more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionScatteringsSpinpolarized/ unpolarized electronsSpin statisticselectron gas in Magnetic FieldFerromagnetic metalsSpin TorqueSpinTorque CurrentSpinTransfer TorqueQuantum Nature of SpinQuestions & Answersmore Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionScatteringsSpinpolarized/ unpolarized electronsSpin statisticselectron gas in Magnetic FieldFerromagnetic metalsSpin TorqueSpinTorque CurrentSpinTransfer TorqueQuantum Nature of SpinQuestions & Answersmore Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11

SpinOrbit Interaction
Spin and Charge TransportRelativistic origin of the spinorbit interactionAn electron moving in an electrical field experiences an effective magnetic field, which acts on the electron magnetic moment (spin). The interaction of the electron magnetic moment with the effective magnetic field is called the spinorbit interaction.The weak and the strong spinorbit interaction can be distinguished from the type electrical field, which induces it:the weak spinorbit interaction, which is induced by an external electrical field or an intrinsic electrical field in a junction or in quantum well. the strong spinorbit interaction, which is induced by atomic nuclear. In the vicinity of the nuclear this field is huge and it can induce a substantial spinorbit interaction.Breaking of orbital symmetry and timeinverse symmetry are two critical factors, which determine the magnitude of the spinorbit interactionContentclick on the chapter for the shortcut(1). Explanation in short(2). Relativistic origin of the SpinOrbit interaction(3). Lorentz transformation(4). Spinorbit interaction in macro world(5). Weak and Strong Spinorbit interaction(6).Magnitude of the Spinorbit interaction.(7). Spinorbit interaction in the centrosymmetric electric field of atomic nuclear(8).SpinOrbit interaction due to the orbital deformation(9). Time inverse symmetry and the spinorbit interaction(10).Enhancement of magnetic field due to the spinorbit interaction(11). gfactor(12).Perpendiculartoplane magnetic anisotropy (PMA)(13). Magnetoelastic effect(14).Voltageinduced spinorbit interaction & VCMA effect(15). Spindependent scatterings(16).Voltageinduced spinorbit interaction & VCMA effect(17). Famous misunderstands and misinterpretations of the SpinOrbit (SO) interaction(18).Spinorbit interaction obtained from the Dirac equations(19). Pauli equation & SpinOrbit interaction(20).Merits and demerits of the use of the Hamiltonian approach for calculations of the SpinOrbit interaction..........Key properties of the spinorbit interactionThe spinorbit interaction is relativistic effect (not quantummechanical).The spinorbit interaction is described by the effective magnetic field H_{SO} of the spinorbit interaction. The H_{SO} is a real magnetic field, which is indistinguishable from a common magnetic field (An exception, see here)Only the electrical field of a nuclear can induced a substantial spinorbit interactionIn the case of a spherical orbital, the electron experiences oppositesign of H_{SO} in equal amounts. As a result, the electron does not experience any spinorbit interactionIn order to experience the spinorbit interaction in the electrical field of an atomic nuclear, two condition should be satisfied.(condition 1): The electron orbital should be deformed. It should not be spherical. It is better it should be not centrosymmetric.(condition 2): The timeinverse symmetry for the orbital should be broken. It means that an external magnetic field should be applied...... What difference the spinorbit interaction does make? What does the spinorbit interaction affect and influence?Effect 1: Enhancement (magnification) of the applied magnetic field. Effect 2: Spindependent scatterings. Effect 3: Spin precession. Spin relaxation. When electron moves across a strong electrical field, the effective magnetic field of the spinorbit interaction causes a spin precession. where it is the case: a electrical current flowing along an interface or a junction. When an electron may move at different angles, it may cause different directions of the precession, therefore the spin relaxation. Which specific changes the spinorbit interaction does?In a nonmagnetic material (paramagnetic or diamagnetic)(Effect 1) gfactor becomes larger than gfactor of an electron in the free space ; (Effect 2)The bulktype Spin Hall effect due to scatterings on nonmagnetic and magnetic impurities (Effect 3) The interfacetype Spin Hall effect due to interface scatterings (Effect 4) spin relaxation becomes larger. Especially for delocalized electrons (conduction electrons) of p symmetry (d or f as well) In a ferromagnetic material(Effect 5) saturation magnetization becomes larger (exchange interaction is enhanced due to the spinorbit interaction) (Effect 6)interfaceinduced perpendicular anisotropy (for example, Co/Pt). It is due to a large difference in the spinorbit enhancement for magnetic field directed along and across the interface (Effect 7) changing the magnetization and magnetization direction due to the