Pyramidal core-shell quantum dot under applied electric and magnetic fields

We have theoretically investigated the electronic states in a core/shell pyramidal quantum dot with GaAs core embedded in AlGaAs matrix. This system has a quite similar recent experimental realization through a cone/shell structure [Phys. Status Solidi-RRL 13, 1800245 (2018)]. The research has been performed within the effective mass approximation taking into account position-dependent effective masses and the presence of external electric and magnetic fields. For the numerical solution of the resulting three-dimensional partial differential equation we have used a finite element method. A detailed study of the conduction band states wave functions and their associated energy levels is presented, with the analysis of the effect of the geometry and the external probes. The calculation of the non-permanent electric polarization via the off-diagonal intraband dipole moment matrix elements allows to consider the related optical response by evaluating the coefficients of light absorption and relative refractive index changes, under different applied magnetic field configurations.

the structure, giving rise to a spatially direct exciton and that through the applied electric field it is possible to polarize the system giving rise to a spatially indirect exciton with changes in lifetime ranging from nanoseconds up to milliseconds. The analysis of the probability distributions shows the evolution between QD and quantum ring induced by the electric field. To date there are no further known developments of this type of cone/shell novel structure nor about similar pyramidal/shell structures. Taking into account the high degree of development that pyramidal QDs have had, we consider that the implementation in the laboratory of a pyramid/shell QD to be viable without much effort. Therefore, using the work from Heyn et al. 32 as a departing point, we have assumed the theoretical investigation of the pyramid/shell QDs as the subject of this research. We will go further and include the effects of a static magnetic field parallel to the vertically applied electric field. We shall focus our attention on the electronic structure, the wave function symmetries, and the intra-band optical absorption. The possible electric-field-induced appearance of indirect excitonic complexes, related to effective spatial separation of electron and hole states is briefly discussed as well. The article has the following organization: The theoretical framework is presented in section II. The section III contains the results and discussion. Finally, in the section IV we outline the conclusions. theoretical framework Figure 1 shows the 3D projection of the structure while a schematic view of the pyramidal core-shell quantum dot (PCSQD) is shown in Fig. 2, with θ labeling the vertex angle, and h i the height of each pyramid. The center of gravity of the PCSQD is assumed to be at z = 0. So, our problem is to study the energy states and their corresponding wave functions for an electron confined in a pyramidal structure like the one shown in Fig. 1 and subjected to the effects of stationary electric or magnetic fields, both applied in the z-axis, parallel to the symmetric axis of the pyramid. Within the framework of the effective mass, the Hamiltonian for this problem, in Cartesian coordinates, takes the form: where e is the electron charge, ⁎ m w b , is the effective mass (b means the barrier region or innermost and outermost pyramid and w means the well region or the pyramid in the center), and V(x, y, z) is the confinement potential for the PCSQD which is V 0 in the innermost and outermost pyramid, zero for the pyramid in the center, and ∞ outside the PCSQD.
The particular gauge chosen to describe the magnetic field in the system implies the conditions  www.nature.com/scientificreports www.nature.com/scientificreports/ for the magnetic vector potential, where → B properly represents the field. The expanded form of the Hamiltonian (Eq. (1)) gives Using the expression of the magnetic potential Eq. (2) the final form of the Hamiltonian Eq. (3) is: The energies and wavefunctions of the bound states can be obtained by solving the Schrödinger equation: i ii The eigenvalues and eigenstates (Eq. (5)) are calculated with the software COMSOL-Multiphysics 33 , which uses a FEM to solve the partial differential equation numerically. A complete description of the COMSOL-Multiphysics licensed software that includes the foundation of the finite element method, the construction of meshes, the discretization of the differential equations, the methods to optimize the processes, the construction of geometries and the convergence criteria can be found in 34,35 . Since Ψ i (x, y, z) is finite, the Dirichlet boundary condition implies that any of its values outside the PCSQD are equal to zero, i.e. the wave functions are zero at the interfaces between the outermost pyramid and the infinite potential region (see Fig. 2). For layered structures such as the one in the current study, the Schrödinger equation interface accounts for the discontinuity in the effective mass by implementing the BenDaniel-Duke boundary conditions.
