In
quantum physics, 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
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
's
atomic energy levels, due to electromagnetic interaction between the electron's
magnetic dipole
In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric ...
, its orbital motion, and the electrostatic field of the positively charged
nucleus. This phenomenon is detectable as a splitting of
spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
s, which can be thought of as a
Zeeman effect
The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel priz ...
product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
and the
strong nuclear force
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
, occurs for
protons and
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s moving inside the nucleus, leading to a shift in their energy levels in the nucleus
shell model
The SHELL model is a conceptual model of human factors that clarifies the scope of aviation human factors and assists in understanding the human factor relationships between aviation system resources/environment (the flying subsystem) and the huma ...
. In the 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 ...
, spin–orbit effects for electrons in
semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
s and other materials are explored for technological applications. The spin–orbit interaction is at the origin of
magnetocrystalline anisotropy
In physics, a ferromagnetic material is said to have magnetocrystalline anisotropy if it takes more energy to magnetize it in certain directions than in others. These directions are usually related to the principal axes of its crystal lattice. I ...
and the
spin Hall effect.
For atoms, energy level splitting produced by the spin–orbit interaction is usually of the same order in size as the relativistic corrections to the
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
and the
zitterbewegung effect. The addition of these three corrections is known as the
fine structure. The interaction between the magnetic field created by the electron and the magnetic moment of the nucleus is a slighter correction to the energy levels known as the
hyperfine structure.
In atomic energy levels
This section presents a relatively simple and quantitative description of the spin–orbit interaction for an electron bound to a
hydrogen-like atom
A hydrogen-like atom (or hydrogenic atom) is any atom or ion with a single valence electron. These atoms are isoelectronic with hydrogen. Examples of hydrogen-like atoms include, but are not limited to, hydrogen itself, all alkali metals such a ...
, up to first order in
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
, using some
semiclassical electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
and non-relativistic quantum mechanics. This gives results that agree reasonably well with observations.
A rigorous calculation of the same result would use
relativistic quantum mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light  ...
, using
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- massive particles, called "Dirac par ...
, and would include
many-body interactions. Achieving an even more precise result would involve calculating small corrections from
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
.
Energy of a magnetic moment
The energy of a magnetic moment in a magnetic field is given by
:
where μ is the
magnetic moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
of the particle, and B is the
magnetic field it experiences.
Magnetic field
We shall deal with the
magnetic field first. Although in the rest frame of the nucleus, there is no magnetic field acting on the electron, there ''is'' one in the rest frame of the electron (see
classical electromagnetism and special relativity). Ignoring for now that this frame is not
inertial
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
, we end up with the equation
:
where v is the velocity of the electron, and E is the electric field it travels through. Here, in the non-relativistic limit, we assume that the Lorentz factor
. Now we know that E is radial, so we can rewrite
.
Also we know that the momentum of the electron
. Substituting this in and changing the order of the cross product gives
:
Next, we express the electric field as the gradient of the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
. Here we make the
central field approximation, that is, that the electrostatic potential is spherically symmetric, so is only a function of radius. This approximation is exact for hydrogen and hydrogen-like systems. Now we can say that
:
where
is the
potential energy of the electron in the central field, and ''e'' is the
elementary charge. Now we remember from classical mechanics that the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of a particle
. Putting it all together, we get
:
It is important to note at this point that B is a positive number multiplied by L, meaning that the
magnetic field is parallel to the
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
al
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of the particle, which is itself perpendicular to the particle's velocity.
Spin magnetic moment of the electron
The
spin magnetic moment
In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. For example, the electron is an elementary spin-1/2 fermion. Quantum electrodynamics gives the ...
of the electron is
:
where
is the spin angular-momentum vector,
is the
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.
The Bohr magneton, in SI units is defined as
\mu_\mat ...
, and
is the electron-spin
g-factor. Here
is a negative constant multiplied by the
spin, so the
spin magnetic moment
In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. For example, the electron is an elementary spin-1/2 fermion. Quantum electrodynamics gives the ...
is antiparallel to the spin angular momentum.
The spin–orbit potential consists of two parts. The Larmor part is connected to the interaction of the spin magnetic moment of the electron with the magnetic field of the nucleus in the co-moving frame of the electron. The second contribution is related to
Thomas precession
In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a pa ...
.
Larmor interaction energy
The Larmor interaction energy is
:
Substituting in this equation expressions for the spin magnetic moment and the magnetic field, one gets
:
Now we have to take into account
Thomas precession
In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a pa ...
correction for the electron's curved trajectory.
Thomas interaction energy
In 1926
Llewellyn Thomas
Llewellyn Hilleth Thomas (21 October 1903 – 20 April 1992) was a British physicist and applied mathematician. He is best known for his contributions to atomic and molecular physics and solid-state physics. His key achievements include calculat ...
relativistically recomputed the doublet separation in the fine structure of the atom. Thomas precession rate
is related to the angular frequency of the orbital motion
of a spinning particle as follows:
:
where
is the
Lorentz factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
of the moving particle. The Hamiltonian producing the spin
precession
is given by
:
To the first order in
, we obtain
:
Total interaction energy
The total spin–orbit potential in an external electrostatic potential takes the form
:
The net effect of Thomas precession is the reduction of the Larmor interaction energy by factor 1/2, which came to be known as the ''Thomas half''.
