Double Quantum Ring under an Intense Nonresonant Laser Field: Zeeman and Spin-Orbit Interaction Effects

Abstract

We theoretically investigate the properties of an electron energy spectrum in a double GaAs-Al_0.3Ga_0.7As quantum ring by using the effective mass and adiabatic approximations, together with a realistic description of the confining potential profile, which is assumed to be deformed due to the application of an intense nonresonant laser field. The effects of the applied magnetic field and spin-orbit interaction are included. We discuss the features of the lowest confined energy levels under a variation of magnetic field strengths and intense laser parameters. The influence of this external probe on the linear optical absorption response associated with interlevel transitions is analyzed by considering both the presence and absence of spin-orbit effects.

1. Introduction

Due to their unique electronic and optical properties, semiconductor quantum rings (QRs) have shown great promise for various opto-electronic applications. These nanostructures are a potential building block for next-generation opto-electronic devices, offering advantages over traditional quantum dots and other semiconductor nanostructures. QRs can efficiently confine charge carriers, generating single photons by recombining an electron-hole pair. This property makes QRs excellent candidates for single-photon emitters. Studies have demonstrated the feasibility of utilizing QRs as efficient single-photon sources [1,2,3]. In addition to single-photon emitters, the unique energy levels in QRs can lead to low-threshold lasing, enabling the development of compact and energy-efficient lasers [4,5]. Furthermore, semiconductor QRs have shown promise for applications in photodetection and photovoltaics. The engineered energy spectrum in QRs can be tailored to match specific wavelengths, making them suitable for wavelength-tunable photodetectors [6,7]. As for photovoltaics, QRs have been studied for their potential to enhance light absorption and charge separation, which could lead to improved efficiency in solar cells [8,9]. Another emerging application of semiconductor QRs is in spintronics. Spin-orbit coupling in QRs can enable efficient manipulation and control of spin states, which is essential for spin-based electronic devices [10,11]. The effects of spin-orbit interaction (SOI) on the opto-electronic properties of QRs have also been investigated [12,13,14,15].

When considering a one-dimensional QR, the optically induced Aharonov-Bohm effect in mesoscopic rings has been reported by Sigurdsson and coworkers [16]. They show, theoretically, that strong electron coupling to circularly polarized photons in non-singly connected nanostructures results in an artificial gauge field that changes the electron phase, which is analogous to the well-known Aharonov-Bohm phase effect. Within the Floquet theory of periodically driven quantum systems, Kozin et al. studied the electronic properties of semiconductor quantum rings with the Rashba spin-orbit interaction irradiated by an off-resonant high-frequency electromagnetic field, finding a relevant modification of all the electronic characteristics of the rings under the dressing field effects (see, for instance, [17] and the references therein). The previous study has been extended to a two-dimensional QR array, demonstrating that a strong high-frequency circularly polarized electromagnetic field can turn a two-dimensional periodic array of interconnected QRs into a topological insulator [18].

Double semiconductor quantum rings (DSQRs) generally consist of two concentric rings with different bandgap materials, leading to interesting quantum confinement effects. The confinement in the radial and azimuthal directions gives rise to a rich energy spectrum and various optical transitions. Several research works have delved into the properties and potential applications of DSQRs. For instance, experimental realizations of these particular nanostructures have appeared at different moments. Kuroda and collaborators first reported on the fabrication of concentric DQRs and investigated the excitonic transitions in those structures [19,20]. Transport in Coulomb-coupled double rings under a magnetic field was measured in Ref. [21], whereas Kim reported on the optical transitions in GaAs-Al${}_{0.3}$Ga${}_{0.7}$As DQRs grown by droplet epitaxy [22].

Theoretically speaking, reports on electronic states in concentric DQRs were published early, for instance, by Li et al., within the framework of the effective mass envelope function approach [23], and by Planelles and Climente, including magnetic field effects [24]. The influence of an applied, intense nonresonant laser field on the electronic and optical properties of semiconductor DSQRs has been dealt in several previous works [25,26,27], and the systematic treatment on optical absorption in concentric DQRs was recently put forward [28].

Here, we aim to present a theoretical investigation of the electronic properties of double concentric QRs by considering the influence of externally applied nonresonant intense lasers and magnetic fields and the effect of spin-orbit coupling. Knowledge about electron states allows for an evaluation of the system’s interlevel linear optical absorption response. The adiabatic approximation is assumed to separate the vertical from the in-plane electron motions for the calculation. The solution for the latter is then tackled by using the finite element method. The article is organized as follows: Section 2 establishes the theoretical model used in this article. Section 3 presents the results, with and without considering SOI, along with the corresponding discussion. Section 4 summarizes the main findings.

