Inquantum mechanics, thespin–orbit interaction (also calledspin–orbit effect orspin–orbit coupling) is arelativistic interaction of a particle'sspin with its motion inside apotential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in anelectron'satomic energy levels, due to electromagnetic interaction between the electron'smagnetic dipole, its orbital motion, and the electrostatic field of the positively chargednucleus. This phenomenon is detectable as a splitting ofspectral lines, which can be thought of as aZeeman effect product of two effects: the apparent magnetic field seen from the electron perspective due to special relativity and the magnetic moment of the electron associated with its intrinsic spin due to quantum mechanics.

For atoms, energy level splitting produced by the spin–orbit interaction is usually of the same order in size as the relativistic corrections to thekinetic energy and thezitterbewegung effect. The addition of these three corrections is known as thefine 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 thehyperfine structure.

A similar effect, due to the relationship betweenangular momentum and thestrong nuclear force, occurs forprotons andneutrons moving inside the nucleus, leading to a shift in their energy levels in thenuclear shell model. In the field ofspintronics, spin–orbit effects for electrons insemiconductors and other materials are explored for technological applications. The spin–orbit interaction is at the origin ofmagnetocrystalline anisotropy and thespin Hall effect.

The interaction was first introduced byLlewellyn Thomas in 1926.^[1]

This section presents a relatively simple and quantitative description of the spin–orbit interaction for an electron bound to ahydrogen-like atom, up to first order inperturbation theory, using somesemiclassicalelectrodynamics and non-relativistic quantum mechanics. This gives results that agree reasonably well with observations.

A rigorous calculation of the same result would userelativistic quantum mechanics, using theDirac equation, and would includemany-body interactions. Achieving an even more precise result would involve calculating small corrections fromquantum electrodynamics.

The energy of a magnetic moment in a magnetic field is given by$${\displaystyle \mathrm{\Delta }H=-\mathit{\mu }\cdot \mathbf{B},}$$whereμ is themagnetic moment of the particle, andB is themagnetic field it experiences.

We shall deal with themagnetic field first. Although in the rest frame of the nucleus, there is no magnetic field acting on the electron, thereis one in the rest frame of the electron (seeclassical electromagnetism and special relativity). Ignoring for now that this frame is notinertial, we end up with the equation$${\displaystyle \mathbf{B}=-\frac{\mathbf{v}\times \mathbf{E}}{c^2},}$$wherev is the velocity of the electron, andE is the electric field it travels through.^[a] Here, in the non-relativistic limit, we assume that the Lorentz factor${\displaystyle \gamma \backsimeq 1}$. Now we know thatE is radial, so we can rewrite$\mathbf{E}=\left|E\right|\frac{\mathbf{r}}{r}$.Also we know that the momentum of the electron${\displaystyle \mathbf{p}=m_{\text{e}}\mathbf{v}}$. Substituting these and changing the order of the cross product (using the identity${\displaystyle \mathbf{A}\times \mathbf{B}=-\mathbf{B}\times \mathbf{A}}$) gives$${\displaystyle \mathbf{B}=\frac{\mathbf{r}\times \mathbf{p}}{m_{\text{e}}{c}^2}\left|\frac{E}{r}\right|.}$$

Next, we express the electric field as the gradient of theelectric potential${\displaystyle \mathbf{E}=-\mathrm{\nabla }V}$. Here we make thecentral 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$${\displaystyle |E|=\left|\frac{\mathrm{\partial }V}{\mathrm{\partial }r}\right|=\frac{1}{e}\frac{\mathrm{\partial }U(r)}{\mathrm{\partial }r},}$$

where${\displaystyle U=-eV}$ is thepotential energy of the electron in the central field, ande is theelementary charge. Now we remember from classical mechanics that theangular momentum of a particle${\displaystyle \mathbf{L}=\mathbf{r}\times \mathbf{p}}$. Putting it all together, we get$${\displaystyle \mathbf{B}=\frac{1}{m_{\text{e}}e{c}^2}\frac{1}{r}\frac{\mathrm{\partial }U(r)}{\mathrm{\partial }r}\mathbf{L}.}$$

It is important to note at this point thatB is a positive number multiplied byL, meaning that themagnetic field is parallel to theorbitalangular momentum of the particle, which is itself perpendicular to the particle's velocity.