Evaluating the energy shift
Thanks to all the above approximations, we can now evaluate the detailed energy shift in this model. Note that ''L''
z and ''S''
z are no longer conserved quantities. In particular, we wish to find a new basis that diagonalizes both ''H''
0 (the non-perturbed Hamiltonian) and Δ''H''. To find out what basis this is, we first define the
total angular momentum
In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
If s is the particle's sp ...
operator
:
Taking the dot product of this with itself, we get
:
(since L and S commute), and therefore
:
It can be shown that the five operators ''H''
0, ''J
2, L
2, S
2'', and ''J''
z all commute with each other and with Δ''H''. Therefore, the basis we were looking for is the simultaneous
eigenbasis
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of these five operators (i.e., the basis where all five are diagonal). Elements of this basis have the five
quantum numbers: ''
'' (the "principal quantum number"),
(the "total angular momentum quantum number"), ''
'' (the "orbital angular momentum quantum number"), ''
'' (the "spin quantum number"), and
(the "''z'' component of total angular momentum").
To evaluate the energies, we note that
:
for hydrogenic wavefunctions (here
is the
Bohr radius
The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
divided by the nuclear charge ''Z''); and
:
Final energy shift
We can now say that
:
where
:
For the exact relativistic result, see the
solutions to the Dirac equation for a hydrogen-like atom.
In solids
A crystalline solid (semiconductor, metal etc.) is characterized by its
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 ' ...
. While on the overall scale (including the core levels) the spin–orbit interaction is still a small perturbation, it may play a relatively more important role if we zoom in to bands close to the
Fermi level (
). The atomic
(spin–orbit) interaction, for example, splits bands that would be otherwise degenerate, and the particular form of this spin–orbit splitting (typically of the order of few to few hundred millielectronvolts) depends on the particular system. The bands of interest can be then described by various effective models, usually based on some perturbative approach. An example of how the atomic spin–orbit interaction influences the band structure of a crystal is explained in the article about
Rashba and
Dresselhaus interactions.
In crystalline solid contained paramagnetic ions, e.g. ions with unclosed d or f atomic subshell, localized electronic states exist. In this case, atomic-like electronic levels structure is shaped by intrinsic magnetic spin–orbit interactions and interactions with
crystalline electric fields. Such structure is named
the fine electronic structure. For
rare-earth
The rare-earth elements (REE), also called the rare-earth metals or (in context) rare-earth oxides or sometimes the lanthanides (yttrium and scandium are usually included as rare earths), are a set of 17 nearly-indistinguishable lustrous silve ...
ions the spin–orbit interactions are much stronger than the
crystal electric field (CEF) interactions. The strong spin–orbit coupling makes ''J'' a relatively good quantum number, because the first excited multiplet is at least ~130 meV (1500 K) above the primary multiplet. The result is that filling it at room temperature (300 K) is negligibly small. In this case, a (2''J'' + 1)-fold degenerated primary multiplet split by an external CEF can be treated as the basic contribution to the analysis of such systems' properties. In the case of approximate calculations for basis
, to determine which is the primary multiplet, the
Hund principles, known from atomic physics, are applied:
* The ground state of the terms' structure has the maximal value ''S'' allowed by the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
.
* The ground state has a maximal allowed ''L'' value, with maximal ''S''.
* The primary multiplet has a corresponding ''J'' = , ''L'' − ''S'', when the shell is less than half full, and ''J'' = ''L'' + ''S'', where the fill is greater.
The ''S'', ''L'' and ''J'' of the ground multiplet are determined by
Hund's rules. The ground multiplet is 2''J'' + 1 degenerated – its degeneracy is removed by CEF interactions and magnetic interactions. CEF interactions and magnetic interactions resemble, somehow,
Stark and
Zeeman effect
The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel priz ...
known from
atomic physics. The energies and eigenfunctions of the discrete fine electronic structure are obtained by diagonalization of the (2''J'' + 1)-dimensional matrix. The fine electronic structure can be directly detected by many different spectroscopic methods, including the
inelastic neutron scattering
Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...
(INS) experiments. The case of strong cubic CEF (for 3''d'' transition-metal ions) interactions form group of levels (e.g. ''T''
2''g'', ''A''
2''g''), which are partially split by spin–orbit interactions and (if occur) lower-symmetry CEF interactions. The energies and eigenfunctions of the discrete fine electronic structure (for the lowest term) are obtained by diagonalization of the (''2L + 1)(2S + 1)''-dimensional matrix. At zero temperature (''T'' = 0 K) only the lowest state is occupied. The magnetic moment at ''T'' = 0 K is equal to the moment of the ground state. It allows the evaluation of the total, spin and orbital moments. The eigenstates and corresponding eigenfunctions
can be found from direct diagonalization of Hamiltonian matrix containing crystal field and spin–orbit interactions. Taking into consideration the thermal population of states, the thermal evolution of the single-ion properties of the compound is established. This technique is based on the equivalent operator theory defined as the CEF widened by thermodynamic and analytical calculations defined as the supplement of the CEF theory by including thermodynamic and analytical calculations.
Examples of effective Hamiltonians
Hole bands of a bulk (3D) zinc-blende semiconductor will be split by
into heavy and light holes (which form a
quadruplet in the
-point of the Brillouin zone) and a split-off band (
doublet). Including two conduction bands (
doublet in the
-point), the system is described by the effective eight-band
model of Kohn and Luttinger. If only top of the valence band is of interest (for example when
, Fermi level measured from the top of the valence band), the proper four-band effective model is
: