Quantum Rings in Electromagnetic Fields

Abstract

This chapter is devoted to optical properties of so-called Aharonov-Bohm quantum rings (quantum rings pierced by a magnetic flux resulting in Aharonov-Bohm oscillations of their electronic spectra) in external electromagnetic fields. It studies two problems. The first problem deals with a single-electron Aharonov-Bohm quantum ring pierced by a magnetic flux and subjected to an in-plane (lateral) electric field. We predict magneto-oscillations of the ring electric dipole moment. These oscillations are accompanied by periodic changes in the selection rules for inter-level optical transitions in the ring allowing control of polarization properties of the associated Terahertz radiation. The second problem treats a single-mode microcavity with an embedded Aharonov-Bohm quantum ring which is pierced by a magnetic flux and subjected to a lateral electric field. We show that external electric and magnetic fields provide additional means of control of the emission spectrum of the system. In particular, when the magnetic flux through the quantum ring is equal to a half-integer number of the magnetic flux quanta, a small change in the lateral electric field allows for tuning of the energy levels of the quantum ring into resonance with the microcavity mode, thus providing an efficient way to control the quantum ring-microcavity coupling strength. Emission spectra of the system are discussed for several combinations of the applied magnetic and electric fields.

Appendix. Analytical Solutions for Small Matrices.

In the limit of weak electric field, $\beta =eER/(\hbar ^2/M_{e}R^2) \ll 1$, the electron ground, first and second excited states are well-described by the following three-by-three system, which is obtained from (13.54) for $\left| m \right| \le 1$

$$\begin{aligned} \begin{pmatrix} \left( f+1 \right) ^{2} &{} \beta &{} 0 \\ \beta &{} f^{2} &{} \beta \\ 0 &{} \beta &{} \left( f-1 \right) ^{2} \end{pmatrix} \begin{pmatrix} c_{+1}^{n} \\ c_{0}^{n} \\ c_{-1}^{n} \end{pmatrix} = \lambda _{n} \begin{pmatrix} c_{+1}^{n} \\ c_{0}^{n} \\ c_{-1}^{n} \end{pmatrix} \text{. } \end{aligned}$$

(13.89)

Here $f=(\varPhi -N\varPhi _0)/\varPhi _0$ with N integer, so that $0 \le f \le 1/2$. The eigenvalues $\lambda _{n}$ of the system (13.89) are the roots of the cubic equation

$$\begin{aligned} \lambda _{n}^{3} - \lambda _{n}^{2} \left( 3 f^{2} + 2 \right) + \lambda _{n} \left( 3 f^{4} + 1 - 2 \beta ^{2} \right) -f^{6} +2f^{4} -f^{2} + 2 f^{2} \beta ^{2} + 2 \beta ^{2} =0 \text{. } \end{aligned}$$

(13.90)

Solving (13.90) we find

$$\begin{aligned} \lambda _{1} = - 2/3 \sqrt{1+ 12 f^{2} +6 \beta ^2} \cos \left( \alpha /3 \right) + f^{2}+2/3 \text{, } \end{aligned}$$

(13.91)

$$\begin{aligned} \lambda _{2} = - 2/3 \sqrt{1+ 12 f^{2} +6 \beta ^2} \cos \left( \alpha /3 - 2 \pi /3 \right) + f^{2}+2/3 \text{, } \end{aligned}$$

(13.92)

$$\begin{aligned} \lambda _{3} = - 2/3 \sqrt{1+ 12 f^{2} +6 \beta ^2} \cos \left( \alpha /3 + 2 \pi /3 \right) + f^{2}+2/3 \text{, } \end{aligned}$$

(13.93)

with

$$\cos {\alpha }=\frac{ 1 - 36 f^2 + 9 \beta ^{2}}{ \left( 1 + 12 f^{2} +6 \beta ^2 \right) ^{3/2}} \text{. }$$

Considering $\beta \ll 1$ (the limit of weak electric field) we expand (13.91–13.93) into Taylor series in f to obtain

$$\begin{aligned} \lambda _{1} = f^2 - 2 \beta ^2 \sum \limits _{n=0}^{\infty } \left( 2f \right) ^{2n} + O(\beta ^4) \text{, } \end{aligned}$$