stress. Magnetostriction (magneto elastic) effect. The stress in a metallic singlecrystal multilayer structure. (Effect 8) Anomalous Hall effect (Effect 9) Changing of the Curie temperature in a thin ferromagnetic due to the interfaceenhanced spinorbit interaction. Q1. The SpinOrbit interaction. What it is ??The SpinOrbit interaction describes the fact that an electron experiences an effective magnetic field when it moves in an electrical field. Q2. The SpinOrbit interaction. How does it affect an electron??The effective magnetic field of the SpinOrbit interaction affects only the electron spin. There are only two interactions: 1) There can be a spin precession 2) There can be a damping of the spin precession, which aligns the electron spin along the effective magnetic field of the spinorbit interaction Important Note 1: The effective magnetic field of the SpinOrbit interaction cannot induce the Lorentz force or the Hall effect.Important Note 2: The effective magnetic field of the SpinOrbit interaction does not interact with the magnetic moment induced by the orbital moment of the electron.Q3.Which electrons may experience the SpinOrbit interaction?All???Only following electrons type (1): Electrons, which occupy an electron state by one. In the case of conduction electrons, they are defined as spinunpolarized and spinunpolarized electrons (See here) The electrons of states, which are occupied by two electrons of opposite spins, do not experience the SpinOrbit interaction. It is because the spin of such states is zero and the timeinverse symmetry is not broken for such state. Example (1) of no SO : deepstate electrons; Example (2) of no SO: conduction electrons of an energy substantially smaller than the Fermi energy (spininactive electrons, See here) type (2): An electron, which has nonzero speed component perpendicular to an applied electrical field. type (3): An electron has a nonzero orbital moment . Fact (1) : ~95% spinorbit related effects in a solid are induced by the electrical field of atomic nuclear and occur due to the electron orbiting the atomic nuclear Fact (2) : The spinorbit interaction is fully described by the effective magnetic field H_{SO} of the spinorbit interaction Example to understand how the spinorbit interaction works in a solid
Magnetoelastic effectThis effect described facts that the elastic stress may enhance the spinorbit interaction and increase the energy of the perpendicular magnetic anisotropy (PMA).see details hereWhen a pressure applied to the film, the atomic orbitals are deformed. There are two types of deformations. (type 1): The orbital becomes more elliptical. (type 2): nuclears are shifted out of the center of the orbital. Both deformation makes the effective magnetic field H_{SO} of the spinorbit interaction larger. In a ferromagnetic material the localized electrons have a noncompensated spin, which creates a magnetic field H_{mag} At an interface between a magnetic and nonmagnetic material, the demagnetization field H_{demag} is created due to uncompensated magnetic moment at the interface. The direction of H_{demag} is perpendicular to the interface and opposite to H_{mag}. The magnetic field H_{inside} inside of the ferromagnetic field equals H_{mag} H_{demag}. The H_{inside} is the total magnetic field except H_{SO}. It includes the external magnetic field Important fact: Additionally, the electron experience the effective magnetic field H_{SO} of the spinorbit interaction, which is always directed along H_{inside}. The magnitude of H_{SO} is proportional to H_{inside} and the degree of the orbital deformation. Without a deformation the orbitals of the localized electrons is nearly spherical and the effective magnetic field H_{SO} of the spinorbit interaction is small. When the pressure applied, the orbitals are deforms in the direction of the applied pressure and the effective magnetic field H_{SO} of the spinorbit interaction increases.
The magnetic energy of an electron equals to a product of the electron spin and H_{inside}+H_{SO}. When magnetization is perpendicular to the film, the orbital deformation is larger, H_{SO} is larger and the magnetic energy is larger. When magnetization is inplane, the orbital deformation is smaller, H_{SO} is smaller and the magnetic energy is smaller. The dependence of the magnetic energy on the magnetization direction is called the magnetic anisotropy. In the case when the difference of the magnetic energy are with respect to the interface, the effect is called the perpendicular magnetic anisotropy (PMA) Relativistic origin of the SpinOrbit interaction
The Theory of Relativity states that a particle moving in an electrical field experiences an effective magnetic field, which is directed perpendicularly to the electrical field and perpendicularly to the particle movement direction. The interaction of this effective magnetic field with the electron spin is called SpinOrbit interaction. It is important to emphasize that the direction and magnitude of the effective magnetic field does not depend either on the particle charge or on the particle spin.