Thespin magnetic moment of the electron is$${\displaystyle {\mathit{\mu }}_S=-g_{\text{s}}{\mu }_{\text{B}}\frac{\mathbf{S}}{\hslash },}$$where${\displaystyle \mathbf{S}}$ is the spin (or intrinsic angular-momentum) vector,${\displaystyle {\mu }_{\text{B}}}$ is theBohr magneton, and${\displaystyle g_{\text{s}}=\mathrm{2.0023...}\approx 2}$ is the electron-sping-factor. Here${\displaystyle \mathit{\mu }}$ is a negative constant multiplied by thespin, so thespin magnetic moment is antiparallel to the spin.

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 toThomas precession.

The Larmor interaction energy is$${\displaystyle \mathrm{\Delta }{H}_{\text{L}}=-\mathit{\mu }\cdot \mathbf{B}.}$$

Substituting in this equation expressions for the spin magnetic moment and the magnetic field, one gets$${\displaystyle \mathrm{\Delta }{H}_{\text{L}}=\frac{g_{\text{s}}{\mu }_{\text{B}}}{\hslash m_{\text{e}}e{c}^2}\frac{1}{r}\frac{\mathrm{\partial }U(r)}{\mathrm{\partial }r}\mathbf{L}\cdot \mathbf{S}\approx \frac{2{\mu }_{\text{B}}}{\hslash m_{\text{e}}e{c}^2}\frac{1}{r}\frac{\mathrm{\partial }U(r)}{\mathrm{\partial }r}\mathbf{L}\cdot \mathbf{S}.}$$

Now we have to take into accountThomas precession correction for the electron's curved trajectory.

In 1926Llewellyn Thomas relativistically recomputed the doublet separation in the fine structure of the atom.^[2] Thomas precession rate${\displaystyle {\mathbf{\Omega }}_{\text{T}}}$ is related to the angular frequency of the orbital motion${\displaystyle \mathit{\omega }}$ of a spinning particle as follows:^[3][4]$${\displaystyle {\mathbf{\Omega }}_{\text{T}}=-\mathit{\omega }(\gamma -1),}$$where${\displaystyle \gamma }$ is theLorentz factor of the moving particle. The Hamiltonian producing the spin precession${\displaystyle {\mathbf{\Omega }}_{\text{T}}}$ is given by$${\displaystyle \mathrm{\Delta }{H}_{\text{T}}={\mathbf{\Omega }}_{\text{T}}\cdot \mathbf{S}.}$$

To the first order in${\displaystyle (v/c{)}^2}$, we obtain$${\displaystyle \mathrm{\Delta }{H}_{\text{T}}=-\frac{{\mu }_{\text{B}}}{\hslash m_{\text{e}}e{c}^2}\frac{1}{r}\frac{\mathrm{\partial }U(r)}{\mathrm{\partial }r}\mathbf{L}\cdot \mathbf{S}.}$$

The total spin–orbit potential in an external electrostatic potential takes the form$${\displaystyle \mathrm{\Delta }H\equiv \mathrm{\Delta }{H}_{\text{L}}+\mathrm{\Delta }{H}_{\text{T}}=\frac{(g_{\text{s}}-1){\mu }_{\text{B}}}{\hslash m_{\text{e}}e{c}^2}\frac{1}{r}\frac{\mathrm{\partial }U(r)}{\mathrm{\partial }r}\mathbf{L}\cdot \mathbf{S}\approx \frac{{\mu }_{\text{B}}}{\hslash m_{\text{e}}e{c}^2}\frac{1}{r}\frac{\mathrm{\partial }U(r)}{\mathrm{\partial }r}\mathbf{L}\cdot \mathbf{S}.}$$The net effect of Thomas precession is the reduction of the Larmor interaction energy by factor of about 1/2, which came to be known as theThomas half.

Thanks to all the above approximations, we can now evaluate the detailed energy shift in this model. Note thatL_z andS_z are no longer conserved quantities. In particular, we wish to find a new basis that diagonalizes bothH₀ (the non-perturbed Hamiltonian) andΔH. To find out what basis this is, we first define thetotal angular momentumoperator$${\displaystyle \mathbf{J}=\mathbf{L}+\mathbf{S}.}$$

Taking thedot product of this with itself, we get$${\displaystyle {\mathbf{J}}^2={\mathbf{L}}^2+{\mathbf{S}}^2+2\mathbf{L}\cdot \mathbf{S}}$$(sinceL andS commute), and therefore$${\displaystyle \mathbf{L}\cdot \mathbf{S}=\frac{1}{2}\left({\mathbf{J}}^2-{\mathbf{L}}^2-{\mathbf{S}}^2\right)}$$

It can be shown that the five operatorsH₀,J²,L²,S², andJ_z all commute with each other and with ΔH. Therefore, the basis we were looking for is the simultaneouseigenbasis of these five operators (i.e., the basis where all five are diagonal). Elements of this basis have the fivequantum numbers:${\displaystyle n}$ (the "principal quantum number"),${\displaystyle j}$ (the "total angular momentum quantum number"),${\displaystyle \ell }$ (the "orbital angular momentum quantum number"),${\displaystyle s}$ (the "spin quantum number"), and${\displaystyle j_z}$ (the "z component of total angular momentum").