2. Theoretical Framework

The system under study is a GaAs double ring embedded in an Al${}_{0.3}$Ga${}_{0.7}$As matrix under the combined effects of a magnetic field in the z-direction and a nonresonant laser field (ILF). The geometry shown in Figure 1a was generated from the profile data of the AFM measurements from the work of Kuroda et al. [19].

Under the effects considered, the Schrödinger equation for a confined electron can be expressed by

$$H\mathit{\psi }(x,y,t)=i\hslash I_2\frac{\partial }{\partial t}\mathit{\psi }(x,y,t),$$

(1)

where $\mathit{\psi }=\left(\begin{array}{c}{\psi }_{\uparrow }\\ {\psi }_{\downarrow }\end{array}\right)$ is the wave function in a vector, with the components representing spin up and spin down. $H=\left(\begin{array}{cc}{H}_{11}& H_{12}\\ H_{21}& H_{22}\end{array}\right)$, ℏ is the reduced Planck constant, and $I_2$ is the $2\times 2$ identity matrix. The Hamiltonian can be written as the sum of several contributions:

$$H=H_0{I}_2+\frac{1}{2}g\mu B{\sigma }_z+H_R+H_D$$

(2)

where

$$H_0=\frac{{\hslash }^2{k}^2}{2m_c^{*}}+V\left(h(x,y)\right).$$

(3)

Here, $h(x,y)$ is the height, and it is a function of the position in the $(x,y)$-plane. Note that in Equation (3), the confinement potential is a function of the QR’s height, which is a function of the coordinates in the $(x,y)$-plane. In this case, we use the adiabatic approximation to convert a 3D problem into a 2D problem, where the confinement potential in the ring region is given by the energy associated with a quantum well of width h with finite potential barriers (the height of the barrier in the whole space is obtained from the product between the band offset and the gap difference between the well and barrier materials). By using the adiabatic approximation, in this study, we propose that the ring height along the z-axis in all of the space is generally much smaller with respect to the dimensions of the ring in the $(x,y)$-plane. More details on the method can be viewed in [29]. In other words, the fast movement occurs along the z-direction, whereas the slow movement occurs along the $(x,y)$-plane.

Since we have cylindrical symmetry, equal radii $\sqrt{x^2+y^2}$ have the same height, $\mathbf{k}=-i\nabla +\frac{e}{\hslash }({\mathbf{A}}_m+{\mathbf{A}}_l)$, g is the Landé factor, ${\mu }_B$ is the Bohr magneton, B is magnetic Field, ${\sigma }_j(j=x,y,z)$ are the j component of the Pauli matrices vector, $m_c^{*}$ is the effective mass with $c=w,b$ for the GaAs double quantum ring and Al${}_{0.3}$Ga${}_{0.7}$As matrix, respectively. The terms $H_R$ and $H_D$ are the Rashba and Dresselhaus spin-orbit interactions, respectively.

This system is taken to be under external magnetic and nonresonant intense laser field (ILF) effects. The incident ILF was defined for elliptical and linear polarizations. The vector potential of the laser field is given by

$${\mathbf{A}}_l\left(t\right)=\left(\mathbf{x}cos\left(\omega t\right)+\kappa \mathbf{y}cos\left(\omega t+\frac{\pi }{2}\right)\right)A_0,$$

(4)

where $\mathbf{x}$ and $\mathbf{y}$ are the unit vectors, $A_0$ is the amplitude, $\omega$ is the frequency of the laser field, and $\kappa =0,1$ is the linear and right circularly polarizations, respectively.

For the applied magnetic field, $\mathbf{B}=B\mathbf{z}$. It uses the symmetry gauge ${\mathbf{A}}_m=\frac{B}{2}(y\mathbf{x}-x\mathbf{y})$, with $\mathbf{B}=-\nabla \times {\mathbf{A}}_m$.

