Electric dipole spin resonance

Theory

Electric dipole spin resonance is the electron spin resonance driven by a resonant AC electric field ${\displaystyle \tilde{\mathbf{E}}}$. Because the Compton length ${\displaystyle {\lambda }_{\mathrm{C}}=\hslash /mc\approx 4\times {10}^{-11}\mathrm{c}\mathrm{m}}$, entering into the Bohr magneton ${\displaystyle {\mu }_{\mathrm{B}}=e{\lambda }_{\mathrm{C}}/2}$ and controlling the coupling of electron spin to AC magnetic field ${\displaystyle \tilde{\mathbf{B}}}$, is much shorter than all characteristic lengths of solid state physics, EDSR can be by orders of magnitude stronger than ESR driven by an AC magnetic field. EDSR is usually strongest in materials without the inversion center where the two-fold degeneracy of the energy spectrum is lifted and time-symmetric Hamiltonians include products of the spin related Pauli matrices ${\displaystyle \mathit{\sigma }}$, as ${\displaystyle \mathbf{S}=(\hslash /2)\sigma }$, and odd powers of the crystal momentum ${\displaystyle \mathbf{k}}$. In such cases electron spin is coupled to the vector-potential ${\displaystyle \tilde{\mathbf{A}}}$ of electromagnetic field. Remarkably, EDSR on free electrons can be observed not only at the spin-resonance frequency ${\displaystyle {\omega }_S}$ but also at its linear combinations with the cyclotron resonance frequency ${\displaystyle {\omega }_C}$. In narrow-gap semiconductors with inversion center EDSR can emerge due direct coupling of electric field ${\displaystyle \tilde{\mathbf{E}}}$ to the anomalous coordinate ${\displaystyle {\mathbf{r}}_{\mathrm{S}\mathrm{O}}}$.

EDSR is allowed both with free carriers and with electrons bound at defects. However, for transitions between Kramers conjugate bound states, its intensity is suppressed by a factor ${\displaystyle \hslash {\omega }_S/\mathrm{\Delta }E}$ where ${\displaystyle \mathrm{\Delta }E}$ is the separation between adjacent levels of the orbital motion.

Simplified theory and physical mechanism

As stated above, various mechanisms of EDSR operate in different crystals. The mechanism of its generically high efficiency is illustrated below as applied to electrons in direct-gap semiconductors of the InSb type. If spin–orbit splitting of energy levels ${\displaystyle {\mathrm{\Delta }}_{\mathrm{s}\mathrm{o}}}$ is comparable to the forbidden gap ${\displaystyle E_{\mathrm{G}}}$, the effective mass of an electron ${\displaystyle m^{\ast }}$ and its g-factor can be evaluated in the framework of the Kane scheme, see k·p perturbation theory.

${\displaystyle m^{\ast }\approx \frac{{\hslash }^2{E}_{\mathrm{G}}}{P^2},|g|\approx \frac{m_0{P}^2}{{\hslash }^2{E}_{\mathrm{G}}}}$,

where ${\displaystyle P\approx 10\text{ eV}\AA }$ is a coupling parameter between the electron an valence bands, and ${\displaystyle m_0}$ is the electron mass in vacuum.

Choosing the spin–orbit coupling mechanism based on the anomalous coordinate ${\displaystyle {\mathit{r}}_{\mathrm{s}\mathrm{o}}}$ under the condition :${\displaystyle {\mathrm{\Delta }}_{\mathrm{s}\mathrm{o}}\approx E_G}$, we have

${\displaystyle r_{\mathrm{s}\mathrm{o}}\approx \frac{{\hslash }^2|g|k}{m_0{E}_{\mathrm{G}}}}$,

where ${\displaystyle k}$ is electron crystal momentum. Then energy of an electron in a AC electric field ${\displaystyle \tilde{E}}$ is

${\displaystyle U=e{r}_{\mathrm{s}\mathrm{o}}\tilde{E}\approx e\tilde{E}\frac{P^2}{E_{\mathrm{G}}^2}k\approx e\tilde{E}\frac{{\hslash }^{2}k}{m^{\ast }{E}_{\mathrm{G}}}.}$

An electron moving in vacuum with a velocity ${\displaystyle \hslash k/m_0}$ in an AC electric field ${\displaystyle \tilde{E}}$ sees, according to the Lorentz transformation an effective magnetic field ${\displaystyle \tilde{B}=v/c\tilde{E}}$. Its energy in this field

${\displaystyle U_v={\mu }_{\mathrm{B}}\tilde{B}=e\tilde{E}\frac{{\hslash }^{2}k}{m_0^2{c}^2},}$

The ratio of these energies

${\displaystyle \frac{U}{U_v}\approx \frac{m_0}{m^{\ast }}\frac{m_0{c}^2}{E_{\mathrm{G}}}}$.

