLÃ©vy *et al.* (1990); Chandrasekhar *et al.* (1991); Mailly *et al.* (1993); Jariwala *et al.* (2001) if the phase coherence of the charge wave functions throughout the whole sample is conserved. Similarly, the presence of SOIs provides the possibility to support persistent spin currents (SC).Sheng and Chang (2006) Hence, without a loss of generality, we construct an ideal-ring model for (in)coherently-rotating excitons under the SOIs. Excitons are found to have a specific spin polarization, and to
possibly form a squeezed complex when the strength of the SOIs is adjusted. Whenever a coherent exciton is viewed as a neutral particle with an electric dipole moment, it acquires a nontrivial dual Aharonov-Casher (AC) phaseAharonov and Casher (1984); Yi *et al.* (1995) when moving in the magnetic field. This implies the possibility to generate persistent dipole currents and spin dipole currents as well. The tunable SOIs open the possibility to coherently manipulate individual particles, and indirectly to manipulate the optical response of an exciton. By controlling the SOIs we can successfully realize a bright-dark sequence that
conceptually could be utilized as an element in an optical switch within the quantum regimes.

The paper is organized as follows. First, we introduce a theoretical model for an exciton in quantum rings subjected both to a magnetic flux and the SOIs. Then we analyze the AB energy spectra, magneto-optical response via the PL spectra, spin orientation, the generation of persistent currents, and the construction of the prototype of a quantum switch. Finally, we give a brief conclusion of the results.

## Ii Hamiltonian

The Hamiltonian of two-dimensional semiconductor quantum-ring excitons with Rashba and Dresselhaus spin-orbit interactions (SOIs) threaded by a magnetic flux perpendicular to the plane of the ring can be written as

(1) |

in which the sum is taken over band indices of the electron and hole bands , and quantum numbers . Correspondingly, the field operator defined by is expressed as

(2) |

where is the Fermi momentum, and are electron and hole annihilation operators, respectively, defined as and , and

(3) |

is the Bloch function constructed by a two-component envelope function and the mutually orthogonal periodic function of the conduction(valance) band in the unit cell. In this manner, the evaluation of the single-particle energies for both an electron and a hole is equivalent to calculating the expectation values of the noninteracting Hamiltonian with the spinor wavefunctions given by . Typically, the Hamiltonian includes the kinetic energy from the canonical momentum , the trapping potential

(4) |

constructed by harmonic and shifted Gaussian wells, the Zeeman interaction , where represents the effective bulk Lande g-factor, and the Rashba and the Dresselhaus SOIs

(5) |

and

(6) |

respectively.

Since the presence of the SOIs couples the spin states, the application of a unitary transformation via the operator

(7) |

diagonalizes the noninteracting system in the subspin spaces.Manuel ValÃn-RodrÃguez *et al.* (2002)
As a result, although and are coupled by the SOIs in a rotating reference frame,
the total angular momentum can be proved to commute with the Hamiltonian, i.e.,
.
Therefore, the spinor wavefunctions in the rotating reference frame for both particles can be constructed by setting

(8) |

Hence we can define the quasi-up and quasi-down wavefunctions for both particles in the lab frame as

(9) | |||||

(10) |

(11) | |||||

(12) |

and can construct an exciton wavefunction via a linear combination of electron-hole pair eigenfunctions

(13) |

While the connection between an electron-hole pair is described via Coulomb interaction, , it is more convenient to deal with the two-particle potential in the momentum representation by writing

(14) |

In this manner, a two-body function in the coordinate space such as , can be reduced to a single-body one in the momentum space by , in which is the first kind Bessel function of zeroth order. Therefore, via Fourier transformation, and writing the single-particle wavefunction in terms of its Fourier conjugate

(15) |

the calculation of Coulomb potential energy can be performed in the momentum space and leads to

(16) |

in which and represent the momentum conservation of electron and hole with state quantum numbers and during the potential scattering.
After analytically integrating the angular part and applying the Weber-Schafheitlin formulaWang *et al.* (1989) for Bessel function integrals, the evaluation of the matrix elements for the direct Coulomb interaction given by

(17) | |||||

results in an analytical expression in terms of radial integrals

(18) | |||||

in which and

(19) |

is a convergent polynomial of the proper fraction with the Gamma function . We note that the expression for the matrix elements (18) is useful independent of the exact radial confinement of the ring system.

### ii.1 Photoluminescence

We next investigate the magneto-optical response of an exciton by exploring the photoluminescence spectrum. PL intensity is proportional to the transition rate that can be calculated via Fermi’s golden rule regarding dipole-allowed light-matter interaction with the quantized electromagnetic fields (in terms of the annihilation and creation operators and ) of mode and polarization . Within the dressed-state representation, the rate of the photon emission process is given by

(20) | |||||

in which the term that is (not) proportion to accounts for the stimulated (spontaneous) emission. To calculate the total spontaneous emission rate, we need to sum over all photon modes and choose to use the continuous representation

(21) |

Then we obtain

(22) |

in which is the refractive coefficient of the material.

