Modeling of anisotropic properties of double quantum rings by the terahertz laser field

The rendering of different shapes of just a single sample of a concentric double quantum ring is demonstrated realizable with a terahertz laser field, that in turn, allows the manipulation of electronic and optical properties of a sample. It is shown that by changing the intensity or frequency of laser field, one can come to a new set of degenerated levels in double quantum rings and switch the charge distribution between the rings. In addition, depending on the direction of an additional static electric field, the linear and quadratic quantum confined Stark effects are observed. The absorption spectrum shifts and the additive absorption coefficient variations affected by laser and electric fields are discussed. Finally, anisotropic electronic and optical properties of isotropic concentric double quantum rings are modeled with the help of terahertz laser field.

detection of time-resolved single-electron tunneling with a precision enhanced by feedback control. These experimental results claim the need to investigate how the coupling between the rings in CDQRs can be influenced both externally and internally. Internally, it has been demonstrated to be realizable by varying the inter-ring distance and aluminium concentration during the calculations of few-electron 20 and impurity-related linear and non-linear optical absorption spectrum 21 respectively, while externally it can be done by applying magnetic and electric fields [22][23][24][25][26][27] and with hydrostatic pressure 28

as well.
A few works of our group were devoted to the study of intense laser field effects in quantum ring structures [29][30][31][32][33] . The current work aims to demonstrate theoretically that inter-ring coupling in semiconductor concentric double quantum rings can be controlled by intense THz laser field and uniform static electric fields. In particular, it is shown that the laser field can rearrange the energy spectrum by eliminating and afterward creating new pairs of degenerated levels, while the static electric field effect in anisotropic laser-dressed confining potential can create both linear and quadratic Stark effects.

Problem
The CDQRs consists of GaAs QRs (well material) separated by Ga 0.7 Al 0.3 As (barrier material). In the absence of laser field the confinement of electron in two-dimensional CDQR structure is modeled according to potential: where V 0 = 257 meV is the height of the potential attributed to the confinement of electrons 34 , r ⊥ denotes the electron position in two-dimensional CDQR and R R R , , 1 in 2 in 1 out , and R 2 out are respectively inner (subscript "1") and outer (subscript "2") radii of inner (superscript "in") and outer (superscript "out") rings. The rings are considered two-dimensional based on the much stronger quantization in growth direction 17 , and the radii of rings are taken in accordance with the sizes of real CDQR structure 15 . It is also supported by works that used two-dimensional models of confinement to make comparisons with experimental data. In ref. 3 authors compared the Aharonov-Bohm oscillations in InAs/GaAs QRs using the two-dimensional parabolic confinement. In addition, although the confinement potential in ref. 17 was defined using the actual shape of CDQRs determined by the atomic force microscopy measurements, mainly the quantized radial motion was considered for the effective mass calculations to compare with photoluminescence data.
In the presence of static electric and THz laser fields the radial and rotational motion of the electron in two-dimensional CDQR can be described by a time-dependent Schrödinger equation: where m = 0.067 m 0 is the effective mass of electron in GaAs 35 , such that m 0 is the rest masses of electron, vector potential A ⊥ (r, t) defines the laser field, e denotes the electron charge, c is the speed of light, F is the electric field strength, and ħ is the reduced Planck constant. The solution of Eq. (2) can be greatly simplified under dipole approximation i.e. when A ⊥ (r, t) ≈ A(t) 36 . For 10 nm wide (in the radial direction) GaAs QRs, it is fulfilled if the laser field frequency ν  1500 THz, and in the current work we will deal with such frequencies. With satisfied dipole approximation, vector potential does not vary in space and the phase-factor transformation 37 can be applied removing the term with A 2 from Eq. (2): ( , ) satisfies the following Schrödinger equation 30,38 : In this work, we are interested in the solution of Eq. (5) in the high-frequency limit ντ  1, where τ is the characteristic transit time of the electron in the structure. It can be obtained by applying the non-perturbative Floquet theory and subsequently keeping only the zero-order terms of Fourier expansions of confinement potential V(r ⊥ ) and wave function φ(r ⊥ , t) 39 . These approximations lead to the following time-independent Schrödinger equation 33 : In Eq. 6 the laser field is considered with fixed linear polarization along the x-axis that results in laser-dressed and electric field influenced effective potential in Eq. (6) and α(t) = −α 0 sin(2πt/T) is the quiver displacement where T is the laser field period. From now on, the peak value α ε ν π = − e m I c ( / ) /(2 ) 0 h 1/4 2 3 will be taken to characterise the laser field effect, where ε h = 10.9 is the high-frequency dielectric constant in GaAs 35 , and the intensity I and ν frequency of laser field are in orders of 1 kW/cm 2 and 1 THz, respectively.
In Fig. 1 the effective potential is presented for the fixed value of α 0 and two different values of electric field strength F. While the electric field results in the tilting of the potential, the laser field decreases the width of well regions along the x-axis in the lower part of the potential and enlarges them in the upper one. In other words, the laser field creates an anisotropy in the confinement potential, which can be continuously controlled by the THz laser field. It is useful to compare our results with those for elliptic core-multishell quantum wires 40 and CDQRs 41 . In these works, the anisotropy was induced by the geometry of the structure, that needs to be controlled during the growth process 40 or by effective mass 41 manipulations. We theoretically demonstrate (also see Fig. 3 for wave functions) the feasibility of it by THz laser field, that is an external influence. The latter effect allows the investigation of the physical properties of quantum rings of different geometries in a single sample of CDQRs. Thus, our results allow the manipulation of different shapes that is important for modeling of experimental studies of CDQRs that in general are not purely circular 42 .  In addition, we are interested in intraband transitions to estimate the optical response of the laser-dressed system. For that reason, the total absorption coefficient is calculated 43 i is the energy difference between the final (f) and initial (i) states, M if defines the dipole matrix element, the Lorentzian parameter is taken equal to Γ = 0.1 meV, and A contains all the other factors 44 is the occupation difference of the ground and final states and is equal to 1, since the final state is vacant and the initial one is the ground state occupied with one electron. Circularly polarized light is considered falling perpendicularly to the plane of the rings.