One of the optical coefficients to be evaluated in this work are the light absorption one, which derives from the imaginary part of the dielectric susceptibility. So, we have: and owing to take into account possible damping effects associated with intraband transitions induced by photon absorption, the Dirac delta term is usually substituted by a Lorentzian term, thanks to the well known relation f i fi f i in which Γ fi (=10 meV in this work) accounts for the corresponding damping rates. In the former expressions, ω represents the incident photon frequency, c is the speed of light in the vacuum, n r is the static value of the refractive index, and  0 is the vacuum permittivity. The quantities E f and E i are, respectively, the energy of the final state and the energy of the initial state of the light-induced intraband transition. Since we are assuming to work in the very low temperature case, the electron density per unit volume is taken to be 2/V, where V represents the PCSQD volume and the 2 indicates the possible spin contributions. This has to do with the situation in which a single electron would be excited towards the conduction band at low T. In this work the electron density was taken as 3 \10 m 22 3 is the unit vector representing the polarization of the -homogeneously intense-incident light (for instance, if the light is circularly polarized in the xy-plane, then ξ → = → ± → e ie / 2 / 2 1 2 , where → e 1 and → e 2 are the unit vectors along the x-and y-direction, respectively).
The general expression for the electric dipole moment matrix element, µ ξ f i , , is the following: in which e is the electron charge and → r is the vector position. In an analogous way, the expression for the coefficient of relative change of the refractive index comes from the real part of the dielectric susceptibility. Its final form reads: 2 fi In Eqs. (6) and (9), the summation is carried out over all possible allowed inter-state transitions.

Results and Discussion
As stated in the previous section, the present study makes use of a FEM to solve the eigenvalues differential equations. In particular, a self-adapting mesh has been used that includes tetrahedra in the volume region, triangles on the surfaces, edge elements at the intersection between two planes, and vertex elements at the intersection between three planes. For a pyramid with h 1 = 5 nm, h 2 = 20 nm, h 3 = 30 nm, and θ = π/2, the used parameters are: 86365 tetrahedra, 11242 triangles, 564 edge elements, and 15 vertex elements, which guarantees a convergence of 0.1 meV for the fifteen lowest states that have been calculated. In this paper we report results for the thirteen lowest energy states. In Fig. 3 the energies of the thirteen lowest confined electron states in a GaAs-Ga 0.7 Al 0.3 As PCSQD are depicted as functions of the innermost pyramid height. Calculations correspond to the situation in which the electric and magnetic field intensities are equal to zero, θ = π/2, h 2 = 20 nm, and h 3 = 30 nm. It is observed that all energy levels have a growing tendency as long as h 1 augments. This is due to the progressive decrease in the volume of the GaAs layer where the electron is confined.
For h 1 = 5 nm the levels (2, 3), (7,8), and (10,11) appear to be degenerate, while for h 1 > 7.4 nm, the degenerate levels are (2, 3), (7,8), and (9,10). This degeneracy comes from the square symmetry of the base of the pyramid, with respect to an axis that passes through its center of gravity and the upper vertex (see Fig. 1). From Fig. 4 -where the projections of the wave functions of the first thirteen confined states onto the xy-plane (with z = 0) and onto the xz-plane (with y = 0) are shown-, it is possible to observe that, for example, the degenerate states Ψ 2 and Ψ 3 have p-like symmetry. The states Ψ 1 and Ψ 5 exhibit s-like symmetry and the state Ψ 4 displays  www.nature.com/scientificreports www.nature.com/scientificreports/ d-like symmetry. The states Ψ 2 and Ψ 3 appear rotated in the xy-plane due to an indeterminacy of the phase, which is typical to the used numerical method. By introducing a very small asymmetry in the dimensions of the base, clearly the Ψ 2 and Ψ 3 states would be oriented along the x and y perpendicular axes. In Fig. 4 the color scale is defined with green corresponding to zero, red to a maximum positive value, and the negative maximum for the blue. For Ψ 1 in the xy-plane, with h 1 = 5 nm, one notices that the wave function