According to the Theory of The Relativity the electric and magnetic field mutually transformed into each other depending on the speed of an observer. For example, if in a coordinate system of static observer there is only a magnetic field, a movable observer will experience this field as both an electrical field and a magnetic field. A particle moving in a static magnetic field experiences an effective electric field. The effective electrical field acts on the particle charge (the Lorentz force, Hall effect) and forces the particle to move along this field. A particle moving in a static electrical field experiences an effective magnetic field. The effective magnetic field acts on the particle magnetic moment (spinorbit interaction) and causes the precession of the magnetic moment around the direction of the effective magnetic field.
Lorentz transformationThe electromagnet field is a relativistic object and it is the Lorentz transformation rules as where E_{static}, H_{static} _{} are the electric and magnetic field in the static coordinate system (reference frame) and E_{move}, H_{move}_{} are the electric and magnetic field in the coordinate system, which moves with a constant speed v. As a result, an electron, which moves in a static magnetic field H_{static}, experience in own reference frame an effective electrical field E_{Hall} , which is called the Hall field (Hall voltage). Similarly, when an electron moves in a static electrical field E_{static}, it experience in own reference frame an effective magnetic field H_{SO} , which is called the effective spinorbit magnetic field For example, when an electron moves in the xdirection in Nonrelativistic case (v<<c),in this case and The Hall field can be calculated as The spinorbit magnetic field can be calculated as Note: An electron should have velocity component perpendicular to a static electrical field E_{static} or a static magnetic field H_{static} in order to experience the magnetic field H_{SO} of spinorbit interaction or the Hall field E_{Hall} An electron always moves along the electrical field (but not perpendicularly). Then, how it can experience the spinorbit interaction?The trajectory of an electron may be very different. There are many cases when an electron moves perpendicularly to an electrical field. For example, when the electron is orbiting around a nuclear. Another example, an electron current along an interface. Usually, there is an electrical field perpendicularly to the interface, but still the electron may move along the interface (See below)
effective spinorbit magnetic field H_{SO}
The Hall Effect and the SpinOrbit interaction are close cousinsthe Hall effect ==== results in ====> an effective electrical field the SpinOrbit interaction ===results in=====> an effective magnetic field Since the spinorbit magnetic field H_{SO} is proportional to 1/c^{2} , the should be very small or almost negligibly small. Should we care about such a tiny effect?it is correct. Usually, the H_{SO} is very small except cases when the electrical field E_{static} is huge. It is the case in close proximity to the nuclear. There the electrical field increase as 1/r, where r is the distance to the nuclear. The nuclear is almost "pointlike" object, therefore the electrical field E_{static} is huge in close proximity of the nuclear. It makes a large H_{SO} Any substantial spinorbit interaction is induced only by an electrical field of a nuclear!!!Other realistic sources of the electrical field in a solid induce only a very weak spinorbit interactionIn close vicinity of a nuclear an electron experiences a very strong electrical field of the nuclear. However, this field is very symmetric and the electron experience the opposite signs of the spinorbit interaction on its path around nuclear. Therefore, the spinorbit interaction cancels itself and the electron experience no spinorbit interaction. An externally applied electrical field or magnetic field or stress field may break the symmetry and the the electron starts to experience very strong effective magnetic field of the spinorbit interaction. For example, when only only 100 Oe of external magnetic field is applied, an electron may experience an effective magnetic field of 10 000 Oe due to the spinorbit interaction. Q. Is the SpinOrbit interaction a quantummechanical effect???
A. No. The SpinOrbit interaction affects both small objects and large objects. The spinorbit interaction exists in the macro world as well.
For example, Figure 5 shows an imaginary case what would happen if the Sun were charged. In this case the magnetic moment of the Earth would interact with the effective magnetic field of the spinorbital interaction induced by this charge. The magnetic moment of the Earth would be aligned accordingly as it is shown in Fig.5.
Weak and Strong Spinorbit interactionTwo types of the SpinOrbit (SO) interactionIn order to experience the Spinorbit interaction an electron should move, it should move and its movement direction should not be along the electrical field. According to the electrical field, which induces the Spinorbit interaction, two types of the Spinorbit interaction can be distinguished: (1) weak SO interaction induced by an external electrical field (SO is proportional to an electron speed). SO is small (2) strong SO interaction induced by a centrosymmetric electrical field of atomic nuclear (SO is proportional to an electron symmetry or an orbital moment of electron). SO can be large
The delocalized (conduction) electrons move simultaneously in the forward direction along lattice and around each atom (nuclear) of the lattice.
The localized electrons (d,f) do not move along lattice. They only rotates around nuclears.
Note: Even though the symmetry of delocalized electrons is s and/or p, they may have a little mixture ofd and f symmetry.