When applying the Kramers-Henneberger transformation [30] to Equation (3), it becomes a time-independent Schrödinger equation

$$H^{\prime }\mathit{\varphi }(x,y)=E{I}_2\mathit{\varphi }(x,y),$$

(5)

where the $H_0$ term becomes

$$H_0^{\prime }=\frac{{\hslash }^2{\nabla }_{x,y}^2}{2m_c^{*}}+i\frac{eB\hslash }{2m_c^{*}}\left(y\mathbf{x}-x\mathbf{y}\right)·\nabla +V_d(x,y).$$

(6)

The term $V_d$ is the laser-dressed potential, and this is calculated as an average of the potential by means of the following integral:

$$V_d=\frac{e^2{B}^2}{8}{\int }_0^{\frac{2\pi }{\omega }}\frac{(X^2\left(t\right)+Y^2\left(t\right))}{m_c^{*}}dt+\frac{\omega }{2\pi }{\int }_0^{\frac{2\pi }{\omega }}V(X\left(t\right),Y\left(t\right))dt,$$

(7)

where $X\left(t\right)=x+c_x{\alpha }_{0}cos\left(\omega t\right)$, $Y\left(t\right)=y+c_y{\alpha }_{0}cos(\omega t+\pi /2)$, and

$$c_x=\left\{\begin{array}{cc}1\hfill & \mathrm{linear}\hfill \\ \frac{1}{\sqrt{2}}\hfill & \mathrm{circular}\hfill \end{array}\right.c_y=\left\{\begin{array}{cc}0\hfill & \mathrm{linear}\hfill \\ \frac{1}{\sqrt{2}}\hfill & \mathrm{circular}.\hfill \end{array}\right.$$

(8)

Here, the last coefficients were defined by convenience to tune the ILF-laser parameter intensity ${\alpha }_0$ for the two polarizations, linear and circular, according to the definitions in Equation (8). Equation (7) is solved via numerical integration. The laser parameter ${\alpha }_0$ is expressed in terms of the geometrical domains ${\alpha }_0^{in}=\frac{eA}{\sqrt{2}{m}_{in}^{*}\omega }\sim \frac{\sqrt{I_0}}{{\omega }^2}$ ($I_0$ is the laser intensity), and ${\alpha }_0^{out}={\alpha }_0^{in}\frac{m_{in}^{*}}{m_{out}^{*}}$ are the inner (GaAs) and outer (AlGaAs) laser parameters of the DQR, respectively. For the laser dressing potential, see the Appendix A at the end.

The terms Rashba and Dresselhaus are obtained from the following two expressions

$$H_R=\alpha {(\mathit{\sigma }\times \mathbf{k})}_z=\alpha ({\sigma }_x{k}_y-{\sigma }_y{k}_x),H_D=\beta ({\sigma }_x{k}_x-{\sigma }_y{k}_y)$$

(9)

When expanding these terms, each of the elements of the matrix H is obtained.

$$\begin{array}{cc}\hfill & H_{11}=H_0+\frac{1}{2}g{\mu }_{B}B\hfill \\ \hfill & H_{22}=H_0-\frac{1}{2}g{\mu }_{B}B\hfill \\ \hfill & H_{12}=(\alpha -i\beta )\frac{\partial }{\partial x}+(\beta -i\alpha )\frac{\partial }{\partial y}+\alpha \frac{eB}{2\hslash }(x-iy)-\beta \frac{eB}{2\hslash }(y-ix)\hfill \\ \hfill & H_{21}=-(\alpha +i\beta )\frac{\partial }{\partial x}-(\beta +i\alpha )\frac{\partial }{\partial y}+\alpha \frac{eB}{2\hslash }(x+iy)-\beta \frac{eB}{2\hslash }(y+ix)\hfill \end{array}$$

(10)

In order to evaluate the optical response of the system, the following expression for the linear absorption coefficient was used:

$${\theta }_{jl}=\omega \sqrt{\frac{{\mu }_0}{{\epsilon }_0\epsilon }}\frac{{\left|M_{jl}\right|}^2{e}^2{\rho }_{jl}\hslash {\Gamma }_{jl}}{{(E_{jl}-\hslash \omega )}^2+{(\hslash {\Gamma }_{jl})}^2}$$

(11)

In this article, the expression for calculating the matrix elements is given by

$$M_{jl}=\int \int \left[{\left({\varphi }_{\uparrow }^{*}\right)}_j\xi {\left({\varphi }_{\uparrow }\right)}_l+{\left({\varphi }_{\downarrow }^{*}\right)}_j\xi {\left({\varphi }_{\downarrow }\right)}_l\right]dxdy$$

(12)

These elements of the transition from state l to j were calculated for different types of polarization, linear ($\xi =x$), right circular $\left(\xi =\frac{x+iy}{\sqrt{2}}\right)$, and left circular $\left(\xi =\frac{x-iy}{\sqrt{2}}\right)$.

Finally, to solve for the uncoupled $xy$-motion of the electrons in the DQR, we used the finite element method, as implemented in the COMSOL Multiphysics software package [31,32,33].