has finite values in the center of the structure (the wave function penetrates the center region), indicating the possibility to find the electron around the gravity center of the PCSQD, whereas for h 1 = 15 nm, the electron will completely confine inside the GaAs layer. When comparing the behavior of Ψ 1 in the xz-plane, it can be seen that for h 1 = 5 nm the electron can be found both in the lateral regions and in the base of the GaAs pyramid, while for h 1 = 15 nm, the probability density concentrates mainly towards the side walls of the central pyramid. Finally, note that the value zero of Ψ 4 along the xz-plane, for both h 1 = 5 nm and h 1 = 15 nm, is consistent with the null value of the wave functions in the xy-plane along the y = 0 line. It is interesting to note from Fig. 4 that the Ψ 7 and Ψ 8 states have exactly the same symmetries as the Ψ 2 and Ψ 3 states and that, like the first two excited states, in the entire range of calculated h 1 -values, Ψ 7 and Ψ 8 are degenerate states. Here it should be noted that Ψ 7 and Ψ 8 double the number of Ψ 2 and Ψ 3 antinodes, which is in accordance with their higher energy values. At h 1 = 7.35 nm an accidental degeneration appears with three states that have the same energy. From Fig. 4(a) it is observed that for h 1 = 5 nm, the calculation of Ψ + Ψ 2 leads to a probability density very similar to that obtained with Ψ 9 2 , while for h 1 = 15 nm, in Fig. 4(b), the calculation of Ψ + Ψ 9 2 10 2 perfectly approximates Ψ 11 2 . This is in agreement with the change of symmetries that is observed in h 1 = 7.35 nm.
Furthermore, the inset in Fig. 3 shows the evolution of the three lowest confined electron states in a PCSQD as h 1 approaches h 2 = 20 nm. That is, as the inner GaAs pyramid thickness approaches zero. Note that for h 1 = 19 nm, the ground state energy reaches the value of the potential barrier (262 meV) and from there the wave functions overflow to the Ga 0.7 Al 0.3 As region. This can be visualized in the fourth column of Fig. 5 where the, z = 0, xy-projections of the wave function for the ground state are presented as the thickness of the GaAs layer tends to zero. In that case, the electron is, actually, confined in a pyramid of Ga 0.7 Al 0.3 As of height h 3 with infinite external potential barriers. For 18.8 nm < h 1 < 19 nm, only the ground state is confined within the GaAs region. Going from h 1 = 18 nm towards h 1 = 19.6 nm it is observed how the system evolves from a 2D-confinement in the GaAs region to a 3D-one in the Ga 0.7 Al 0.3 As structure. Figure 6 shows the nonzero transition matrix elements (dipole moment divided by the electron charge, ) between the ground state (Ψ i , i = 1) and the first twelve excited states (Ψ j , j = 2, ..., 13) in a GaAs-Ga 0.7 Al 0.3 As PCSQD as functions of the structure's innermost height (h 1 ), for zero magnetic and electric fields, keeping fixed the other structure dimensions. In Fig. 6(a) the results are for circular polarization of the incoming light in the xy-plane while in Fig. 6(b) they correspond to linear polarization along the z-axis. From Fig. 6(a) we realize that nonzero off-diagonal matrix elements are present only for j = 2 and j = 3. The reason why, , comes from the fact that when calculating M x 1,4 , Ψ 1 is an even function with respect to xz-plane with y = 0 (Ψ 1 (x, −y, z) = Ψ 1 (x, y, z)), while with respect to the same plane, Ψ 4 is an odd function (Ψ 4 ; see first row of Fig. 4(a,b)]. When calculating M y 1,4 , Ψ 1 is an even function with respect to yz-plane with x = 0 (Ψ 1 (−x, y, z) = Ψ 1 (x, y, z)), whilst with respect to the same plane, Ψ 4 is an odd function (Ψ 4 ; see first row of Fig. 4(a,b)]. Because the wave functions Ψ 1 and Ψ 5 are even with respect to the xz-plane with y = 0 and the yz-plane with x = 0, it is obtained that 5 1 ,5 and, consequently, = ± M 0 x iy 1,5 [see first row of Fig. 4(a,b)]. The symmetry arguments used to justify the null values of the matrix elements for the transitions Ψ 1 → Ψ 4 and Ψ 1 → Ψ 5 are the same arguments that can be applied to justify the non-zero values of the transitions Ψ 1 → Ψ 2 and Ψ 1 → Ψ 3 . To discuss the reasons why the z-polarization induces or suppresses certain transitions, the symmetry properties of the wave functions are useful as well, taking into account, basically, the projections on the xy-plane; given that, because of the height of the pyramids, all the excited states under consideration have only one or two