Electric dipole spin resonance (EDSR) is a method to control the
magnetic moment
In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
s inside a material using
quantum mechanical
Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of a ...
effects like the
spin–orbit interaction
In quantum mechanics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin– ...
. Mainly, EDSR allows to flip the orientation of the magnetic moments through the use of
electromagnetic radiation
In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength ...
at
resonant
Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximu ...
frequencies. EDSR was first proposed by
Emmanuel Rashba
Emmanuel I. Rashba (October 30, 1927 – January 12, 2025) was a Soviet-American theoretical physicist of Jewish origin who worked in Ukraine, Russia and in the United States. Rashba is known for his contributions to different areas of condensed ...
.
Computer hardware
Computer hardware includes the physical parts of a computer, such as the central processing unit (CPU), random-access memory (RAM), motherboard, computer data storage, graphics card, sound card, and computer case. It includes external devices ...
employs the
electron charge
C, or c, is the third letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''cee'' (pronounced ), plural ''cees''.
History
"C ...
in
transistors
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch electrical signals and electric power, power. It is one of the basic building blocks of modern electronics. It is composed of semicondu ...
to process information and the electron magnetic moment or
spin
Spin or spinning most often refers to:
* Spin (physics) or particle spin, a fundamental property of elementary particles
* Spin quantum number, a number which defines the value of a particle's spin
* Spinning (textiles), the creation of yarn or thr ...
for
magnetic storage
Magnetic storage or magnetic recording is the storage of data on a magnetized medium. Magnetic storage uses different patterns of magnetisation in a magnetizable material to store data and is a form of non-volatile memory. The information is acc ...
devices. The emergent field of
spintronics
Spintronics (a portmanteau meaning spin transport electronics), also known as spin electronics, is the study of the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental electronic charge, in solid-st ...
aims in unifying the operations of these subsystems. For achieving this goal, the electron spin should be operated by electric fields. EDSR allows to use the electric component of
AC fields to manipulate both charge and spin.
Introduction
Free electrons possess
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
and magnetic moment
whose absolute value is about one
Bohr magneton
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
In SI units, the Bohr magneton is defined as
\mu_\mat ...
.
The standard
electron spin resonance
Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spin ...
, also known as electron paramagnetic resonance (EPR), is due to the coupling of
electron magnetic moment
In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The value of the electron magne ...
to the external magnetic field
through the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
describing its
Larmor precession
Sir Joseph Larmor (; 11 July 1857 – 19 May 1942) was an Irish mathematician and physicist who made breakthroughs in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influential work was ...
. The magnetic moment is related to electron
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
as
, where
is the
g-factor and
is the reduced
Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
. For a free electron in vacuum
. As the electron is a
spin-1/2
In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one f ...
particle, the spin operator can take only two values:
. So, Larmor interaction has quantized energy levels in a time-independent magnetic field as the energy is equal to
. In the same way, under a resonant AC magnetic field
at the frequency
, results in electron paramagnetic resonance, that is, the signal gets absorbed strongly at this frequency as it produces transitions between spin values.
Coupling electron spin to electric fields in atoms
In atoms, electron orbital and spin dynamics are coupled to the electric field of the
proton
A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s in the
atomic nucleus
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
according to the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
. An electron moving in a static electric field
sees, according to the
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
s of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
, a complementary magnetic field
in the electron
frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale have been specified in physical space. It ...
. However, for slow electrons with
this field is weak and the effect is small. This coupling is known as the
spin–orbit interaction
In quantum mechanics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin– ...
and gives
corrections to the atomic energies about the order of the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
squared
, where
. However, this constant appears in combination with the atomic number
as
, and this product is larger for massive atoms, already of the order of unity in the middle of the
periodic table
The periodic table, also known as the periodic table of the elements, is an ordered arrangement of the chemical elements into rows (" periods") and columns (" groups"). It is an icon of chemistry and is widely used in physics and other s ...
. This enhancement of the coupling between the orbital and spin dynamics in massive atoms originates from the strong attraction to the nucleus and the large electron speeds. While this mechanism is also expected to couple electron spin to the electric component of electromagnetic fields, such an effect has been probably never observed in
atomic spectroscopy
In physics, atomic spectroscopy is the study of the electromagnetic radiation absorbed and emitted by atoms. Since unique elements have unique emission spectra, atomic spectroscopy is applied for determination of elemental compositions. It can ...
.
Basic mechanisms in crystals
Most important, spin–orbit interaction in atoms translates into
spin–orbit coupling in crystals. It becomes an essential part of the
band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
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 enhanced, of the order of unity, for materials with heavy
ion
An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by convent ...
s or with specific asymmetries.