3. Results and Discussion

The parameters used in this study are ${\mu }_0=1.257\times {10}^{-6}$ Tm/A, $m^{*}\left(GaAs\right)=0.067m_0$, $m^{*}\left(A{l}_{x}G{a}_{1-x}As\right)=0.093m_0$, $V\left(A{l}_{x}G{a}_{1-x}As\right)=262$ meV, $\epsilon =12.58$, $\hslash \Gamma =0.5$ meV, Landé factor $g=-2.15$, Rashba coefficient $\alpha =5.4$ meV·nm, and Dresselhaus coefficient $\beta =10.8$ meV·nm. Here, $m_0$ is the free electron mass.

Figure 1 shows a diagram of the GaAs-AlGaAs double quantum ring, with the profile taken from the work of Kuroda et al. [19] (in that study, the profile was obtained through AFM characterization measurements). An azimuthal symmetry was assumed, which allowed for a reduction in computational cost since the solution in the azimuthal angle co-ordinate is known analytically. Thus, the problem is reduced in numerical terms to the solution of a two-variable differential equation. A three-dimensional diagram of the double ring is shown in panel 1(a), where it can be seen that the variable height along the radial direction is small compared to the dimensions of the (x, y)-plane, which allows for obtaining adequate solutions by using the adiabatic approximation. In panel 1(a), the magnetic field vector (in the z-direction) is also shown, as well as two more vectors representing the polarization of the intense nonresonant laser, that is, in the x-direction (linear) and right circular. A cross-section of the confinement potential due to the variable height, without and with the presence of a magnetic field, B = 0 and 5 T, respectively, is presented in panel 1(b). It can be observed that the magnetic field generates significant changes in confinement in the outer ring. In panel 1(c), the laser-dressed potential for the two values of the laser parameter and in the absence of the fields for comparison purposes is reported. The value of alpha0 = 10 nm generates a lower barrier (approx. 190 meV) when compared to 262 meV without laser effects, and this also eliminates the double ring behavior of the geometry. For the higher laser field, a double ring appears with the appearance of a barrier in the area where, fundamentally, there is greater height for the first ring. These effects exhibited under the presence of the laser allow for treating the system as if it has another geometry (reference article).

The energies of a double GaAs-AlGaAs ring as a function of the magnetic field, both with and without SOI effects, are shown in Figure 2a and Figure 2b, respectively. In Figure 2a, the electron is confined in the outer ring, which is precisely the one that presents changes in its effective potential due to the magnetic field, as observed in Figure 1b, where the increase in potential generally leads to higher energy, with oscillations between states. In Figure 2b, with SOI activated, the states for spin up and down are differentiated. From Equation (2), it can be seen that the Zeeman term only appears in the presence of a magnetic field, unlike the SOI Rashba and Dresselhaus terms, which appear even without a magnetic field. This is evident in Figure 2b, where for B = 0, the first excited state of Figure 1b appears as two distinct states, even in the absence of a magnetic field. The inclusion of a magnetic field further accentuates the SOI effects; for example, the ground state is quasi-degenerate for very small magnetic fields (B <= 0.5). As the intensity of the field increases, the separation of the states becomes more pronounced, including the appearance of oscillations between the ground state and the first excited state. The complexity of the spectrum under spin effects is not only due to the emergence of new states or breaking degeneracies; note that for Figure 2a, as the magnetic field increases, the ground state has a symmetry that comes from higher states in the absence of a field. In contrast, when spin is taken into account, the ground and first excited states are separated from the higher states, indicating that the symmetry for these two states does not become that of the higher energy states. This means that spin-orbit interaction not only causes changes in energy values but also exhibits important qualitative changes.

Figure 3 displays the energy as a function of the magnetic field in a double GaAs/AlGaAs ring, both without (first column) and with (second column) spin effects. Additionally, the presence of an intense nonresonant laser with linear polarization (x-direction) is shown for two laser parameter values: 10 nm (first row) and 20 nm (second row). Using an ILF with linear polarization destroys the cylindrical symmetry of the problem, which is most evident in the lowest states, where it is observed that, for instance, when the laser intensity is increased (Figure 3c,d), more states lose the characteristic oscillation that occurs in the absence of a laser. These changes in the energy spectra are equivalent to the changes in the geometry of the rings, but in this case, they result from the change in confinement created by the laser. Figure 3d shows that while spin effects differentiate the states as the field increases, a large value of the laser parameter inhibits the separation between the lowest energy states.