antinodes along the z-direction. In the case of a single node for the excited state, it appears displaced along the z-direction with respect to the ground state and when two nodes appear, clearly the corresponding wave function has opposite symmetry, along the z-direction, to that of the ground state. The increasing character of the matrix elements for circular polarization, shown in Fig. 6(a), results from the fact that, as the height of the innermost pyramid (h 1 ) increases, the region where there is the highest probability of finding the electron moves away from the origin and thereby increases the overlap between the wave functions (see comparatively Ψ 1 and Ψ 2 in the first Figure 5. Pictorial view of the wave function projections (onto z = 0 and y = 0 planes) for the ground state in a GaAs-Ga 0.7 Al 0.3 As pyramidal core-shell quantum dot, when h 1 → h 2 . The setup of the structure is as in Fig. 4. The green color corresponds to a zero value whereas the red one is associated to the positive maxima. (2020) 10:8961 | https://doi.org/10.1038/s41598-020-65442-x www.nature.com/scientificreports www.nature.com/scientificreports/ rows of Fig. 4(a,b)). A similar behavior occurs for the matrix element M z 1,5 in Fig. 6(b) (see comparatively Ψ 1 and Ψ 5 in the second rows of Fig. 4(a,b)).
In Fig. 7 we are presenting the energies of the first thirteen bounded states for an electron confined in a GaAs-Ga 0.7 Al 0.3 As PCSQD as a function of the applied electric field strength. The results are for zero magnetic field and fixed dimensions of the structure. From the figure it is possible to observe some features that can be highlighted, such as: (i) throughout the whole range of the electric field intensity, the Ψ 2 and Ψ 3 states are doubly degenerate, the same as Ψ 7 and Ψ 8 states; (ii) for electric field strengths smaller than 65 kV/cm, the Ψ 10 and Ψ 11 states are degenerate and, for that specific value of the electric field they exchange symmetry with the Ψ 9 state, giving rise to the doubly degenerated Ψ 9 and Ψ 10 states, for field values greater than 65 kV/cm; (iii) for F = −5.82 kV/cm an accidental degeneracy appears between Ψ 4 and Ψ 5 states, which is transferred to states Ψ 5 and Ψ 6 at F = 20.5 kV/ cm; (iv) for F = 65.16 kV/cm a threefold degeneracy appears between Ψ 9 , Ψ 10 , and Ψ 11 states; and, finally, (v) for F > 50 kV/cm, the behavior of the lowest eight states is linear and decreasing, thus showing a saturation effect with the electric field. It is important to note that the negatively oriented electric fields push the electronic states towards the top vertex of the pyramid, while the positive fields push them toward the bottom plane of the pyramid  www.nature.com/scientificreports www.nature.com/scientificreports/ (see Fig. 1). Bearing in mind that at the top vertex of the pyramid the electronic states interact with four planes while at the bottom of the pyramid the interaction is with only one plane, this explains the reason why the energy curves are more sensitive to the electric field in the F > 0 regime. Examining, for example, the ground state, it is clearly noted that, for a finite value of the field, the energy curve is not symmetric with respect to F = 0. This is due, as already said, to the fact that the number of planes with which the particle interacts goes from four to one when the field changes from negative to positive values. The decreasing nature of this state with |F| is explained by the displacement towards lower energies of the bottom of the potential well related with the superimposition of the linear potential from the field with the confining potential of the structure. Figure 8 shows the projections, on the z = 0 and y = 0 planes, of the first thirteen wave functions for an electron confined in a PCSQD with fixed values of the geometry, zero magnetic field, and considering two values of the applied electric field. In 8(a) the electric field pushes the carriers towards the apical region of the pyramid whereas in 8(b) these are pushed towards the pyramid base (see Figs. 1 and 2). Some of the main characteristics observed from the figure are the following: (i) For F = −100 kV/cm and F = +100 kV/cm, both the ground state and the first two excited states preserve their symmetries; something that is consistent with the absence of anticrossings between these states in Fig. 7, (ii) the states Ψ 2 and Ψ 3 are degenerate with p-like symmetry, (iii) when going from F = −100 