(13.94)

$$\begin{aligned} \lambda _{2} = 1+f^{2}+\beta ^{2} \left[ 1- \sum \limits _{n=0}^{\infty } \frac{\left( -1 \right) ^{n} \left( 2n \right) !}{\left( 1-2n \right) \left( n! \right) ^{2}} \left( \frac{f}{\beta ^{2}} \right) ^{2n} \right] + O(\beta ^4)\text{, } \end{aligned}$$

(13.95)

$$\begin{aligned} \lambda _{3} = 1+f^{2}+\beta ^{2} \left[ 1+ \sum \limits _{n=0}^{\infty } \frac{\left( -1 \right) ^{n} \left( 2n \right) !}{\left( 1-2n \right) \left( n! \right) ^{2}} \left( \frac{f}{\beta ^{2}} \right) ^{2n} \right] + O(\beta ^4)\text{. } \end{aligned}$$

(13.96)

It can be shown that (13.95, 13.96) coincide with the results of the perturbation theory in eER for quasi-degenerate states [96] if the coupling to the states with $|m|>1$ is neglected.

The energy spectrum given by (13.91–13.93) is plotted in Fig. 13.19. It is nearly indistinguishable from the energy spectrum, which was obtained by numerical diagonalization of the $23 \times 23$ system in Sect. 13.4 for the same value of $\beta$. A small discrepancy between the plotted energy spectra is noticeable only for the first and second excited states. The energy spectrum obtained by numerical diagonalization of the $23 \times 23$ system is slightly shifted towards the smaller energies. This shift occurs because the considered $3 \times 3$ matrix does not take into account the coupling between the $m=\pm 1$ and $m=\pm 2$ states. For the infinite system and $f=0$, perturbation theory up to the second order in $\beta$ yields

$$\begin{aligned} \lambda _{1}=-2 \beta ^{2} \text{, } \lambda _{2}=1 - \beta ^{2}/3\text{, } \lambda _{3}=1 + 5\beta ^{2}/3\text{, } \end{aligned}$$

(13.97)

whereas from (13.94–13.96) one gets

$$\begin{aligned} \lambda _{1}=-2 \beta ^{2} \text{, } \lambda _{2}=1 \text{, } \lambda _{3}=1 + 2\beta ^{2} \text{. } \end{aligned}$$

(13.98)

The $\lambda _{2}$ and $\lambda _{3}$ values in (13.97) differ from the values in (13.98) by $-\beta ^{2}/3$ which corresponds to the repulsion between the $m=\pm 1$ and $m=\pm 2$ states calculated using the second order perturbation theory.

Fig. 13.19

The normalized energy spectrum as a function of dimensionless parameter f for $\beta =0.1$. Dashed line—the result of analytical solution of the $3 \times 3$ system. Solid line—the result of numerical diagonalization of the $23 \times 23$ system. A horizontal line is shown to indicate $\lambda =0$ value

When $f=1/2$, and in the absence of a lateral electric field, the $m=0$ and $m=-1$ states are degenerate with energy $\varepsilon _{1} \left( 0 \right) /4$, i.e. $\lambda _1=\lambda _2=1/4$, whereas the $m=+1$ state energy is nine times larger ($\lambda _3=9/4$). The contribution from this remote state can be neglected, and the electron ground and first excited states are well-described by the following two-by-two system, which contains $c_{-1}$ and $c_{0}$ coefficients only,

$$\begin{aligned} \begin{pmatrix} f^{2} &{} \beta \\ \beta &{} \left( f-1 \right) ^{2} \end{pmatrix} \begin{pmatrix} c_{0}^{n} \\ c_{-1}^{n} \end{pmatrix} = \lambda _{n} \begin{pmatrix} c_{0}^{n} \\ c_{-1}^{n} \end{pmatrix} \text{. } \end{aligned}$$

(13.99)

The eigenvalues $\lambda _{n}$ of the system (13.99) are the roots of the quadratic equation

$$\begin{aligned} \lambda _{n}^2 -\lambda _{n} \left( 2 f^2 - 2f +1 \right) +f^{4} -2f^{3} +f^{2} -\beta ^2 =0 \text{. } \end{aligned}$$