. Magnitude of the Spinorbit interaction.Generally, the magnitude of the spininteraction in a solid is small!!!. For example, when an electron moves across an external electrical field: (a moderate electrical field + a moderate electron speed = a very small spinorbit interaction) Except for an electron, which moves in a close vicinity of an atomic nuclear a very strong electrical field + a moderate electron speed = a strong spinorbit interaction
Example 1.
Even in the of the highestpossible electron speed in solid and largestpossible applied electrical field, the effective magnetic field of the spinorbit interaction is small!! Estimation: Maximum electron speed + Maximum applied electrical fieldElectron Speed: Saturation Velocity :1E7 m/s (GaAs Si ) It is maximum drift speed of electrons in a solid. Experimentally I have measured the saturation velocity (See here). An electron can not go faster, because above the saturation velocity the electron intensively illuminate phonons. It is similar to the case when a supersonic plane flies faster than the speed of sound.The applied voltage: Breakdown voltage: 5E5 V/cm(GaAs, Si) It is maximum voltage, which could be applied to a semiconductor (a oxide). For higher voltage the avalanche breakdown occurs. Result: The effective magnetic field of the spinorbit interaction is only 0.5 Gauss It is too small!!! For example, Earth's magnetic field at at the Earth's surface ranges from 0.250.65 Gauss. Example 2. An electron rotating around a nuclear.Electron Speed: linear speed of electron rotating around atom : ~2.1E6 m/s The applied voltage:Coulomb Electrical Field in H atom at 1st orbital (r=0.053 nm) 5.1E9 V/cm The high voltage is because the orbit is very close to the nuclear !!!Result: The effective magnetic field of the spinorbit interaction is 125 kGauss=12.5 TIt is rather large!!!. Such large magnetic field can only be obtained by a superconducting magnet. It is important: An electron may experience such large magnetic field only when it is very near to the nuclear and only when the electron is rotating around the nuclear.
Q. Both localized (d,f) and delocalized (s,p) electrons are rotating around nuclears (atoms), is it sufficient for them to experience a strong spinorbit interaction? A. No. It is far not sufficient. There are several other conditions the electron should satisfy in order to experience the spinorbit interaction: The following describes the reasons why an electron does not experience the spinorbit interaction when the electron orbit is spherical and why it does experience the spinorbit interaction for other shapes of the orbital.
Spinorbit interaction in the centrosymmetric electric field of atomic nuclear.
Q. How is it possible that an electron, while rotating around a nuclear, does not experience the SpinOrbit Interaction ???? A. It is because, for the spherical orbit an electrons makes an equal number of rotations in two opposite directions. Since for opposite rotation directions the directions of the effective magnetic field of the spinorbit interaction are opposite, an electron does not experience any spinorbit interaction.
When an electron rotates in a spherical orbital (sorbital), it does not experience any spinorbital interaction. For the sorbital the orbital moment is zero. That means that the electron rotates in clockwise and anti clockwise directions an equal number of times. The effective magnetic field the electron experiences due the spinorbit interaction is zero, because it is fully compensated during rotations in opposite directions. When an electron rotates in an elliptical orbital (d or porbital), the orbital momentum is not zero and the electron rotates in one direction more than in the opposite direction. In this case the electron experiences the effective magnetic field due to the spinorbit interaction.
The following is important Q. In case of sorbital a half of rotations an electron experience the field of the spinorbit interaction in one direction and on another half in the opposite direction. This case should be different from the case when the does not experience the spinorbit interaction at all. Therefore, the spinorbit interaction still does affect the electron of sorbital. Is it correct? A. No, it is not correct. The spinorbit interaction does not affect an electron of sorbital for the following reason: An electron is an elementary particle, which could not be divided into the parts (See here). Therefore, it is impossible that the spin of one part of the electron rotates in one direction and the spin of another part rotates in a different direction. A whole electron has only one direction of the spin. In the case when along the length of electron (the meanfree path) or along the electron orbit the magnetic field changes, the spin interact with an average magnetic field. It is important there is always one defined spin direction for one electron.
Q. How to make an electron to rotate in one direction more than in the opposite direction??? How to make the spinorbit interaction stronger?? Simple Answer: It is necessary to deform the electron orbital.