kV/cm to F = +100 kV/cm the Ψ 5 and Ψ 6 states go to occupy the positions of the Ψ 4 and Ψ 5 , respectively, and the Ψ 4 state occupies the position of the Ψ 6 one, consistently with the Fig. 7. It can be noticed that, at F = 20 kV/cm, Ψ 4 exchanges symmetry with Ψ 6 , (iv) For F = −100 kV/cm and F = +100 kV/ cm, the numerical method used introduces a phase of ±π/4 for Ψ 2 and Ψ 3 states in the z = 0 plane. This phase is also present in the Ψ 10 and Ψ 11 states at F = −100 kV/cm and Ψ 12 and Ψ 13 states at F = +100 kV/cm, (v) when comparing Ψ 1 , Ψ 2 , and Ψ 3 , in Fig. 8(a,b), it is clearly seen how, in the first case, the states are displaced towards the pyramid apex while in the second case they are directed towards the pyramid basal plane, (vi) the presence of only one antinode in the z-direction of the Ψ 1 , Ψ 2 , and Ψ 3 states -given the odd symmetry of Ψ 2 and Ψ 3 with respect to a ±π/4 rotated plane =, ensures that over the entire range of applied electrical fields, there is a non-zero value of the matrix elements for xy-circularly polarized incident radiation, as will be seen below, and (vii) in general, for all excited states, the energy is higher at F = −100 kV/cm with respect to F = +100 kV/cm, due to the greater interaction with the lateral planes at the pyramid apex. Note that the electric field implies a remarkable change of the wave function characteristics. It can be affirmed that for negative electric field strengths, the electronic probability is distributed in a 3D-region whereas for sufficiently high positive electric fields, the spatial distribution of the states primarily locates nearby the pyramid basal plane. With the electric field, the system evolves from a three-dimensional quantum dot (for negative fields) to a two-dimensional quantum dot (for positive fields).
When going from negative to positive electric field values, the superposition between Ψ 1 and Ψ 2 (Ψ 1 and Ψ 3 ) states increases along with the increase in the spatial extent of the states. This justifies the ever increasing character of transition matrix elements ± M x iy 1,2 and ± M x iy 1,3 in Fig. 9(a). For ≅ F 100 kV/cm, a saturation effect of these matrix elements is observed due to the lateral potential barriers influence on the WFs. For F = 66 kV/cm, it is observed that the transitions Ψ 1 → Ψ 10 , Ψ 11 are transformed into Ψ 1 → Ψ 9 , Ψ 10 , which is in agreement with the crossing observed in Fig. 7 for such electric field value, at E = 96 meV. For circular polarization (see Fig. 9(a)) and F = −14 kV/cm only the Ψ 1 → Ψ 2 , Ψ 3 transitions are present, the other transitions are suppressed, this despite the fact that the symmetries in each z-plane are preserved with the electric field, but they change as the plane moves in that direction giving rise to contributions that cancel each other out. In Fig. 9(b), for z-polarized incident light, it is observed, for example, that the Ψ 1 → Ψ 4 transition is transformed into the Ψ 1 → Ψ 5 transition, and this finally becomes the Ψ 1 → Ψ 6 transition. This behavior is in agreement with the observed crossings between the Ψ 4 , Ψ 5 , and Ψ 6 states in Fig. 7 for F = −6.4 kV/cm and F = 19.5 kV/cm. This situation is also evident for other permitted transitions either with circular or linear incident polarized light, as shown in the two panels of Fig. 9. The increase of M z 1,4 in the negative range of applied electrical fields is due to the fact that initially, for F = −100 kV/cm, the www.nature.com/scientificreports www.nature.com/scientificreports/ maximum probability of both states is located at the apex of the pyramid; as F grows towards zero, the Ψ 1 state extends over the entire central pyramid while Ψ 4 remains almost static at the apex of the pyramid (note that the Ψ 4 state has two antinodes in the z-direction, which guarantees the non-null value of the matrix element). The curve reaches a maximum at F = −23 kV/cm where precisely the ground state has its maximum spatial distribution. For F > 0, where the character of the transition is Ψ 1 → Ψ 5 and then Ψ 1 → Ψ 6 , the decreasing behavior of M z 1,5 and M z 1,6 is due to the fact that the ground state is compressed towards the base of the pyramid and the excited state in question undergoes a progressive displacement towards the base of the pyramid as the field grows. Similar analysis, based on the distributions of the wave functions and their symmetries, explain the behavior of the other matrix elements.
