Electric dipole spin resonance

In quantum mechanics, electrons possess electric charge $e$ and magnetic moment ${\boldsymbol {\mu }}$ whose absolute value equals to the Bohr magneton $\mu _{B}$ . Contemporary computational techniques employ electron charge in transistors to process information and electron spin in magnetic storage devices to accumulate it. The emergent field of spintronics aims in unifying operations of these subsystems. For achieving this goal, electron spin should be operated by electric fields, and electric dipole spin resonance (EDSR) is one of the most efficient tools for achieving this goal. It allows performing spin flip transitions by resonant $ac$ electric fields.

Theory

Electron spin resonance, also known as electron paramagnetic resonance, is due to the coupling of electron magnetic moment ${\boldsymbol {\mu }}$ to the external magnetic field ${\boldsymbol {B}}$ through the Hamiltonian $H=-({\boldsymbol {\mu }}\cdot {\boldsymbol {B}})$ describing its Larmor precession. Magnetic moment ${\boldsymbol {\mu }}$ is related to electron angular momentum ${\boldsymbol {S}}$ as ${\boldsymbol {\mu }}=g{\mu _{B}}{\boldsymbol {S}}$ , where $g$ is a Landé g-factor. For a free electron in vacuum $g\approx 2$ . Larmor interaction quantizes electron spin energy levels in a dc magnetic field $B$ as $E_{\pm }=\pm {\frac {1}{2}}g\mu _{B}B$ , and a resonant ac magnetic field ${\tilde {{\boldsymbol {B}}}}(t)$ of a frequency $\omega _{S}=g\mu _{B}B/\hbar$ results in electron paramagnetic resonance.

Coupling electron spin to electric fields in vacuum and atoms

An electron moving in vacuum in an ac electric field ${{\tilde E}}(t)$ sees, according to the Lorentz transformation, an ac magnetic field ${{\tilde B}}(t)\approx (v/c){{\tilde E}}(t)$ in its center of mass system. However, for slow electrons with $v/c\ll 1$ this field is so weak that its effect can be neglected. In atoms, electron orbital and spin dynamics are coupled because of the electric field of nuclei as it follows from the Dirac equation. This coupling, known as spin-orbit interaction, is small in the fine-structure constant $e^{2}/\hbar c\approx 1/137$ . However, this constant appears in a combination with the atomic number $Z$ as $Ze^{2}/\hbar c$ ,^[1] and this product is of the order of unity already in the middle of the periodic table. This enhancement of the coupling between the orbital and spin dynamics originates from strong electric fields and electron velocities near nuclei. While this mechanism is also expected to couple electron spin to ac electric fields, such an effect has been probably never observed in atomic spectroscopy.

Basic mechanisms of Electric Dipole Spin Resonance in crystals

Most important, this enhanced spin-orbit coupling in atoms translates into spin-orbit coupling in crystals formed from such atoms or ions that may be strong even for slow electrons. It becomes an essential part of the band structure of their energy spectrum. The ratio of the spin-orbit splitting of the bands to the forbidden gap becomes a parameter that evaluates the effect of spin-orbit coupling, and it is generically of the order of unity.

As a result, even slow electrons in solids experience strong spin-orbit coupling. This means that the Hamiltonian of a free electron in a crystal includes, side by side with the electron quasimomentum ${\boldsymbol {k}}={\boldsymbol {p}}/\hbar$ also Pauli matrices ${\boldsymbol {\sigma }}$ , and the terms including both of them are not small. Coupling of electron spin to the external electromagnetic field can be found by the substitution ${\boldsymbol {k}}\rightarrow {\boldsymbol {k}}-(e/\hbar c){\boldsymbol {A}}$ as is required by the gauge invariance of the theory, here ${\boldsymbol {A}}$ is the vector potential, and the substitution is known as Peierls substitution. And because the electric field ${\boldsymbol {E}}=-{\frac {1}{c}}\partial {\boldsymbol {A}}/\partial t$ , it becomes coupled to the electron spin and can produce spin-flip transitions.

Electric Dipole Spin Resonance (EDSR) is the electron spin resonance driven by a resonant ac electric field ${{\tilde {\boldsymbol {E}}}}$ . Because the Compton length $\lambda _{C}=\hbar /mc\approx 4\times 10^{{-11}}$ cm entering into the Bohr magneton $\mu _{B}=e\lambda _{C}/2$ and controlling the coupling of electron spin to ac magnetic field ${{\tilde {\boldsymbol {B}}}}$ is much shorter than all characteristic lengths of solid state physics, EDSR can be by orders of magnitude stronger than the electron paramagnetic resonance driven by ac magnetic field. EDSR was proposed by Rashba.^[2]

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 Pauli matrices ${\boldsymbol {\sigma }}$ and odd powers of the quasimomemtum ${\boldsymbol {k}}$ . In such cases electron spin is coupled to the vector-potential ${{\tilde {\boldsymbol {A}}}}$ of electromagnetic field. Remarkably, EDSR on free electrons can be observed not only at the spin-resonance frequency $\omega _{S}$ but also at its linear combinations with the cyclotron resonance frequency $\omega _{C}$ . In narrow-gap semiconductors with inversion center EDSR can emerge due direct coupling of electric field ${{\tilde {\boldsymbol {E}}}}$ to the anomalous coordinate ${\boldsymbol {r}}_{{SO}}$ , see Spin-orbit interaction.

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 $\hbar \omega _{S}/\Delta E$ where $\Delta E$ is the separation between adjacent levels of the orbital motion.

Origin of high intensity of EDSR: 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 $\Delta_{so}$ is comparable to the forbidden gap $E_{G}$ , the effective mass of an electron $m^{*}$ and its g-factor can be evaluated in the framework of the Kane scheme,^[3]^[4] see [[<math>k\cdot p</math> perturbation theory]]

$m^*\approx\frac{\hbar^2E_G}{P^2},\,\,\,|g|\approx\frac{m_0P^2}{\hbar^2E_G}$ ,

where $P\approx 10 eV\AA$ is a coupling parameter between the electron an valence bands, and $m_{0}$ is the electron mass in vacuum.

Choosing the spin-orbit coupling mechanism based on the anomalous coordinate ${\boldsymbol{r}_{so}}$ (see spin-orbit coupling) under the condition $\Delta_{so}\approx E_G$ , we have

$r_{so}\approx\frac{\hbar^2|g|k}{m_0E_G}$

where $k$ is electron quasimomentum. Then energy of an electron in a ac electric field ${\tilde E}$ is

$U=er_{so}{\tilde E}\approx e{\tilde E}\frac{P^2}{E_G^2}k\approx e{\tilde E}\frac{\hbar^2k}{m^*E_G}.$

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

$U_v=\mu_B{\tilde B}=e{\tilde E}\frac{\hbar^2k}{m_0^2c^2},$

where $\mu _{B}$ is the Bohr magneton and $c$ is the speed of light. The ratio of these energies

$\frac{U}{U_v}\approx\frac{m_0}{m^*}\frac{m_0c^2}{E_G}$ .

This expression shows explicitly where the dominance of EDSR over the electron paramagnetic resonance comes from. The numerator $m_0c^2\approx$ 0.5MeV of the second factor is a half of the Dirac gap while $E_{G}$ is of atomic scale, $E_G\approx$ 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 $Ze^{2}/\hbar c$ of the atomic number $Z$ and the fine-structure constant $e^{2}/\hbar c\approx 1/137$ 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 electron paramagnetic resonance is usually the dominant mechanism.

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^[5] observed a motionally narrowed EDSR line at $\omega _{S}$ frequency against a background of a wide cyclotron resonance band. MacCombe et al.^[6] working with high quality InSb observed isotropic EDSR driven by the $({\boldsymbol {r}}_{{SO}}\cdot {{\tilde {{\boldsymbol {E}}}}})$ mechanism at the combinational frequency $\omega _{C}+\omega _{S}$ where $\omega _{C}$ is the cyclotron resonance frequency. Strongly anisotropic EDSR band due to inversion-asymmetry $k^{3}$ Dresselhaus spin-orbit coupling was observed in InSb at the spin-flip frequency $\omega _{S}$ by Dobrowolska et al.^[7] 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 ${{\boldsymbol E}}({{\boldsymbol r}})$ which mixes wave functions of different valleys.^[8] Infrared EDSR observed in semimagnetic semiconductor Cd $_{{1-x}}$ Mn $_{x}$ Se^[9] was ascribed^[10] 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,^[11] 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 and are 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 ${\boldsymbol {B}}(t)$ practically cannot address individual electron spins at such a scale, but individual spins can be well addressed by time dependent electric fields ${\boldsymbol {E}}(t)$ produced by nanoscale gates. All basic mechanisms of EDSR listed above are operating in quantum dots,^[12] but in A $_{3}$ B $_{5}$ compounds also the hyperfine coupling of electron spins to nuclear spins plays an essential role.^[13]^[14]^[15] For achieving fast qubits operated by EDSR^[16] are needed nanostructures with strong spin-orbit coupling. For the Rashba interaction $H_{R}=\alpha (\sigma _{x}k_{y}-\sigma _{y}k_{x})$ the strength of interaction is characterized by the coefficient $\alpha$ . In InSb quantum wires the magnitude of $\alpha$ of the atomic scale of about 1 eV $\AA$ has been already achieved.^[17] A different way for achieving fast spin qubits based on quantum dots operated by EDSR is using nanomagnets producing inhomogeneous magnetic fields,^[18] see Spin-orbit interaction.

References

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↑ E. I. Rashba, Cyclotron and combined resonances in a perpendicular field, Sov. Phys. Solid State 2, 1109 -1122 (1960)
↑ E. O. Kane, J. Phys. Chem. Phys. 1, 249 (1957)
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↑ R. L. Bell, Electric Dipole Spin Transitions in InSb, Phys. Rev. Lett. 9, 52 (1962)
↑ B. D. McCombe, S. G. Bishop, and R. Kaplan, Combined Resonance and Electron g values in InSb, Phys. Rev. Lett. 18, 748 (1967).
↑ M. Dobrowolska, Y. Chen, J. K. Furdyna, and S. Rodriguez, Effects of Photon-Momentum and Magnetic-Field Reversal on the Far-Infrared Electric-Dipole Spin Resonance in InSb, Phys. Rev. Lett. 51, 134 (1983).
↑ E. M. Gershenzon, N. M. Pevin, I. T. Semenov, and M. S. Fogelson, Electric-Dipole Excitation of Spin Resonance in Compensated n-Type Ge,Soviet Physics-Semiconductors 10, 104-105 (1976).
↑ M. Dobrowolska, A. Witowski, J. K. Furdyna, T. Ichiguchi, H. D. Drew, and P. A. Wolff, Far-infrared observation of the electric-dipole spin resonance of donor electrons in Cd $_{{1-x}}$ Mn $_{x}$ Se, Physical Review B 29, 6652 (1984).
↑ L. S. Khazan, Yu. G. Rubo, and V. I. Sheka, Exchange-induced optical spin transitions in semimagnetic semiconductors, Physical Review B 47, 13180 (1993).
↑ V. V. Kveder, V. Ya. Kravchenko, T. R. Mchedlidze, Yu. A. Osip'yan, D. E. Khmel'nitskii, A. I. Shalynin, Combined resonance at dislocations in silicon, JETP Lett. 43, 255 (1986).
↑ C. Kloeffel and D. Loss, Prospects for Spin-Based Quantum Computing in Quantum Dots, Annual Review of Condensed Matter Physics, 4, p.51-81 (2013)
↑ E. A. Laird, C. Barthel, E. I. Rashba, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Hyperfine-mediated gate-driven electron spin resonance, Phys. Rev. Lett. 99, 246601 (2007).
↑ E. I. Rashba, Theory of electric dipole spin resonance in quantum dots: Mean field theory with Gaussian fluctuations and beyond, Phys. Rev. B 78, 195302 (2008)
↑ M. Shafiei, K. C. Nowack, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, Resolving Spin-Orbit- and Hyperfine-Mediated Electric Dipole Spin Resonance in a Quantum Dot, Phys. Rev. Lett. 110, 107601 (2013)
↑ J. W. G. van den Berg, S. Nadj-Perge, V. S. Pribiag, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Fast Spin-Orbit Qubit in an Indium Antimonide Nanowire, Phys. Rev. Lett. 110, 066806 (2013)
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