Theory of electric dipole spin resonance in a parabolic quantum well

A theory of electric dipole spin resonance (EDSR), that is caused by various mechanisms of spin-orbit coupling, is developed as applied to free electrons in a parabolic quantum well. Choosing a parabolic shape of the well has allowed us to find explicit expressions for the EDSR intensity and its dependence on the magnetic field direction in terms of the basic parameters of the Hamiltonian. By using these expressions, we have investigated and compared the effect of specific mechanisms of spin orbit (SO) coupling and different polarizations of ac electric field on the intensity of EDSR. It is our basic assumption that the SO coupling energy is small compared with all different competing energies (the confinement energy, and the cyclotron and Zeeman energies) that allowed us to describe all SO coupling mechanisms in the framework of the same general approach. For this purpose, we have developed an operator formalism for calculating matrix elements of the transitions between different quantum levels. To make these calculations efficient enough and to derive explicit and concise expressions for the EDSR intensity, we have established a set of remarkable identities relating the eigenfrequencies and the angles defining the spatial orientation of the quantizing magnetic field $\mathbf{B}(\theta ,\varphi )$. Applicability of these identities is not restricted by EDSR and we expect them to be useful for the general theory of parabolic quantum wells. The angular dependences of the EDSR intensity, found for various SO coupling mechanisms, show a fine structure consisting of alternating up and down cusps originating from repopulating different quantum levels and their spin sublevels. Angular dependences of the EDSR intensity are indicative of the relative contributions of the competing mechanisms of SO coupling. Our results show that electrical manipulating electron spins in quantum wells is generally highly efficient, especially by an in-plane ac electric field.

Figure 1

(Color online) The energy spectrum of electrons confined in a parabolic quantum well in a tilted magnetic field. Here, ${\omega }_0$ is the confinement frequency, ${\omega }_c=eB∕mc$ is the cyclotron frequency, $\theta$ is the polar angle of the field $\mathbf{B}$, ${\omega }_{\xi }$ and ${\omega }_{\eta }$ are the frequencies of two eigenmodes ${\omega }_{\zeta }$, $\zeta =\xi ,\eta$. (a) The dependence of ${\omega }_{\zeta }$ on the ratio ${\omega }_c∕{\omega }_0$ for three values of $\theta$. Cyclotron-like modes are shown by dotted lines and confinement-like modes by full lines. For $\theta =0$ these modes intersect, while for $\theta \ne 0$ they interchange at ${\omega }_c={\omega }_0$. (b) The dependence of ${\omega }_{\zeta }$ on $\theta$ for three values of ${\omega }_c∕{\omega }_0$. The modes retain their identity in the whole interval $0⩽\theta ⩽\pi ∕2$, and ${\omega }_{\eta }>{\omega }_{\xi }$ for ${\omega }_c<{\omega }_0$ while ${\omega }_{\eta }<{\omega }_{\xi }$ for ${\omega }_c>{\omega }_0$. At the degeneracy point, ${\omega }_c={\omega }_0$, the two ${\omega }_{\eta ,\xi }$ branches have no specific identity. They are separatrices dividing the plane into one $\eta$ (gray) and two $\xi$ (white) regions.

Figure 2

(Color online) The difference of the filling factors, $\Delta \nu$, of the two components of a spin doublet of an isolated energy level as a function of the total filling factor of this level $0⩽\nu ⩽2$. For a small level width $\Gamma$, its shape is close to a triangular form, and an up cusp develops at $\nu =1$. In a plot including adjacent energy levels, down cusps at the end of this interval will arise. With increasing $\Gamma$, the cusps are smeared. However, at $\Gamma$ as large as $\Gamma =\hslash {\omega }_s∕2$, pronounced extrema at $\nu =1$ and $\nu =0,2$ are still seen.

Figure 3

(Color online) The dependence of the EDSR intensity on the polar angle $\theta$ of a magnetic field $\mathbf{B}$ for Rashba spin-orbit coupling, Eq. (4). The electric field is perpendicular to the confinement plane, $\stackrel{\mathrm{̃}}{\mathbf{E}}\Vert \widehat{\mathbf{z}}$. The ratio of spin-flip and cyclotron frequencies, ${\omega }_s∕{\omega }_c=-0.17$, is typical of InAs; ${\omega }_0$ being the confinement frequency. The calculations were performed for three values of the level width $\Gamma$, as specified in the figure. For $\mathbf{B}\Vert \widehat{\mathbf{z}}$, electrons completely populate the ground level, $n_{\xi }=n_{\eta }=0$. (a) Strong confinement, ${\omega }_c∕{\omega }_0=0.5$. With increasing $\theta$ electrons populate higher cyclotron-like $\xi$ levels, while they remain at the ground $\eta$ level. (b) Weak confinement, ${\omega }_c∕{\omega }_0=2$. With increasing $\theta$ electrons populate higher $\eta$ levels, while they remain at the ground $\xi$ level. In both cases, the principal peak originates from the resonance ${\omega }_{\zeta }=\mid {\omega }_s\mid$ with $\zeta =\xi$ for (a) and $\zeta =\eta$ for (b). The fine structure of the spectrum, distinctly seen for small level width $\Gamma =0.05{\omega }_0$, originates from populating higher $n_{\zeta }$ levels in a tilted magnetic field $\mathbf{B}$. Up cusps are marked by up arrows and down cusps by down arrows. In (b), the inflexion point seen between the $\nu =5$ cusp and the EDSR maximum reflects an abrupt change in the population of intersecting levels at ${\omega }_{\eta }=\mid {\omega }_s\mid$. Insets show the dependence of the chemical potential $\eta \left(\nu \right)$ on the filling factor $\nu$ and the quantum numbers of partially populated states, $n_{\xi }$ or $n_{\eta }$, that contribute to spin-flip transitions inside the various regions of filling factor values.