In Figs. 10-12, we present the study of the applied magnetic field effects on the electronic states in a GaAs-Ga 0.7 Al 0.3 As PCSQD. The magnetic field is applied in the z-direction, which coincides with the symmetry axis of the heterostructure. This guarantees that the symmetry of the states in the different planes where z = const. is preserved. In Fig. 10, the energies for the first thirteen confined states are reported, in Fig. 11 the figures correspond  www.nature.com/scientificreports www.nature.com/scientificreports/ to results proportional to the dipole matrix elements considering circular and linear polarization for the incident radiation. Finally, Fig. 12 contains the projections of the wave functions on the z = 0 and y = 0 planes.
In Fig. 10 it is observed that the first relevant effect of the magnetic field is the breakdown of degeneracy for all reported states. Besides, much of the corresponding off-diagonal dipole moment matrix elements appear to be different to zero, as functions of B, as can be readily noticed from Fig. 11, and will be discussed below.
To interpret this situation, we resort to the wave functions and probability densities depicted in Fig. 12. Note that in Fig. 12(a), at zero magnetic fields, the states Ψ 2 and Ψ 3 (which correspond to real wave functions) have the same configuration of nodes and antinodes and are characterized by being rotated with respect to each other at an angle of 90°, taking the symmetry axis as the axis of rotation. This, as previously analyzed, explains the degeneration of the states. The same situation is valid for the Ψ 7 and Ψ 8 states of Fig. 12(a).
When the magnetic field is turned on (B = 30 T), one may observe that the wave functions become complex, with real and imaginary components, as represented in Fig. 12(b,c). Analyzing the Ψ 1 and Ψ 4 states (which at zero magnetic field correspond to the Ψ 2 and Ψ 3 states), it is observed that both the real and imaginary part of both states are displaced towards the base of the pyramid. It is also appreciated that while for Ψ 1 the real part of the wave function is always positive, in the case of Ψ 4 there are three regions of maximum positive and three regions of maximum negative contributions. In the case of the imaginary parts, the projections in the z = 0 plane show a positive maximum and a negative maximum for the Ψ 1 state while for Ψ 4 there are three positive and three negative maxima. The combination of the real and imaginary parts, which corresponds to the probability density (as shown in Fig. 12(d)), leads to the fact that the Ψ 1 state (of lower energy) is located in the region near the axis of the pyramid with a wide volumetric distribution of the probability density. In the same manner, for the Ψ 4 state (of larger energy), the electron tends to concentrate in a thin layer near the base of the pyramid, with well-defined maxima near the vertices of the square cross section. A similar situation is exhibited by the Ψ 7 and Ψ 8 states which, when the magnetic field is turned on until B = 30 T, evolve to become the Ψ 10 and Ψ 6 states, respectively.
Then, a second point to highlight in Fig. 10 is the presence of ground state oscillations as the magnetic field increases. Note from Fig. 11 that, for B = 30 T, the ground state (Ψ 1 ) has a real part whose symmetry coincides with that of the ground state (Ψ 1 ) at B = 0 and that the imaginary part of Ψ 1 at B = 30 T has the same p-like symmetry of Ψ 2 at B = 0. This explains the change in symmetry presented by the ground state at B = 21.2 T. As a third aspect, note also the presence of anticrossings between states which are induced by the magnetic field effects. Near B = 15 T, an anticrossing appears between the states that have been labeled as Ψ 3 and Ψ 7 at zero magnetic field. At B = 21 T there is another anticrossing, this time between Ψ 5 and Ψ 9 . Finally, Fig. 10 shows multiple accidental degeneracies. For example, the ground state presents accidental degeneracy at B = 21 T. What is most relevant to our investigation is that all these crossings or anticrossings between states are reflected in changes in the symmetry of the wave functions and, consequently, in changes in the selection rules for optical transitions between states.