The deformation or distortion of the electron orbital can be done be an external electrical field or stress.See VCMA effectThe orbital can be distorted by an electrical field. In this case, the electron experiences the effective magnetic field due to the spinorbit interaction. When the orbital is distorted by an external electrical field, the existence of the effective magnetic field due to the spinorbit interaction is called the Rashba effect. When the orbital is distorted by an axial crystal field, the existence of the effective magnetic field due to the spinorbit interaction is called the Dresselhaus effect. Note: The external magnetic field may deform the orbit. However, the deformation is very small. The magnetic field has another important function for the SO. The magnetic field breaks the timeinverse symmetry, which is a key condition for SO to occur (See below). Direct (weak) and indirect (strong) spinorbit interaction (SO)There are two kinds of the spinorbit interaction in a crystal lattice. In both cases an electron experiences an effective magnetic field of the spinorbit interaction. Direct (weak) An electron moves perpendicularly to an electrical field. The electrical field directly induces the magnetic field. For example, such electron movement across an electrical field is possible in in a quantum well. The electrical field could be an externallyapplied electrical field, an axial crystal field or/and an electrical field across interface or junction due to a charge accumulation. Only delocalized (conduction) electrons may experience the direct SO. Indirect (strong) In this case the magnetic field of the spinorbit interaction is induced not by an external electrical field, but by the electrical field of a nuclear. The external electrical field just deforms the electron orbital making the spinorbit interaction stronger. In contrary to the direct SO, in the case of the indirect SO it is not necessary for an electron to move along the crystal lattice. Therefore, the indirect spinorbit interaction may experience localized electrons, delocalized (conduction) electrons and standingwave electrons. In contrast to direct SO, the indirect SO can occurs only when the timeinverse symmetry is broken. It can be broken by an external magnetic field or a local magnetic field. (See below)
The spinorbit interaction in compound metals and semiconductors.When a crystal consists of different atoms, the electrons are distributed asymmetrically. Some electron orbit is shifted from a cation to be closer to anion. , the orbital becomes deformed. That causes a stronger spinorbit interaction. This is reason, for example, why the spinorbit interaction is significantly stronger in GaAs than in Si. In an ionic crystal the covalent electrons are nearlyfully transformed from a cation to a anion and the electron orbital becomes again more centersymmetrical with a weak spinorbit interaction. This is reason, for example, why the spinorbit interaction is significantly weaker in ZnO than in GaAs. Q. Why the spin orbit interaction is larger in a heavy element with a larger atomic number??Simple answer: The strength of the spinorbit interaction is directly proportional to the electric field of the nuclear. The nuclear charge is larger for an element of a larger atomic number. Therefore, the electrical field of the nuclear and the spinorbit interaction, which is induced by this field, becomes larger as well. Another reason: the screening by inner electrons becomes weaker and asymmetrical (See below) Screening of spinorbit interactions by inner electrons.
Because of the screening of an electrical field of a nuclear by inner electrons , the strength of spinorbit interaction reduces. The effects of screening: (effect 1) The spinorbit interaction (SO), which is induced by a anion, is smaller than the SO, which is induced by a cation. Since there are more electrons in the vicinity of an anion than in the vicinity of cation, the screening of nuclear field of anion is larger. Therefore, the spinorbit interaction induced by the nuclear of anion is smaller. (effect 2) In atoms of unfilled inner shells the spinorbit interaction is stronger. In the case when the inner shell of atom is not fully filled, the screening of the nuclear by the electrons of the inner shell is not centrosymmetric. It makes the spinorbit interaction stronger. It is good to know.1. As Note: The orbital moment as well as the electron speed are different at different points of the Brillouin zone.
2. The magnetic moment of an electron is a quantum mechanical sum of magnetic moments induced by the spin and induced by the orbital moment. The effective magnetic field acts only on spin and it does not effect the magnetic moment due to the orbital moment.
SpinOrbit interaction due to the orbital deformation
The effective magnetic field of the spinorbit interaction for localized electrons due to a deformation of electron orbit may be very large. It may reach 130 kOe and larger. The effective magnetic field for the delocalized electrons is smaller, but still it may be large.The type of orbit deformation, which may enlarge the spinorbit interaction (1) The electron orbit should be deformed along one direction (2) The electron orbit should be deformed asymmetrically in respect to its nuclear
The orbital is significantly deformed in compound materials with covalent bonding (like GaAs). Therefore, they have a larger spinorbit interaction. In materials with ionic bonding, the orbital is less deformed and they have a smaller spinorbit interaction (like ZnO). The p , d and f orbitals are inherently asymmetrical. For each individual p , d and f orbital, the spinorbit interaction may be strong. For each individual p or d or f orbital, the timeinverse symmetry is broken. However, in a nonmagnetic metal or a semiconductor, where the timeinverse symmetry is not broken, the electron wavefunction is a combination of the wave functions of different moments and it is more symmetric. Therefore, in a crystal the spinorbit interaction of electrons of p or d or f symmetry may be not as strong as in the case of a separated atom.
Enhancement of magnetic field due to the spinorbit interaction
Along an applied external magnetic field, an additional magnetic field is induced due to the spinorbit interaction. Therefore, an electron experience a larger magnetic field than externally applied due to the spinorbit interactionThis is the most important property of the spinorbit interaction !!!.This property determines how the spinorbit interaction affects electrons in a solidIn fact, it is the joint work of two relativistic effects: 1) the Lorentz force 2) the spinorbit interaction  The Lorentz force, which is induced by an external magnetic field, deforms the electron orbital and breaks the timeinverse symmetry;  Because of the broken timeinverse symmetry, the strong effective magnetic field is induced by the spinorbit interaction.
Time inverse symmetry and the spinorbit interaction
Timeinverse symmetry: not broken Average effective magnetic field of the spinorbit interaction: zero
When the timeinverse symmetry in the material is not broken, there is an equal probability that electron circulating around the nuclear in the clockwise and anti clockwise directions. Since the electron experiences equal and opposite effective magnetic field of the spinorbit interaction, in the average the electron does not experiences any effective magnetic field of the spinorbit interaction. (See Fig. above) Even in the case when the orbital moment of the electron is not zero, when the timeinverse symmetry in the material is not broken, there is an equal probability for an electron to occupy the orbit with opposite orbital moment and again the average effective magnetic field of the spinorbit interaction: zero note: in this case the spinorbit interaction affects the spin relaxation 
Timeinverse symmetry: broken
Average effective magnetic field of the spinorbit interaction: nonzero
An external magnetic field (or exchange field) breaks the timeinverse symmetry and it causes a nonzero average effective magnetic field of the spinorbit interaction in the direction of the external magnetic field. Since the electron moves around the nuclear, it experiences the Lorentz force in the magnetic field. The Lorentz force is in opposite directions for electron moving in the clockwise and anticlockwise directions around the magnetic field. The Lorentz force modifies the orbital of electrons. When an electron moves in the anticlockwise direction, it moves closer to the nuclear and it experiences the larger electrical field from the nuclear and the larger corresponded effective magnetic field of the the spinorbit interaction. When an electron moves in the clockwise direction, it moves more distant from the nuclear and it experiences the smaller electrical field from the nuclear and the smaller corresponded effective magnetic field of the the spinorbit interaction. In the average, the average the electron experiences a nonzero effective magnetic field of the the spinorbit interaction in the direction of the external magnetic field. note: The effective electrical field of the Lorentz force can not induce the spinorbit interaction, because of its relativistic nature (See here)
Fig. 13 shows the diamagnetic response of the atom to the external magnetic field. Therefore, a material with the largest diamagnetic constant should have the largest spinorbit interaction. Notice: all electrons have the diamagnetic response shown in Fig. 13, including electrons of the inner orbitals and electrons of the inert gases. However, the electrons of the the external orbitals have uncompensated spin and only they experiences the spinorbit interaction.
Enhancement of magnetic field due to the spinorbit interaction
When a magnetic applied to the material, it breaks the time inverse symmetry. As result, the electron starts to experience nonzero effective field of the spinorbit interaction. The effective magnetic field of the spinorbit interaction is in the same direction as the applied external magnetic field. The total magnetic field, which the electron experiences, becomes larger. In some cases, the total effective magnetic field may be a significantly larger than the external magnetic field.
Both the external applied magnetic field and the exchange field can be amplified due to the spinorbit interaction. The induced effective magnetic field of the spinorbit interaction may be significantly different for different directions of the applied external magnetic field. It is the largest in the direction, in which the electron orbit is deformed (See Fig. 14).
