Spin Resonance in a Quantum Dot at the Edge of a Topological Insulator with the Inclusion of Continuum States

Abstract

The dynamics of electronic states under the action of a periodic electric field applied to a quantum dot created by magnetic barriers at the one-dimensional edge of a two-dimensional topological insulator based on a HgTe/CdTe quantum well has been studied. A configuration with two discrete levels is considered and transitions to continuum states above barriers are taken into account. The frequency of oscillations of the discrete level populations has been calculated for various field amplitudes. It has been shown numerically and analytically that the inclusion of the continuous spectrum leads to a decrease in the total population of discrete levels in time. The characteristic times of this decrease corresponding to transitions to continuum have been determined. The dynamics of average values of the energy, spin projections, coordinates, local probability density, and local spin density has been calculated. Time-averaged local probability density current that describes the quantum dot escape at transitions to the continuous spectrum has been calculated. The results of this work can be useful for the design of new generations of nanoelectronics and spintronics devices based on topological insulators.

Appendices

APPENDIX A

Here, we discuss the derivation of the Hamiltonian (2) for the interaction of edge states with magnetic barriers. We use the results of work [11], where the interaction Hamiltonian was obtained in the basis of edge states, i.e., two-component spinors that are the eigenfunctions of the Hamiltonian (1). The authors of [11] studied the interaction of the spin with a magnetic impurity, which can be considered as a strongly localized magnetic barrier. The spin part of the exchange interaction operator is independent of the spatial profile of the interaction potential if it remains sufficiently short-range, which is also true for our model. We use Eq. (6) from the supplementary material in [11] for the interaction operator H_ex with the magnetic impurity (barrier):

$${{H}_{{{\text{ex}}}}} = \left( {\begin{array}{*{20}{c}} {{{J}_{z}}{{S}^{z}} + {{J}_{{{\text{an}}}}}{{S}^{ - }} + J_{{{\text{an}}}}^{*}{{S}^{ + }}}&{J_{ \bot }^{*}{{S}^{ - }}} \\ {{{J}_{ \bot }}{{S}^{ + }}}&{ - ({{J}_{z}}{{S}^{z}} + {{J}_{{{\text{an}}}}}{{S}^{ - }} + J_{{{\text{an}}}}^{*}{{S}^{ + }})} \end{array}} \right)f(y).$$

(22)

Here, f(y) is the function describing the spatial localization of the magnetic barrier along the edge, which is parallel in our model to the y axis. In [11], as well as in [7, 8], the function f(y) was approximated by a delta function. In our model, we assume a smoother profile of the magnetization with a finite width and a finite height; this profile is approximated by a step function, which better corresponds to real magnets, as we believe. In this case, the interaction with a magnet still has an exchange, i.e., local and short-range nature. The magnetization components of the barrier are denoted as S^z and S^± = S ^x ± iS ^y. The exchange coupling constants are denoted as J_z, J_⊥ = $J_{ \bot }^{*}$ (because J_⊥ is real according to [11]), and an addition from the anisotropic component of the exchange interaction with the parameter J_an. The authors of [11] accepted the estimate |J_an| ≪ |J_⊥| ≪ |J_z|; for this reason, the anisotropic term in our model can be omitted and we set J_an = 0. Further, the magnetization of the barrier in our problem lies in the xy plane; i.e., the z component of the magnetization is S ^z = 0. In our problem, the interaction Hamiltonian (22) with the magnetic barrier in the basis of two-component eigenfunctions of the Hamiltonian of edge states (1) acquires the form:

$${{H}_{{{\text{ex}}}}} = \left( {\begin{array}{*{20}{c}} 0&{J_{ \bot }^{*}{{S}^{ - }}} \\ {J_{ \bot }^{{}}{{S}^{ + }}}&0 \end{array}} \right)f(y).$$

(23)

Taking into account that J_⊥ is real, denoting the barrier magnetization projections as |J_⊥|S ^x = M cosθ, |J_⊥|S ^y = M sinθ, and taking into account that S^± = S ^x ± iS ^y, Eq. (23) is represented in the form

$${{H}_{{{\text{ex}}}}} = M\left( {\begin{array}{*{20}{c}} 0&{\cos \theta - i\sin \theta } \\ {\cos \theta + i\sin \theta }&0 \end{array}} \right)f(y),$$