To evaluate the energies, we note that$${\displaystyle ⟨\frac{1}{r^3}⟩=\frac{2}{a^3{n}^3\ell (\ell +1)(2\ell +1)}}$$for hydrogenic wavefunctions (here${\displaystyle a=\hslash /(Z\alpha m_{\text{e}}c)}$ is theBohr radius divided by the nuclear chargeZ); and$${\displaystyle ⟨\mathbf{L}\cdot \mathbf{S}⟩=\frac{1}{2}(⟨{\mathbf{J}}^2⟩-⟨{\mathbf{L}}^2⟩-⟨{\mathbf{S}}^2⟩)=\frac{{\hslash }^2}{2}(j(j+1)-\ell (\ell +1)-s(s+1)).}$$

We can now say that$${\displaystyle \mathrm{\Delta }E=\frac{\beta }{2}(j(j+1)-\ell (\ell +1)-s(s+1)),}$$where the spin-orbit coupling constant is$${\displaystyle \beta =\beta (n,l)=Z^4\frac{{\mu }_0}{4\pi }{g}_{\text{s}}{\mu }_{\text{B}}^2\frac{1}{n^3{a}_0^3\ell (\ell +1/2)(\ell +1)}.}$$

For the exact relativistic result, see thesolutions to the Dirac equation for a hydrogen-like atom.

The derivation above calculates the interaction energy in the (momentaneous) rest frame of the electron and in this reference frame there's a magnetic field that's absent in the rest frame of the nucleus.

Another approach is to calculate it in the rest frame of the nucleus, see for example George P. Fisher:Electric Dipole Moment of a Moving Magnetic Dipole (1971).^[5] However the rest frame calculation is sometimes avoided, because one has to account forhidden momentum.^[6]

In solid state physics and particle physics,Mott scattering describes the scattering of electrons out of an impurity which includes the spin-orbit effects.^[7][8] It is analogous to theCoulomb scattering (Rutherford scattering) with the addition of spin-orbit coupling. In particle physics, it is due to relativistic corrections.^[7]

This sectionmay be too technical for most readers to understand. Pleasehelp improve it tomake it understandable to non-experts, without removing the technical details.(December 2017) (Learn how and when to remove this message)

A crystalline solid (semiconductor, metal etc.) is characterized by itsband structure. 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 theFermi level (${\displaystyle E_{\text{F}}}$). The atomic${\displaystyle \mathbf{L}\cdot \mathbf{S}}$ (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 aboutRashba andDresselhaus interactions.

In crystalline solid contained paramagnetic ions, e.g. ions with unclosed d or f atomic subshell, localized electronic states exist.^[9][10] In this case, atomic-like electronic levels structure is shaped by intrinsic magnetic spin–orbit interactions and interactions withcrystalline electric fields.^[11] Such structure is namedthe fine electronic structure. Forrare-earth ions the spin–orbit interactions are much stronger than thecrystal electric field (CEF) interactions.^[12] The strong spin–orbit coupling makesJ 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(2J + 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${\displaystyle |J,J_z⟩}$, to determine which is the primary multiplet, theHund principles, known from atomic physics, are applied:

• The ground state of the terms' structure has the maximal valueS allowed by thePauli exclusion principle.
• The ground state has a maximal allowedL value, with maximalS.
• The primary multiplet has a correspondingJ = |L −S| when the shell is less than half full, andJ =L +S, where the fill is greater.

TheS,L andJ of the ground multiplet are determined byHund's rules. The ground multiplet is2J + 1 degenerated – its degeneracy is removed by CEF interactions and magnetic interactions. CEF interactions and magnetic interactions resemble, somehow, theStark and theZeeman effect known fromatomic physics. The energies and eigenfunctions of the discrete fine electronic structure are obtained by diagonalization of the (2J + 1)-dimensional matrix. The fine electronic structure can be directly detected by many different spectroscopic methods, including theinelastic neutron scattering (INS) experiments. The case of strong cubic CEF^[13] (for 3d transition-metal ions) interactions form group of levels (e.g.T_2g,A_2g), 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 atT = 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${\displaystyle |{\mathrm{\Gamma }}_n⟩}$ 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^[14] defined as the CEF widened by thermodynamic and analytical calculations defined as the supplement of the CEF theory by including thermodynamic and analytical calculations.

Hole bands of a bulk (3D) zinc-blende semiconductor will be split by${\displaystyle {\mathrm{\Delta }}_0}$ into heavy and light holes (which form a${\displaystyle {\mathrm{\Gamma }}_8}$ quadruplet in the${\displaystyle \mathrm{\Gamma }}$-point of the Brillouin zone) and a split-off band (${\displaystyle {\mathrm{\Gamma }}_7}$ doublet). Including two conduction bands (${\displaystyle {\mathrm{\Gamma }}_6}$ doublet in the${\displaystyle \mathrm{\Gamma }}$-point), the system is described by the effective eight-bandmodel of Kohn and Luttinger. If only top of the valence band is of interest (for example when${\displaystyle E_{\text{F}}\ll {\mathrm{\Delta }}_0}$, Fermi level measured from the top of the valence band), the proper four-band effective model is$${\displaystyle H_{\text{KL}}(k_{\text{x}},k_{\text{y}},k_{\text{z}})=-\frac{{\hslash }^2}{2m}\left[\left({\gamma }_1+\frac{5}{2}{\gamma }_2\right)k^2-2{\gamma }_2\left(J_{\text{x}}^2{k}_{\text{x}}^2+J_{\text{y}}^2{k}_{\text{y}}^2+J_{\text{z}}^2{k}_{\text{z}}^2\right)-2{\gamma }_3\sum _{m\ne n}{J}_m{J}_n{k}_m{k}_n\right]}$$where${\displaystyle {\gamma }_{1,2,3}}$ are the Luttinger parameters (analogous to the single effective mass of a one-band model of electrons) and${\displaystyle J_{\text{x},\text{y},\text{z}}}$ are angular momentum 3/2 matrices (${\displaystyle m}$ is the free electron mass). In combination with magnetization, this type of spin–orbit interaction will distort the electronic bands depending on the magnetization direction, thereby causingmagnetocrystalline anisotropy (a special type ofmagnetic anisotropy).If the semiconductor moreover lacks the inversion symmetry, the hole bands will exhibit cubic Dresselhaus splitting. Within the four bands (light and heavy holes), the dominant term is$${\displaystyle H_{\text{D}3}=b_{41}^{8\text{v}8\text{v}}[(k_{\text{x}}{k}_{\text{y}}^2-k_{\text{x}}{k}_{\text{z}}^2)J_{\text{x}}+(k_{\text{y}}{k}_{\text{z}}^2-k_{\text{y}}{k}_{\text{x}}^2)J_{\text{y}}+(k_{\text{z}}{k}_{\text{x}}^2-k_{\text{z}}{k}_{\text{y}}^2)J_{\text{z}}]}$$

where the material parameter${\displaystyle b_{41}^{8\text{v}8\text{v}}=-81.93\text{meV}\cdot {\text{nm}}^3}$ for GaAs (see pp. 72 in Winkler's book, according to more recent data the Dresselhaus constant in GaAs is 9 eVų;^[15] the total Hamiltonian will be${\displaystyle H_{\text{KL}}+H_{\text{D}3}}$).Two-dimensional electron gas in an asymmetric quantum well (or heterostructure) will feel the Rashba interaction. The appropriate two-band effective Hamiltonian is$${\displaystyle H_0+H_{\text{R}}=\frac{{\hslash }^2{k}^2}{2m^{\ast }}{\sigma }_0+\alpha (k_{\text{y}}{\sigma }_{\text{x}}-k_{\text{x}}{\sigma }_{\text{y}})}$$where${\displaystyle {\sigma }_0}$ is the 2 × 2 identity matrix,${\displaystyle {\sigma }_{\text{x},\text{y}}}$ the Pauli matrices and${\displaystyle m^{\ast }}$ the electron effective mass. The spin–orbit part of the Hamiltonian,${\displaystyle H_{\text{R}}}$ is parametrized by${\displaystyle \alpha }$, sometimes called the Rashba parameter (its definition somewhat varies), which is related to the structure asymmetry.