Methods
The laser-dressed eigenvalues E d and eigenvectors ψ d (r ⊥ ) are found numerically in COMSOL Multiphysics software 45 , using the finite element method. Meshing is done with triangular elements, and Lagrangian shape functions are used 46 . A square is taken as a computational domain with side size of = .
L R 2 8 2 out . This value is found sufficient to avoid eigenfunction traces outside of it. In the presence of the fields, the fourth order Lagrangian shape functions are used, and the domain is meshed with "Extremely fine" option of "General physics" calibration node. In the absence of the fields, third order Lagrangian shape functions and "Extra fine" option 45 is used.

Results and Discussion
Degenerated laser-dressed energy spectrum and intraband absorption. Figure 2 illustrates the influence of the laser field on the energy spectrum in the absence of electric field. The inset columns are the wave functions of bound states for the lowest α 0 = 0 and highest α 0 = 3 nm values of laser field parameter. The deformation of the confinement potential Fig. 1(a) brings up all the energy levels with the augmentation of α 0 , meaning that all the considered ten energy levels are positioned lower enough in the confining potential. Another influence of the laser field is the rearrangement of energy levels: at first, it eliminates the original degeneracy caused by the cylindrical symmetry of confining potential and then it makes new pairs of degenerated levels. In the absence of laser field, the following pairs form degenerated couples: third and fourth, fifth and sixth, seventh and eighth, and ninth and tenth. Viewing the forms of wave functions in Fig. 2(a) at α 0 = 3 nm, one sees that laser field leads to new combinations: first and third, fourth and fifth, and sixth and ninth (second and seventh pair is not that close to like other pairs, but their wave functions have forms similar enough to consider them as ones having a tendency to degenerate afterwards). The reason is the laser field that changes the symmetry axes from the diagonal of the square to the x and y-axes seen, resulting from the shape modification of confining potential observed in Fig. 1. Also the energy spectrum in Fig. 2(a) is full of crossing points and has only one anti-crossing event, shown in the enlarged graph in Fig. 2(b). For example, the first excited state wave function is symmetric with respect to the y-axis, while the third one shows antisymmetry, which means that they can cross, much like the terms of the diatomic molecule 47 . Meanwhile, anti-crossing occurs between the ground and the first excited level, because in the case of crossing the ground state must have zeros, that is not allowed 48 .
The evolution of wave function shapes with the increment of α 0 is depicted in Fig. 3 for the first, third, fourth and fifth states. As expected from the results in Fig. 1, the distribution of wave functions is anisotropic. It localizes along the y-axis, as α 0 is increased. Besides, the ground state wave function gradually moves from the inner ring to the outer one and the wave functions show the tendency to accumulate along the y-axis. The point is that for low lying states the contraction of the well width is the biggest along the polarization direction of laser field (x-axis) and is almost unchanged along the y-axis, where the probability to find the electron turns bigger. The obtained modification of electron localization between the rings can be useful for the manipulation transport properties of QR arrays in optoelectronic devices: for example, the proper value of α 0 can shift the electronic cloud to the outer (inner) ring, thus turning on (off) the tunneling between the rings.
In order to study the possibility of the charge delocalization by the THz laser field in CDQR structure, it is interesting to investigate laser field influence on electron probability density (PD) distribution for the ground state by varying the barrier region width L B between the inner and outer ring and α 0 at the same time. For that reason, the ratio of probability densities in the outer ring and the inner ring -r = (∫ outer |ψ d | 2 dr ⊥ )/(∫ inner |ψ d | 2 dr ⊥ ) is calculated. The PD is considered to be fully delocalized to the outer ring, once the r > 5 × 10 2 . Under this condition, the map of PD delocalization points is presented in Fig. 4. It can be observed that smaller values of L B require bigger ones for α 0 to reach delocalization, and vice versa. The reason for this lays in the coupling of the QRs, which is stronger if the QRs are closer. In addition, the values of r at which delocalization occurs do not depend on any fixed ratio L B /α 0 .
The Δ if energies dependence on α 0 are shown in Fig. 5(a), where the area of the circles is directly proportional to the dipole matrix element modulus square. The corresponding α(Ω) absorption coefficient dependence on incident photon energy Ω by gradually changing values of α 0 is shown in Fig. 5(b) in the absence of electric field F = 0. The allowed transitions are 1 → 3, 1 → 4, 1 → 7, 1 → 8, 1 → 9, 1 → 10. The results for 1 → 9 and 1 → 10 are not presented, since their contribution is much weaker compared with other transitions. The selection rule that defines this transitions is based on the symmetry of wave functions of the excited states, that must not have antisymmetry or symmetry with respect to both of the coordinate axes; otherwise the M if matrix element is zero. The Δ if curves of 1 → 3 and 1 → 4, 1 → 7 and 1 → 8 pairs start from the same value. It is an expected result, as long as in the absence of laser field the mentioned states are degenerated. Starting from the α 0 = 1.3 nm the 1 → 3 transition has the biggest value of dipole matrix element modulus. Nevertheless, this very issue does not make the   Figure 5(b) demonstrates that 1 → 4, 1 → 7 and 1 → 8 transitions have absorption coefficients greater than 1 → 3, although related |M if | 2 is much smaller. This is a consequence of Ω  factor in Eq. (7). Besides, 1 → 3 shows the redshift, 1 → 4, 1 → 8 transitions undergo a blueshift, and 1 → 7 one in the [0, 1.3 nm] interval demonstrates redshift and subsequently only the blueshift of the absorption spectrum. These spectrum shifts are caused by the results for Δ if as shown in Fig. 5(a).

Stark effect in laser-dressed states.
In this section, we consider static electric field effect on already laser-dressed CDQRs. Figure 6 explores the influence on the energy levels of the electric field applied in different directions. The direction is defined by β = ∠ˆû e ( , ) x angle, where û and ê x are unit vectors of electric field and laser field polarization, respectively. In case of an electric field directed along the x-axis (β = 0°) quadratic Stark effect 49 is observed for both values of α 0 = 1.5 nm; 3 nm, and all the energy levels decrease as a reason of effective confining potential tilting demonstrated in Fig. 1(b). In addition, since û vector direction is also symmetrical one for the only laser field affected potential in Fig. 1(a), related wave functions have symmetry or antisymmetry with respect to the x-axis, and energy spectrum can reveal both crossing and anti-crossing points. On the other hand, for β = 45° case the direction of û does not follow the symmetry of V d (r ⊥ ) potential energy. This implies that the related wave functions do not have any distinct symmetry or antisymmetry. Thus, the energy levels cannot express any crossing behavior. If the direction of electric field is perpendicular to ê x (β = 90°) energy levels start to become linear functions of F, that are more clearly observed for α 0 = 3 nm, or in other words, when the anisotropy of the confining potential is greater. There are experimental studies that pointed out the importance of anti-crossing features that can be used to measure the degeneracy in coupled quantum systems. For instance, studies in refs 50,51 demonstrated the possibility to measure the tunnel splitting in a double QD charged with a single electron. Also, by magnetophotoluminescence spectroscopy the existence of a hole-spin-mixing term directly related to the anti-crossings in the excitonic spectrum was obtained in electric field influenced InAs QD molecules (see ref. 52 ). In the context of the mentioned works, we show that one can effectively control the anti-crossings in the energy spectrum of CDQRs with electric field once the anisotropy is achieved.
Besides that, the electric field can serve as a potential tool to manipulate the optical response of the CDQR system. In the presence of electric field all the transitions are allowed, since M if matrix element never becomes zero. The oscillator strengths 27,53 of the most intensive transitions are demonstrated in Fig. 7. The results are respectively related to the energy spectra in Fig. 6. While 1 → 2 is observed as the most probable transition one for all the values of F in β = 0° case, the appearance of a non-zero angle changes the scenario. For α 0 = 1.5 nm and β = 45° values, although the highest value is obtained for O 12 at F = 0 ( Fig. 7(b)), in [0.25 kV/cm, 1.25 kV/cm] range probabilities of other transitions  Fig. 7(c) and (f). Now 1 → 5 is the most probable one for all the considered values of F. Only in the absence of electric field 1 → 2 turns out to be the most intensive one. Figure 8 the combined influence of laser and electric fields by changing the direction of electric field of F = 1.5 kV/cm strength and keeping the polarization vector ê x of laser field fixed is considered. As Fig. 8 shows, the energy levels mostly have extrema for β = 90°, with the exception of the second excited energy level that has maxima at β = 24° and β = 156° and minimum at β = 90°, and the fourth one that together with the minima at β = 13° and β = 167° are almost constant in [72°, 108°] interval. The appearance of the extrema and the region of invariance can be attributed to the complex distribution of electron cloud throughout the variation of β. In addition, the observed symmetry with respect to the β = 90° point, is caused by the mirror symmetry of the dressed potential in Fig. 1(a) with respect to the x− and y− axis, which means that at angles β and 180° − β electric field affects identically.