We evaluate Eq. (22) by rotating the reference frame such that the dipole vector points in the direction, labeled by , and the wave vector of the laser pulse is . To satisfy the orthogonal criteria, one field polarization vector can be chosen to lie in the - plane having coordinate components , hence the second one takes the form . As a consequence, the PL intensity of the dipole transition results in

(23) |

Evaluation of Eq. (23) in semiconductor nanostructures can be carried out by defining the one-particle polarization operator given by

(24) |

In what follows we only consider direct optical transitions. We apply a approximation and expand around the band bottom to simulate a strong recombination that occurs at the -point in III-V heterostructures. Eventually, the polarization operator for annihilating an electron-hole pair is written as

(25) |

However, the applied magnetic flux causes a cross effect with the time-dependent electromagnetic fields. Let be the static field, the external perturbation involves , which means, while , the matrix element of the annihilation of an electron-hole pair should be constructed via , which is proportional to , in which denotes the vacuum state with exciton vacancies. Therefore, in the presence of crossed external fields, the matrix element of the dipole transition takes the form

(26) | |||||

While the velocity gauge field can be equivalently converted to in the length gauge and the integration of the periodic functions over the unit cell can be dealt with as in the bulk, Eq. (26) has been proven to be adequate for the calculation of the transition rate for a spontaneous recombination.

Extensive exploration of the SOI associated excitonic PL spectrum is made feasible by replacing the spin degenerate charge wavefunctions in Eq. (26) with the spinors of Eqs. (9-12). While the recombination destroys the excitonic states, the energy is compensated in the way of photon radiation whenever the particles in the electron-hole pair have opposite rotational parity and spin orientations. In other words, the effective recombination of bright excitons is associated with

(27) | |||||

Substituting Eq. (27) into the dipole transition of Eq. (23), we find that for excitonic recombination between and states, . At finite temperatures, there is a considerable possibility to detect photon emissions from high-level recombination and therefore the total PL intensity that sums up contributions from states with energies by the classical Boltzmann distribution can be expressed as

(28) |

## Iii Persistent Currents

The charge current density is the expectation value of the charge current operator , where is the charge density operator. Eventually, in matrix representation, it can be written as . Similarly, the spin density operator defined by , where denotes Pauli matrices, leads to the spin current density operator and gives the spin current density . Since the velocity of a particle of a corresponding Hamiltonian is given by , for an electron in an ideal 1D ring threaded by a perpendicular magnetic flux under the influence of SOIs, it moves along the circumference of the ring with the velocity

(29) |

This implies that the current densities can be evaluated from the mean values of the Hamiltonian eigenstates . Then after some arithmetic one obtains the persistent charge current

(30) |

together with the spin currents

(31) |

in the presence of the SOIs, where , with as the flux quantum denoting the spin-up(-down) state, respectively and is the angle between the spin vector and the z-axis.

When assuming ballistic collisions in an ideal ring, the thermal equilibrium properties as well as the generation of persistent currents of the spinor excitons can be investigated analytically. In a hard-ring 1D confinement, an electron has the energy

(32) |

Eqs. (30) and (31) lead to the charge current and the spin current being expressed as

(33) | |||||

Similarly, the eigenenergy of a hole can be written as

(34) |

Correspondingly, the persistent currents are

(35) | |||||

To find the magnitudes of , we need to examine the spin orientation of single particles. While writing and , we obtain

(36) |

and find that . Similarly, by writing and , we obtain

(37) |

and find that . The two-particle exciton system has to be investigated in a four dimensional space. Spanned by the four-component basis , the spin orientation of the up(down)-pair and mixed-pair exciton is represented by

(38) |

(39) |

Eqs. (36-39) reveal that the single-particle picture is approximately valid to determine the SC of an incoherent exciton, but below we show that for the coherent case, the SC is related to the resultant of the two-particle spin vectors.

A coherent exciton can be viewed as a neutral particle with an electric dipole moment ; thus it is also referred to as a dipolar exciton. Moving in the magnetic field , the exciton acquires a nontrivial dual Aharnov-Casher phase defined by

(40) |

As a result, an equilibrium persistent dipole current can be defined by

(41) |

For simplicity, setting , a dipole can be viewed as a complex with mass , where is set as an effective length threaded by the magnetic field.

Coherent excitons can be classified into two classes. The first class consists of full up(down)-pair particles and their energies are represented by

(42) |

whereas the second class consists of excitons with mixed-pair particles having energies

(43) |

In both cases, represents the total angular momentum of the exciton, , , and are used to represent the effective SOIs and Lande g-factors of the exciton.

The dipole currents (DC) and spin dipole currents (SDC) for effective spin up and down excitons can then be expressed in an organized way as

(44) |

and

(45) |

in which and