This expression shows explicitly where the dominance of EDSR over the electron paramagnetic resonance comes from. The numerator ${\displaystyle m_0{c}^2\approx 0.5\mathrm{M}\mathrm{e}\mathrm{V}}$ of the second factor is a half of the Dirac gap while ${\displaystyle E_{\mathrm{G}}}$ is of atomic scale, ${\displaystyle E_{\mathrm{G}}\approx }$1eV. The physical mechanism behind the enhancement is based on the fact that inside crystals electrons move in strong field of nuclei, and in the middle of the periodic table the product ${\displaystyle Z\alpha }$ of the atomic number ${\displaystyle Z}$ and the fine-structure constant ${\displaystyle \alpha }$ is of the order of unity, and it is this product that plays the role of the effective coupling constant, cf. spin–orbit coupling. However, one should bear in mind that the above arguments based on effective mass approximation are not applicable to electrons localized in deep centers of the atomic scale. For them the EPR is usually the dominant mechanism.

Inhomogeneous Zeeman coupling mechanism

Above mechanisms of spin–orbit coupling in solids originated from the Thomas interaction and couple spin matrices ${\displaystyle \mathit{\sigma }}$ to electronic momentum ${\displaystyle \mathbf{k}}$. However, the Zeeman interaction

${\displaystyle H_{\mathrm{Z}}(\mathbf{r})=-\mathit{\mu }\cdot \mathbf{B}(\mathbf{r})}$

in an inhomogeneous magnetic field ${\displaystyle \mathbf{B}(\mathbf{r})}$ produces a different mechanism of spin–orbit interaction through coupling the Pauli matrices ${\displaystyle \mathit{\sigma }}$ to the electron coordinate ${\displaystyle \mathbf{r}}$. The magnetic field can be both a macroscopic inhomogeneous field or a microscopic fast-oscillating field inside ferro- or antiferromagnets changing at the scale of a lattice constant.

Experiment

EDSR was first observed experimentally with free carriers in indium antimonide (InSb), a semiconductor with strong spin–orbit coupling. Observations made under different experimental conditions allowed demonstrate and investigate various mechanisms of EDSR. In a dirty material, Bell observed a motionally narrowed EDSR line at ${\displaystyle {\omega }_S}$ frequency against a background of a wide cyclotron resonance band. MacCombe et al. working with high quality InSb observed isotropic EDSR driven by the ${\displaystyle ({\mathbf{r}}_{\mathrm{s}\mathrm{o}}\cdot \tilde{\mathbf{E}})}$ mechanism at the combinational frequency ${\displaystyle {\omega }_{\mathrm{C}}+{\omega }_S}$ where ${\displaystyle {\omega }_{\mathrm{C}}}$ is the cyclotron frequency. Strongly anisotropic EDSR band due to inversion-asymmetry ${\displaystyle k^3}$ Dresselhaus spin–orbit coupling was observed in InSb at the spin-flip frequency ${\displaystyle {\omega }_S}$ by Dobrowolska et al. spin–orbit coupling in n-Ge that manifests itself through strongly anisotropic electron g-factor results in EDSR through breaking translational symmetry by inhomogeneous electric fields which mixes wave functions of different valleys. Infrared EDSR observed in semimagnetic semiconductor Cd${\displaystyle {}_{1-x}}$Mn${\displaystyle {}_x}$Se was ascribed to spin–orbit coupling through inhomogeneous exchange field. EDSR with free and trapped charge carriers was observed and studied at a large variety of three-dimensional (3D) systems including dislocations in Si, an element with notoriously weak spin–orbit coupling. All above experiments were performed in the bulk of three-dimensional (3D) systems.

Applications

Principal applications of EDSR are expected in quantum computing and semiconductor spintronics, currently focused on low-dimensional systems. One of its main goals is fast manipulation of individual electron spins at a nanometer scale, e.g., in quantum dots of about 50 nm size. Such dots can serve as qubits of quantum computing circuits. Time-dependent magnetic fields practically cannot address individual electron spins at such a scale, but individual spins can be well addressed by time dependent electric fields produced by nanoscale gates. All basic mechanisms of EDSR listed above are operating in quantum dots, but in A${\displaystyle {}_3}$B${\displaystyle {}_5}$ compounds also the hyperfine coupling of electron spins to nuclear spins plays an essential role. For achieving fast qubits operated by EDSR are needed nanostructures with strong spin–orbit coupling. For the Rashba spin–orbit coupling

${\displaystyle H_{\mathrm{R}}={\alpha }_{\mathrm{R}}({\sigma }_x{k}_y-{\sigma }_y{k}_x)}$,

the strength of interaction is characterized by the coefficient ${\displaystyle {\alpha }_{\mathrm{R}}}$. In InSb quantum wires the magnitude of ${\displaystyle {\alpha }_{\mathrm{R}}}$ of the atomic scale of about 1 eV${\displaystyle \AA }$ has been already achieved. A different way for achieving fast spin qubits based on quantum dots operated by EDSR is using nanomagnets producing inhomogeneous magnetic fields.