And finally, Fig. 9(a) shows all the most intensive transitions to undergo blueshift of the absorption spectrum in the [0°, 90°] interval and redshift in [90°, 180°]. The related absorption coefficient is demonstrated in Fig. 9

Final Remarks
We have demonstrated that the electronic and optical properties of CDQR system can be readily controlled with THz laser and static electric fields. Particularly, it is calculated that the intense THz laser field permits to study double QRs of different geometries in a single sample of CDQRs, that is an important finding for modeling of experimental studies. In addition, by changing the characteristic parameter α 0 of the laser field, one can come to a new set of degenerated levels in laser-dressed CDQR and manage the distribution of electron cloud between the rings. The impact of inter-ring barrier width L B variation on electron PD in the rings shows that delocalization of PD from the inner to the outer ring does not depend on the fixed value of L B /α 0 ratio.
The selection rule that defines the intraband transitions of the circularly polarized light in only laser field influenced CDQR system are shown to allow only the transitions from the ground state to the excited states that do not have antisymmetry or symmetry with respect to both coordinate axes. Also, with the augmentation of α 0 both the blue-and redshifts of the absorption spectrum are observed.
The addition of a static electric field on the energy spectrum results in linear and quadratic Stark effects, caused by the anisotropic modification of confining potential by the laser field. Linear Stark effect is observed for the electric field perpendicular to laser field polarization and is more pronounced for the larger anisotropy of the confining potential. For the quadratic Stark effect, the direction parallel to laser field is favorable. Also, the electric field direction drastically changes the crossing and anti-crossing behaviors of the energy levels. Moreover, electric field removes the laser field-induced selection rule and allows all the intraband transitions.
Besides that, it is shown that the electric field influence on the laser-dressed CDQRs allows to readily control the anti-crossings in the energy spectrum. In addition, if the electric field is parallel to laser polarization the biggest oscillator strength (absorption intensity) has 1 → 2 transition. In case of a perpendicular direction of the electric field, 1 → 5 transitions have the biggest intensity. Electric field direction changing also affects the absorption spectrum, making mainly blueshifts in [0°, 90°] and redshifts in [90°, 180°] orientations. It is worth to note that the THz laser field can in principle control any anisotropic (induced by the geometry, effective mass, defects, etc.) properties of the CDQR nanostructure. We believe that the results are useful and will open up new possibilities to the improved design and characterization of new devices based on CDQR, such as THz detectors, efficient solar cells, photon emitters, to cite a few. Figure 9. Δ if energy dependence on β (a) and corresponding absorption coefficient (in arbitrary units) dependence on incident photon energy Ω for different directions (defined by β) of F = 1.5 kV/cm electric field (b). Laser field parameter is fixed to α 0 = 1.5 nm and the area of the circles in Fig. 9(a) is proportional to the respective dipole matrix element |M if | 2 .