(13.100)

Solving (13.100) we find

$$\begin{aligned} \lambda _{1,2}= f^{2}-f+1/2 \mp \sqrt{f^{2}-f + \beta ^{2} + 1/4} \text{, } \end{aligned}$$

(13.101)

yielding for $f=1/2$ the eigenvalue difference $\lambda _2-\lambda _1=2\beta$, corresponding to the energy splitting of eER as expected from the perturbation theory for degenerate states. The energy spectrum given by (13.101) is plotted in Fig. 13.20 together with two lowest eigenvalues of the $23 \times 23$ system demonstrating a spectacular accuracy of the approximate solution for $\beta =0.1$.

Let us now return to the three-by-three matrix and examine how its eigenvectors are modified with changing f. Near the point $f=0$ it is convenient to write the eigenvectors of the system (13.89) in the following form

$$\begin{aligned} \begin{pmatrix} c_{+1}^{n} \\ c_{0}^{n} \\ c_{-1}^{n} \end{pmatrix} = A_{n} \begin{pmatrix} \left[ \lambda _{n} - \left( f -1 \right) ^{2} \right] \left( \lambda _{n} -f^2 \right) - \beta ^{2} \\ \left[ \lambda _{n} - \left( f-1 \right) ^{2} \right] \beta \\ \beta ^{2} \end{pmatrix} \text{, } \end{aligned}$$

(13.102)

where $A_{n}$ denotes the normalization constant corresponding to the eigenvalue $\lambda _n$ and (13.102) is valid only for $\beta \ne 0$. For $f=0$ in the limit of weak electric field ($\beta \ll 1$) we obtain

$$\begin{aligned} \begin{pmatrix} c_{+1}^{1} \\ c_{0}^{1} \\ c_{-1}^{1} \end{pmatrix} = \frac{ \left( 1+ 1 \sqrt{1+8 \beta ^{2}} + 8 \beta ^{2} \right) ^{-1/2}}{\sqrt{2}} \begin{pmatrix} - 2 \beta \\ 1 + \sqrt{1+8 \beta ^{2}} \\ - 2 \beta \end{pmatrix} {\mathop {\longrightarrow }\limits ^{\beta \rightarrow 0}} \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \text{, } \end{aligned}$$

(13.103)

$$\begin{aligned} \begin{pmatrix} c_{+1}^{2} \\ c_{0}^{2} \\ c_{-1}^{2} \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} - 1 \\ 0 \\ 1 \end{pmatrix} \text{, } \end{aligned}$$

(13.104)

$$\begin{aligned} \begin{pmatrix} c_{+1}^{3} \\ c_{0}^{3} \\ c_{-1}^{3} \end{pmatrix} = \frac{ \left( 1- 1 \sqrt{1+8 \beta ^{2}} + 8 \beta ^{2} \right) ^{-1/2}}{\sqrt{2}} \begin{pmatrix} 2 \beta \\ \sqrt{1+8 \beta ^{2}} - 1 \\ 2 \beta \end{pmatrix} {\mathop {\longrightarrow }\limits ^{\beta \rightarrow 0}} \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} \text{. } \end{aligned}$$

(13.105)

From (13.103–13.105) one can see that for $f=0$ and $\beta \ll 1$ the electron ground state is almost a pure $m=0$ state, whereas the angular dependencies of the wavefunctions of the first and second excited states are well-described by $\sin \varphi$ and $\cos \varphi$ respectively.

Fig. 13.20

The normalized energy spectrum as a function of dimensionless parameter f for $\beta =0.1$. Dashed line - the result of analytical solution of the $2 \times 2$ system. Solid line - the result of numerical diagonalization of the $23 \times 23$ system. A horizontal line is shown to indicate $\lambda =0$ value

The structure of eigenfunctions near $f=1/2$ is best understood from (13.99), which yields

$$\begin{aligned} \begin{pmatrix} c_{0}^{1} \\ c_{-1}^{1} \end{pmatrix} = A \begin{pmatrix} \beta \\ 1/2 - f - \sqrt{f^{2}-f + \beta ^{2} + 1/4} \end{pmatrix} \text{, } \end{aligned}$$