In Figure 4, energy is plotted as a function of the laser parameter, which is related to the intensity of the electromagnetic wave used. In addition, a magnetic field of 5 T was applied in the z-direction. In order to observe the changes induced by the spin in the spectrum, it is reported both in the absence (a) and under the effects of spin (b). The selected polarization increases the values of the barriers so that, as the laser parameter increases, confinement is induced for the lowest states along the y-axis. In Figure 4a, it can be seen that a higher laser intensity favors the separation of some states, which, in the absence of ILF, are quasi-degenerate. When spin interaction effects are included (Figure 4b), it becomes evident that for higher values of the laser parameter, the effect of ILF predominates over the state separation caused by spin. This is evident in the spectrum for the states that are separated in the absence of ILF and then come together when the laser parameter is increased.

Figure 5 displays the energy as a function of the magnetic field for a double GaAs/AlGaAs ring, both without (first column) and with (second column) spin effects. Additionally, the presence of an intense nonresonant laser with right circular polarization is shown for two values of the laser parameter: 10 nm (first row) and 20 nm (second row). These results highlight that the inclusion of ILF with circular polarization does not result in any loss of symmetry in the structure, unlike when linear polarization is included, which causes a loss of azimuthal symmetry. As such, the typical oscillations of the states as a function of the magnetic field observed in quantum rings are preserved with circular polarization. In accordance with the effective potential generated by the ILF, the energies are higher when the laser parameter is increased, and a lower density of states per unit energy is also observed. This has important implications for potential applications since the intraband transition energies can be fine-tuned through the inclusion of ILF.

In Figure 6, the squared matrix elements for transitions 1–2 and 1–3 are displayed as a function of the magnetic field in the first and second columns, respectively. The first row shows a calculation that incorporates both the SOI and the Zeeman effects, whereas the second row shows a calculation without the spin effect. In each of the four panels of Figure 6, we can observe the complementarity between right and left circular polarization since, in general, when one is activated, the other loses its magnitude. For the specific values of the magnetic field, there is even a drastic shift from one to the other. Note that in Figure 6a,b, the matrix elements either strengthen or weaken gradually, exhibiting behavior closer to what is expected for this phenomenon. The changes are only abrupt at specific values of the magnetic field; for example, in Figure 6a, at 4.1 T, there is an activation/deactivation of the transition depending on whether the polarization of the incident beam is right or left circular. In Figure 6c,d, when the contribution of spin is not included, the value of the matrix elements remains practically constant with the increase in the magnetic field, only it is observed that the type of circular polarization activates the transition for different ranges of the field. In Figure 6b, we observe a loss of complementarity between the two polarizations used since transition 1–3 becomes only a part of the structure of a more complex spectrum with the appearance of more states due to spin interaction. These results allow us to highlight the importance of including spin effects in this type of work since the quantitative and qualitative changes are not negligible.

Finally, in Figure 7, the results obtained for the linear absorption coefficient due to the 1–2 transition are reported without the inclusion of spin interaction (Figure 7a) and with the contribution of spin interaction (Figure 7b). Three values of the magnetic field were taken, which were applied in the z-direction, B = 0, 2.5, 5, T. The continuous lines represent right circular polarization, and the dashed lines represent left circular polarization. In Figure 7a, it is highlighted that in the absence of a magnetic field, the result is the same for any of the polarizations, no matter the kind of polarization. In contrast, when the SOI and Zeeman effects are included, in the absence of a magnetic field $B=0$, the results are different depending on the polarization. Note that the absorption values are reduced when the effects of spin-orbit interaction are considered.

4. Conclusions

We have investigated the behavior of electron energy levels in a double GaAs-AlGaAs quantum ring within the effective mass approximation through a realistic confining potential—which was taken from experimental reports—and dressed by the influence of an intense nonresonant laser field. The model also assumes the presence of an externally applied magnetic field and includes the effects of Zeeman and spin-orbit interaction. The latter is considered both in the Rashba and Dresselhaus formulations.

As expected, spin-orbit coupling causes a splitting of the degenerate levels, even at a zero magnetic field, but keeps the ground state oscillations, which actually disappear when an intense laser field acts upon the system. A linearly polarized laser field also removes some degeneracies, but in combination with spin-orbit interaction, the spectrum becomes completely resolved. The application of a circularly polarized laser field keeps the oscillatory variation of the ground state as a function of the magnetic field while raising up the whole spectrum by almost 10 meV.

Spin-orbit influence is responsible for a significant modification of the electric dipole moment off-diagonal matrix elements that are associated with transitions from the ground to the first two excited states. This is reflected in the linear optical absorption coefficient in the form of a noticeable reduction in resonant peak amplitude.