gfactorwiki page about gfactor is here The gfactor describes the ratio between the spin or the orbital moment and the magnetic moment of an electron For an electron in free space the value of g equals to 2.002319
 There are two cases where the gfactor is used and it can be measured: 1) Ferromagnetic resonance and electron paramagnetic resonance. The gfactor describes the precession frequency (Larmor frequency) of the spin in an external magnetic field. The external magnetic field is applied at an angle with respect to the spin direction. 2) Zeeman effect. The gfactor describes the energy difference for electrons, which spins are along and opposite to the direction of the magnetic field.  Important notice: The gfactor, which is measured from the ferromagnetic or paramagnetic resonance, is not always same as the gfactor, which is measured from the Zeeman spliting. The reason of the difference: In a solid there is no precession of the orbital moment in a magnetic field (See here) , but the orbital moment contributes to the Zeeman spliting.  Orbital moment and the gfactorIn atoms when the spin is compensated and the magnetic moment is only due to the orbital moment, the gfactor equals to 1. The gfactor of atoms of gas is between 2 and 1. In crystal: 1) The orbital moment of localized and delocalized electrons in a crystal does not contribute to the ferromagnetic or paramagnetic resonance, because the external electron orbits are fixed by the crystal structure and the interactions with neighbor atoms. 2) orbital moment is contributes to the Zeeman splitting.  The spinorbit interaction and the gfactornonmagnetic materials (paramagnetic and diamagnetic materials) When an external magnetic field is applied, an electron in crystal experiences a larger magnetic field, because the effective magnetic field is enlarged due to the spinorbit interaction. Even though in the reality the effective electron gfactor does not change and only the effective magnetic field changes due to the spinorbit interaction, it is convenient to assume the gfactor of the material is changed, but the magnetic field remains unchanged. Therefore, the Larmor frequency can be calculated as where k_{SO} is coefficient, which described the enhancement of the magnetic field due to the spinorbit interaction. From Eq. (g4), the Larmor frequency is calculated as where the gfactor is Often the gfactor is defined and measured for the external magnetic field strength H instead of the magnetic induction B. In this case the effective gfactor can be used gfactor and Specific magnetic susceptibility for nonmagnetic materials
Content gfactor conduction band (bulk): GaAs : 0.3 (300 K) 0.45 (50 K) InAs: 15 InP: 1.5 GaSb=8 InSb=51.3 nSi: =1.9985 pSi=2
Cu=
Specific magnetic susceptibility (CGSemu=Siunit/4pi)
Paramagnetic (Si unit)
Diamagnetic (Si unit)
EPR for ge=2 9.3882 GHz> 3.35 kG
ferromagnetic metals In ferromagnetic metals gfactor, saturated magnetization and width of FMR peak in ferromagnetic metals. Click to extend
;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;FMR ;;;;;;;;;;;;;;;;;;;;;;;;; relaxation parameters for YIG width of FMR resonance
Q. Why the spinorbital enhancement of the magnetic field cannot be included into Magnetic susceptibilityIn the of paramagnetic metals, the spinorbit interaction does not produced any additional magnetic field or magnetization inside material. It only makes larger the effective magnetic field, which the electron experiences. Note: Both the magnetic susceptibility and the spinorbit interaction enhance the effective magnetic field, which an electron experiences. Except ferromagnetic metals, the enhancement due to the magnetic susceptibility is much weaker than the enhancement due to the spinorbit interaction. For example (See above), in diamagnetic materials the enhancement is only about 0.001 %, in paramagnetic it is 0.01 %. As can be evaluated from the gfactor, the enhancement due to the spinorbit interaction is 110 % for the most of materials. In the case of IIIV semiconductors (GaAs,InAs), the enhancement may be more than 100 %.
Perpendiculartoplane magnetic anisotropy (PMA)Detailed description of the PMA is hereExample 1: Fe/Pt, Co/Pt
The magnetization of a singlematerial ferromagnetic film is in the film plane. In the case when the film consists of a thin layers of different metals, the magnetization could be out of plane. The example of such multilayered films are Co/Pt, Fe(fcc)/Pt, Co/Tb, Fe(fcc)/Tb. Since the strength of the spinorbit interaction depends of the shape of the electron orbit in a crystal, the perpendiculartoplane magnetic anisotropy only a feature of a specific crystal orientation and only a specific crystal orientations of the interfaces. For example, in all abovementioned cases the perpendiculartoplane magnetic anisotropy is feature of only fcc(111) interfaces or very similar hcp interfaces Perpendiculartoplane magnetic anisotropy occurs due to a strong effective magnetic field of the spinorbit interaction in the vicinity of the contact. The enhancement of the effective field of the spinorbit interaction occurs because of a deformation of the orbital of the ferromagnetic and nonmagnetic metals in the close vicinity of the interface.
In the bulk of the metals, the shape of the orbitals are close to a sphere (shown as the red and bluecolored spheres). In the vicinity of the contact, the orbitals are deformed. In the bulk: There is an exchange interaction in the bulk of Fe, but there is no exchange interaction in the bulk of Pt. The blue arrows show the effective magnetic field of the exchange interaction is shown The effective magnetic field of the exchange interaction breaks the timeinverse symmetry. It induces the effective magnetic field of the the exchange interaction (red arrows) along the direction of the exchange field. The effective magnetic field of the spinorbit interaction is small in the bulk of the Fe, because its nearspherical orbital. In the bulk of Pt the effective field of the spinorbit interaction is zero, because the timeinverse symmetry is not broken in there. In the vicinity of the interface:  the exchange field in Fe becomes smaller, because there are less Fe atoms in the surrounding.  the exchange field in Pt is nonzero, because of the magnetic proximity effect or the interlayer exchange coupling (See here) the effective magnetic field of the spinorbit interaction is nonzero in Pt. It can be larger than this field in Fe, because the electrical charge of the Pt nuclear (+78) is larger than charge of Fe nuclear (+26) and the distance between atoms nearly the same.