Figure 4

The dependence of the EDSR intensity on the magnetic field direction for a (001) quantum well. The electric field is perpendicular to the QW plane, $\stackrel{\mathrm{̃}}{\mathbf{E}}\Vert \widehat{\mathbf{z}}$. The electron concentration is low, hence, for all $\theta$ values electrons populate only the ground level, $n_{\xi },n_{\eta }=0$. The level width, $\Gamma =0.2{\omega }_0$; ${\omega }_c∕{\omega }_0=0.5$, ${\omega }_s∕{\omega }_c=-0.17$. (a) 2D Dresselhaus spin-orbit coupling mechanism, Eq. (3). The fourfold symmetry of the figure reflects the symmetry of the square of the spin-flip transition matrix element. (b) Interference of 2D Dresselhaus and Rashba spin-orbit coupling mechanisms. The twofold symmetry of the figure reflects the ${\mathbf{C}}_{2v}$ symmetry of the Hamiltonian ${\widehat{H}}_D$; coupling constants have been chosen as ${\alpha }_D={\alpha }_R$.

Figure 5

The dependence of the EDSR intensity on the magnetic field direction for a (001) quantum well. The electric field is in the QW plane, $\stackrel{\mathrm{̃}}{\mathbf{E}}\perp \widehat{\mathbf{z}}$. The electron concentration is low; parameter values are the same as in Fig. 4. (a) and (b)—Rashba spin-orbit coupling, Eq. (4). For (a), the electric field $\stackrel{\mathrm{̃}}{\mathbf{E}}$ is parallel to the projection ${\mathbf{B}}_{\perp }$ of the field $\mathbf{B}$ onto the confinement plane, $\stackrel{\mathrm{̃}}{\mathbf{E}}\Vert {\mathbf{B}}_{\perp }$, hence, the intensity is rotationally symmetric. For (b), the electric field is parallel to the (110) crystallographic axis, $\stackrel{\mathrm{̃}}{\mathbf{E}}\Vert \left(110\right)$. (c) and (d)—2D Dresselhaus spin-orbit coupling, Eq. (3). For (c), the electric field $\stackrel{\mathrm{̃}}{\mathbf{E}}$ is parallel to ${\mathbf{B}}_{\perp }$, $\stackrel{\mathrm{̃}}{\mathbf{E}}\Vert {\mathbf{B}}_{\perp }$, hence, the intensity shows fourfold symmetry. For (d), $\stackrel{\mathrm{̃}}{\mathbf{E}}\Vert \left(110\right)$, and the gross features are similar to those of figure (b). For (e), $\stackrel{\mathrm{̃}}{\mathbf{E}}\Vert {\mathbf{B}}_{\perp }$, the two spin-orbit coupling mechanisms with ${\alpha }_D={\alpha }_R$ interfere; the symmetry is lowered because of their interference.

Figure 6

(Color online) The dependence of the EDSR intensity on the polar angle $\theta$ of a magnetic field $\mathbf{B}$ for Dresselhaus spin-orbit coupling, Eq. (2), for two values of the azimuth $\varphi$. The electric field is perpendicular to the QW plane, $\stackrel{\mathrm{̃}}{\mathbf{E}}\Vert \widehat{\mathbf{z}}$. The notations and basic parameter values are the same as in Fig. 3. Similarly to Fig. 3, the spectra show the principal maximum at ${\omega }_{\zeta }=\mid {\omega }_s\mid$ and the fine structure of up and down cusps indicating repopulation of different energy levels with $\theta$ changing. The major distinctions from Fig. 3 include (i) a minimum at $\theta =0.42\times \pi ∕2$ originating from a zero in the effective coupling constant ${\alpha }_D^{\mathrm{eff}}(n_{\xi },n_{\eta })$ of Eq. (67), (ii) additional minima in panels (c) and (d) originating from the zeros of functions $F_c\left(\theta \right)$ and $F_s\left(\theta \right)$ at $\theta =0.82\times \pi ∕2$ and $\theta =0.61\times \pi ∕2$, respectively, and (iii) smearing of the main peak in panel (a) because of the slow angular dependence of the envelope function shown in the inset to that panel. The insets to panels (a) and (b) demonstrate that the results found for the strong confinement limit still retain reasonable accuracy for ${\omega }_c={\omega }_0∕2$. The insets to panels (c) and (d) demonstrate the positions of the zeros of functions $F_c$ and $F_s$.