Above expressions for spin–orbit interaction couple spin matrices${\displaystyle \mathbf{J}}$ and${\displaystyle \mathit{\sigma }}$ to the quasi-momentum${\displaystyle \mathbf{k}}$, and to the vector potential${\displaystyle \mathbf{A}}$ of an AC electric field through thePeierls substitution$\mathbf{k}=-i\mathrm{\nabla }-\frac{e}{\hslash c}\mathbf{A}$. They are lower order terms of the Luttinger–Kohnk·p perturbation theory in powers of${\displaystyle k}$. Next terms of this expansion also produce terms that couple spin operators of the electron coordinate${\displaystyle \mathbf{r}}$. Indeed, a cross product${\displaystyle (\mathit{\sigma }\times \mathbf{k})}$ isinvariant with respect to time inversion. In cubic crystals, it has a symmetry of a vector and acquires a meaning of a spin–orbit contribution${\displaystyle {\mathit{r}}_{\text{SO}}}$ to the operator of coordinate. For electrons in semiconductors with a narrow gap${\displaystyle E_{\mathrm{G}}}$ between the conduction and heavy hole bands, Yafet derived the equation^[16][17]$${\displaystyle {\mathbf{r}}_{\text{SO}}=\frac{{\hslash }^{2}g}{4m_0}\left(\frac{1}{E_{\mathrm{G}}}+\frac{1}{E_{\mathrm{G}}+{\mathrm{\Delta }}_0}\right)(\mathit{\sigma }\times \mathbf{k})}$$where${\displaystyle m_0}$ is a free electron mass, and${\displaystyle g}$ is a${\displaystyle g}$-factor properly renormalized for spin–orbit interaction. This operator couples electron spin${\displaystyle \mathbf{S}=\frac{1}{2}\mathit{\sigma }}$ directly to the electric field${\displaystyle \mathbf{E}}$ through the interaction energy${\displaystyle -e({\mathbf{r}}_{\text{SO}}\cdot \mathbf{E})}$.

Electric dipole spin resonance (EDSR) is the coupling of the electron spin with an oscillating electric field. Similar to theelectron spin resonance (ESR) in which electrons can be excited with an electromagnetic wave with the energy given by theZeeman effect, in EDSR the resonance can be achieved if the frequency is related to the energy band splitting given by the spin–orbit coupling in solids. While in ESR the coupling is obtained via the magnetic part of the EM wave with the electron magnetic moment, the ESDR is the coupling of the electric part with the spin and motion of the electrons. This mechanism has been proposed for controlling the spin of electrons inquantum dots and othermesoscopic systems.^[18]

• Angular momentum coupling
• Angular momentum diagrams (quantum mechanics)
• Electric dipole spin resonance
• Kugel–Khomskii coupling
• Lamb shift
• Relativistic angular momentum
• Spherical basis
• Stark effect

• Condon, Edward U. & Shortley, G. H. (1935).The Theory of Atomic Spectra. Cambridge University Press.ISBN 978-0-521-09209-8.{{cite book}}:ISBN / Date incompatibility (help)
• Griffiths, David J. (2004).Introduction to Quantum Mechanics (2nd ed.). Prentice Hall.
• Landau, Lev;Lifshitz, Evgeny."§72. Fine structure of atomic levels".Quantum Mechanics: Non-Relativistic Theory, Volume 3.
• Yu, Peter Y.;Cardona, Manuel (1995).Fundamentals of Semiconductors. Springer.
• Winkler, Roland (2003).Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems. Springer.

• Manchon, Aurelien; Koo, Hyun Cheol; Nitta, Junsaku; Frolov, SM; Duine, RA (2015). "New perspectives for Rashba spin–orbit coupling".Nature.14 (9):871–82.arXiv:1507.02408.Bibcode:2015NatMa..14..871M.doi:10.1038/nmat4360.PMID 26288976.S2CID 24116488.
• Rashba, Emmanuel I. (2016)."Spin–orbit coupling goes global"(PDF).Journal of Physics: Condensed Matter.28 (42) 421004.Bibcode:2016JPCM...28P1004R.doi:10.1088/0953-8984/28/42/421004.PMID 27556280.S2CID 206047842.

• Atomic physics
• Magnetism
• Spintronics

• CS1 errors: ISBN date
• Articles with short description
• Short description is different from Wikidata
• Wikipedia articles that are too technical from December 2017
• All articles that are too technical