As a result, even slow electrons in solids experience strong spin–orbit coupling. This means that the Hamiltonian of an electron in a crystal includes a coupling between the electron
crystal momentum
In solid-state physics, crystal momentum or quasimomentum is a Momentum#Momentum in quantum mechanics, momentum-like Vector (geometric), vector associated with electrons in a Crystal structure, crystal lattice. It is defined by the associated Rec ...
and the electron spin. The coupling to the external electric field can be found by substituting the momentum in the kinetic energy as
, where
is the
magnetic vector potential
In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the ma ...
, as it is required by the
gauge invariance
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
of electromagnetism. The substitution is known as
Peierls substitution. Thus, the electric field
becomes coupled to the electron spin and its manipulation may produce transitions between spin values.
Theory
Electric dipole spin resonance is the electron spin resonance driven by a resonant ''AC''
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
. Because the
Compton length , entering into the Bohr magneton
and controlling the coupling of electron spin to ''AC'' magnetic field
, is much shorter than all characteristic lengths of
solid state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state p ...
, EDSR can be by orders of magnitude stronger than EPR 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
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
, as
, and odd powers of the crystal momentum
. In such cases electron spin is coupled to the vector-potential
of electromagnetic field. Remarkably, EDSR on free electrons can be observed not only at the spin-resonance frequency
but also at its linear combinations with the
cyclotron resonance frequency
. In narrow-gap semiconductors with inversion center EDSR can emerge due direct coupling of electric field
to the anomalous coordinate
.
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
where
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
is comparable to the forbidden gap
, the effective mass of an electron
and its ''g''-factor can be evaluated in the framework of the Kane scheme,
see
k·p perturbation theory.
:
,
where
is a coupling parameter between the electron an valence bands, and
is the electron mass in vacuum.
Choosing the
spin–orbit coupling mechanism based on the anomalous coordinate
under the condition :
, we have
:
,
where
is electron crystal momentum. Then energy of an electron in a ''AC'' electric field
is
:
An electron moving in vacuum with a velocity
in an AC electric field
sees, according to the
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
an effective magnetic field
. Its energy in this field
:
The ratio of these energies
:
.
This expression shows explicitly where the dominance of EDSR over the
electron paramagnetic resonance
Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spin ...
comes from. The numerator
of the second factor is a half of the Dirac gap while
is of atomic scale,
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 periodic table, also known as the periodic table of the elements, is an ordered arrangement of the chemical elements into rows (" periods") and columns (" groups"). It is an icon of chemistry and is widely used in physics and other s ...
the product
of the atomic number
and the fine-structure constant
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
to electronic momentum
. However, the Zeeman interaction
:
in an inhomogeneous magnetic field
produces a different mechanism of spin–orbit interaction through coupling the Pauli matrices
to the electron coordinate
. 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
Indium antimonide (InSb) is a crystalline compound made from the elements indium (In) and antimony (Sb). It is a narrow- gap semiconductor material from the III- V group used in infrared detectors, including thermal imaging cameras, FLIR sy ...
(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
frequency against a background of a wide
cyclotron resonance band. MacCombe et al.
working with high quality InSb observed isotropic EDSR driven by the
mechanism at the combinational frequency
where
is the cyclotron frequency. Strongly anisotropic EDSR band due to inversion-asymmetry
Dresselhaus spin–orbit coupling was observed in InSb at the spin-flip frequency
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
Mn
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
A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using s ...
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
Quantum dots (QDs) or semiconductor nanocrystals are semiconductor particles a few nanometres in size with optical and electronic properties that differ from those of larger particles via quantum mechanical effects. They are a central topic i ...
of about 50 nm size. Such dots can serve as
qubits
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
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
B
compounds also the
hyperfine coupling
In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate electronic energy levels and the resulting splittings in those electronic energy levels of atoms, molecules, and ions, due to electromagnetic multipole inte ...
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
:
,
the strength of interaction is characterized by the coefficient
. In InSb
quantum wires the magnitude of
of the atomic scale of about 1 eV
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.
See also
*
Fine electronic structure
*
Stark effect
The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several compon ...
*
Zeeman effect
The Zeeman effect () is the splitting of a spectral line into several components in the presence of a static magnetic field. It is caused by the interaction of the magnetic field with the magnetic moment of the atomic electron associated with ...
*
Electron electric dipole moment
The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field:
: U = - \mathbf d_ \cdot \mathbf E.
The electron's electric dipole moment (EDM ...
References
Further reading
*
*
*{{cite book, author1=G. L. Bir, author2=G. E. Pikus, title=Symmetry and Strain Induced Effects in Semiconductors, publisher=Wiley, location=New York, year=1975, isbn=978-0470073216
Quantum mechanics