This can be clearly seen in Fig. 11, where the squared absolute expected values of the dipole matrix elements with ξ = x ± iy (for circular polarization) and ξ = z (for linear polarization) are presented for transitions between the ground state and the first twelve excited states and between the first excited state and the next eleven excited states. Unlike the cases discussed in Figs. 6 and 9, here it has been necessary to include transitions from the first excited state given the crossing between Ψ 1 and Ψ 2 at B = 21 T. The complex character, with real and imaginary parts, of the wave functions, explains the absolutely different response presented by the system to light with right and left circular polarization, as seen from Fig. 11(a,b). Note that each line in the three panels of this figure is composed of transitions between multiple different states. Each open symbol indicates the change between states involved in transitions. Each open symbol appears for magnetic field and energy values that correspond to the www.nature.com/scientificreports www.nature.com/scientificreports/ crossings between states in Fig. 10. Looking at Fig. 11(c) for example, one may observe that, under field conditions, essentially four well-defined transitions appear while at B = 30 T only two transitions will be noticeable. This evidences a remarkable change in selection rules as the magnetic field grows. In the case of Fig. 11(a), at B = 0 there are four well-defined transitions that evolve into four others, but between different energy states.
The optical coefficients. In Fig. 13, the light absorption (a) and relative refraction index change (b) coefficients appear plotted as functions of the z-polarized incident photon energy and the applied magnetic field. The calculations considered the situation with zero applied electric field and kept constant the geometry and dimensions of the structure. Note that at B = 0, the peak of greater amplitude, both in the case of the absorption coefficient and the relative refractive index changes, is given for the 1 → 5 transition. This is consistent with the content of Fig. 11(c) where the most significant value of M z i j , at B = 0 is, precisely, given for the 1 → 5 transition. Besides, For B = 30 T, in Fig. 13(a) two peaks with approximately the same intensity are observed. This is despite the fact that in Fig. 11(c) the 1 → 9 transition has a matrix element smaller than that of the 2 → 7 transition, whose energy is lower than the one corresponding to the 1 → 9 transition.
Taking into account that the magnitude of α(ω) is proportional to the product | | E M fi z fi 2 , at B = 30 T the 2 → 7 and 1 → 9 transitions are proportional to 398.7 nm 2 meV and 401 nm 2 meV, respectively. Additionally, it can be seen from Fig. 13(a) that the 1 → 5 and 1 → 9 transitions, at B = 0 evolve to the 2 → 5 and 2 → 7 transitions at B = 30 T, which is consistent with the anti-crossing taking place near B = 20 T with E 65 = 80 meV between the Ψ 5 and Ψ 6 states.
On the other hand, when observing Fig. 13(b), one may see that the relative refractive index changes peak amplitudes exactly follow the behavior presented by those corresponding to the optical absorption coefficient in Fig. 13(a). This comes from the fact that for a particular i → j transition, the coefficient of relative refraction index change is an odd function with respect to the transition energy, E fi = E f − E i , the same at which the absorption coefficient shows the resonant peak structure. Also, it is clear that the Δn/n r coefficient has a maximum and a minimum localized at E p = E fi − ℏΓ and E p = E fi + ℏΓ, respectively. Additionally, taking into account that the magnitude of the two resonant peaks of Δn/n r are proportional to M z i j , 2 , the reason why the peaks of the 2 → 7 transition are significantly greater than the peaks of 1 → 9 at B = 30 T is explained.

conclusions
We have performed the investigation of electron states in core-shell pyramidal quantum dots considering the effect of externally applied electric and magnetic fields to the structure. The results of the calculation included modifications of the system size and geometry as well. Accordingly, we present a detailed discussion about the properties of energies and wave functions under different configurations, making emphasis in those related with the symmetry of states and how they are modified by the application of the external probes, showing both crossings and anticrossings in their evolution as functions of the field strengths. Regarding this, the study finds that a number of inter-state transitions can become forbidden, and the presence of an external probe, with its associated degeneracy breaking, activates some of them.
The information about the electronic structure allows to evaluate the coefficients of light absorption and relative refractive index change associated to allowed transitions between the lowest confined states. We comment