(24)

or

$$\begin{gathered} {{H}_{{{\text{ex}}}}} = Mf(y)\left( {\cos (\theta )\left( {\begin{array}{*{20}{c}} 0&1 \\ 1&0 \end{array}} \right)} \right. \\ \left. { + \sin \theta \left( {\begin{array}{*{20}{c}} 0&{ - i} \\ i&0 \end{array}} \right)} \right), \\ \end{gathered} $$

(25)

i.e., in the form

$${{H}_{{{\text{ex}}}}} = Mf(y)({{\sigma }_{x}}\cos \theta + {{\sigma }_{y}}\sin \theta ),$$

(26)

which describe the interaction of edge states with each of the magnetic barriers in the Hamiltonian (2) of our model.

APPENDIX B

A spatially uniform time-alternating electric field (i.e., considered in the quasistationary approximation) can be included in the Hamiltonian by means of scalar or vector potentials. Both approaches are applied to edge states in TIs and similar structures. In particular, we used a scalar potential in [12], a vector potential was used in [29], and contributions from the scalar and vector potentials were taken into account in [30, 31]. In this appendix, we show that the transition from the scalar to vector potential, which results in the appearance of the spin operator σ_z in the Hamiltonian, does not change the matrix element responsible for the Rabi frequency for transitions between two discrete levels in our system.

When the vector potential A(t) is used, the alternating electric field F(t) = –∂A/c∂t is included in the Hamiltonian by means of the usual substitution k → k – eA/$\hbar$c. After this substitution for our problem with the Hamiltonian (2) and the periodic electric field that is directed along the y axis and has the strength F sinωt, we obtain the perturbation operator V₁(t) in the form

$$\frac{{{{V}_{1}}(t)}}{F} = - \frac{A}{{\hbar \omega }}{{\sigma }_{z}}\cos \omega t.$$

(27)

Here, F is the amplitude of the electric field measured in meV per nanometer, A = 360 meV nm is the constant in the Hamiltonian (2), and the frequency of the electric field ω satisfies the resonance condition (12). In this form, the electric-field-induced perturbation includes the spin operator σ_z and does not contain the coordinate operator, as in the gauge with the scalar potential used in the Hamiltonian (11), for which the perturbation operator V₂(t) has the form

$${{V}_{2}}(t){\text{/}}F = y\sin \omega t.$$

(28)

The right-hand sides of both operators (27) and (28) include quantities with a dimension of coordinate. Consequently, the matrix elements of these operators without a time-dependent factor can be denoted as

$$y_{{mn}}^{{(1)}} = {{y}_{0}} \cdot {{({{\sigma }_{z}})}_{{mn}}},\quad y_{{mn}}^{{(2)}} = {{y}_{{mn}}}.$$

(29)

Here, the amplitude y₀ = A/$\hbar$ω coincides with the right-hand side of inequality (19), which illustrates the characteristic scale of oscillations of the average coordinate value for dynamics in the basis of edge states. Thus, both approaches with the scalar and vector potential give the same set of the characteristic amplitudes for the electric-field-induced perturbation.

The quantitative comparison of two expressions in Eqs. (29) for the matrix elements between two discrete levels E₁ and E₂ gives the following result. The matrix element of the coordinate in the second expression in Eqs. (29) is numerically obtained as |$y_{{12}}^{{(2)}}$| = 13.19 nm. To estimate the first expression in Eqs. (29), it is necessary to know y₀, which is equal to 18.71 nm in our problem, and the matrix element of the spin projection operator σ_z. The numerical calculation for a pair of discrete levels at the chosen parameters of the problem gives |σ_z|₁₂ = 0.705. The substitution of this value into the first expression in Eqs. (29) yields the absolute value of the matrix element |$y_{{12}}^{{(1)}}$| = 13.19 nm, which coincides with the value obtained from the gauge with the scalar potential. The coincidence of the results obtained in approaches with the spin and coordinate operators is not accident because the velocity operator for this model, according to Eq. (18), is proportional to the spin projection operator σ_z; i.e., the dynamics of the coordinate and z spin projection are closely related. We believe that good agreement between two approaches makes it possible to use any of them to study Rabi oscillations. Therefore, the scalar potential can be used to describe the spatially uniform time-alternating electric field in the Hamiltonian (11).