(13.106)

$$\begin{aligned} \begin{pmatrix} c_{0}^{2} \\ c_{-1}^{2} \end{pmatrix} = A \begin{pmatrix} f - 1/2 + \sqrt{f^{2}-f+ \beta ^{2} + 1/4 } \\ \beta \end{pmatrix} \text{. } \end{aligned}$$

(13.107)

Here A is the normalization constant and $\beta \ne 0$. For $f=1/2$ we get

$$\begin{aligned} \begin{pmatrix} c_{0}^{1} \\ c_{-1}^{1} \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ - 1 \end{pmatrix} \text{, } \begin{pmatrix} c_{0}^{2} \\ c_{-1}^{2} \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \text{. } \end{aligned}$$

(13.108)

From (13.108) one can see that for $f=1/2$ the angular dependencies of the ground and first excited states wavefunctions are described by $\sin \left( \varphi /2 \right)$ and $\cos \left( \varphi /2 \right)$ respectively.

Fig. 13.21

Magnetic flux dependence of the wavefunction coefficients $\left| c_{0} \right| ^{2}$ (solid line), $\left| c_{-1} \right| ^{2}$ (dotted line), and $\left| c_{+1} \right| ^{2}$ (dashed line): a for the ground state; b for the first excited state; c for the second excited state

Figure 13.21 shows the magnetic flux dependencies of the coefficients $\left| c_{0} \right| ^{2}$, $\left| c_{-1} \right| ^{2}$, and $\left| c_{+1} \right| ^{2}$ for the electron ground, first and second excited states. From these plots one can see that the electron ground state is almost a pure $m=0$ state in a wide region $0 \le f \lesssim 1/4$. An admixture of the $m=-1$ wavefunction increases smoothly as one approaches the point of degeneracy $f=1/2$. Finally, when $f=1/2$, the ground state wavefunction is expressed as a difference of the $m=-1$ and $m=0$ wavefunctions. The first and the second excited states behave differently. In a small region near the point $f=0$ the electron first and second excited states wavefunctions consist of a strong mixture of the $m=-1$ and $m=+1$ functions with a tiny admixture of the $m=0$ function. In particular, when $f=0$ the first and second excited states eigenfunctions with good accuracy can be expressed as the difference and the sum of the $m=-1$ and $m=+1$ functions respectively. Optical transitions between these states and the ground state are only allowed if the polarization of the associated optical excitations is either perpendicular (for the first excited state) or parallel (for the second excited state) to the direction of the applied in-plane electric field. Away from the $f=0$ region, only the coefficient $c_{-1}$ (in the case of the first excited state) or $c_{+1}$ (in the case of the second excited state) remains in the (13.102), which now describes almost pure $m=+1$ and $m=-1$ states. When f exceeds 1 / 4 the first excited state starts to contain a noticeable ad-mixture of $m=0$ function, as discussed above, and for $f=1/2$ the first excited state eigenfunction is expressed as a sum of the $m=-1$ and $m=0$ wavefunctions in equal proportions, whereas the second excited state remains an almost pure $m=+1$ state.

The same trend in the evolution of wavefunctions of the three lowest energy states with changing the flux through the ring can be seen from perturbation theory. For $f=0$, the degeneracy between the first and second excited states is removed in the second order in eER only. Nevertheless, as a result of the degeneracy, the introduction of any weak perturbation drastically modifies the wavefunctions corresponding to these states, turning them from the eigenstates of the angular momentum operator to the sine and cosine functions. With a slight increase of f, so that $f>\beta ^2$, the first and the second excited states, which are not degenerate anymore for $f \ne 0$, become governed mainly by the diagonal terms of the Hamiltonian, which do not mix the $m=-1$ and $m=+1$ functions. When $f=1/2$, the $m=-1$ and $m=0$ states are degenerate in the absence of the electric field. This degeneracy is removed in the first order in eER. The off-diagonal matrix elements connecting $m=-1$ and $m=0$ functions remain of the same order of magnitude as the difference between the diagonal terms of the Hamiltonian across a broad range of f values near $f=1/2$. This results in strong mixing of the $m=-1$ and $m=0$ components in the eigenfunctions of the ground and first excited states for $1/4 \lesssim f \le 1/2$.