Dependence on the magnetization direction  exchange field does not depend on the magnetization direction.  the effective magnetic field of the spinorbit interaction significantly depends on the magnetization direction. Because of the deformation of the orbitals, it is large, when the magnetization is perpendicular to the plane, and it is small, when the magnetization is in the film plane (For the case shown in Fig. 15)
Example 2: CoFeB/MgO, Fe/MgO
It was found experimentally (See here Ikeda et al. Nature Material 2010) that the magnetization of a Fe or FeCoB thin film on MgO depends on the thickness of the CoFeB layer.
It should be noticed that the magnetizations of a thin Fe(bcc)(001) on Cu(bcc)(001) is also is perpendicular to plane. Similar to CoFeB/MgO, this film is tensilestrained with only a little strainrelaxation.
It might be possible that this kind of the perpendiculartofilm magnetic anisotropy is a feature of a tensilestrained thin CoFe (001) films.
Since the strength of the spinorbit interaction depends of the shape of the electron orbit in a crystal, the perpendiculartoplane magnetic anisotropy only a feature of a specific crystal orientation and only a specific crystal orientations of the interfaces. For example, the magnetization of a thin Co(hcp) or Co (fcc) film on MgO or Cu is inplane.
Magnetoelastic effect (Villari effect) and the Spinorbit interactionThis effect described facts that the elastic stress may enhance the spinorbit interaction and increase the energy of the perpendicular magnetic anisotropy (PMA)
Wikipedia page is hereWhen a pressure applied to the film, the atomic orbitals are deformed. There are two types of deformations. (type 1): The orbital becomes more elliptical. (type 2): nuclears are shifted out of the center of the orbital. Both deformation makes the effective magnetic field H_{SO} of the spinorbit interaction larger. Without a deformation the orbitals of the localized electrons is nearly spherical and the effective magnetic field H_{SO} of the spinorbit interaction is small. When the pressure applied, the orbitals are deforms in the direction of the applied pressure and the effective magnetic field H_{SO} of the spinorbit interaction increases. In a ferromagnetic material the localized electrons have a noncompensated spin, which creates a magnetic field H_{mag} At an interface between a magnetic and nonmagnetic material, the demagnetization field H_{demag} is created due to uncompensated magnetic moment at the interface. The direction of H_{demag} is perpendicular to the interface and opposite to H_{mag}. The magnetic field H_{inside} inside of the ferromagnetic field equals H_{mag} H_{demag} Additionally, the electron experience the effective magnetic field H_{SO} of the spinorbit interaction, which is always directed along H_{inside}. The magnitude of H_{SO} is proportional to H_{inside} and the degree of the orbital deformation. The magnetic energy of an electron equals to a product of the electron spin and H_{inside}+H_{SO}. When magnetization is perpendicular to the film, the orbital deformation is larger, H_{SO} is larger and the magnetic energy is larger. When magnetization is inplane, the orbital deformation is smaller, H_{SO} is smaller and the magnetic energy is smaller. The dependence of the magnetic energy on the magnetization direction is called the magnetic anisotropy. In the case when the difference of the magnetic energy are with respect to the interface, the effect is called the perpendicular magnetic anisotropy (PMA)
Strains The perpendiculartoplain magnetization may also increase (decrease) due to strain.
When a thin film is grown on a substrate of different lattice constant, the film is strained. When the lattice parameter of the film is larger than that of the substrate, the strains are tensile. The effective magnetic field of the spinorbit interaction, which induced by the strains, is directed perpendicularly to the film plane. When the lattice parameter of the film is larger than that of the substrate, the strains are compressive. The effective magnetic field of the spinorbit interaction, which induced by the strains, is directed in plane. Lattice constant of some metals. Click to expand
Fe (BCC) = 2.870Å (along [110] 2.03 Å ) Ta (BCC)= 3.310 Å (along [110] 2.34 Å ) Cr (BCC) =2.880 A (along [110] 2.036 Å ) V (BCC)=3.020 A (along [110] 2.135 Å ) W(BCC) =3.160 A Cu (BCC metastable) =2.88 Å
Co (hcp) =2.59 Å Ru (hcp) =2.700 Å Ti(hcp)= 2.950 A
Pt (FCC) =3.920 Å (a/2=1.96) Au (FCC)=4.080 Å (a/2=2.040Å) Cu(FCC) =3.610 Al (FCC)=4.050 (a/2=2.025) Pd (FCC) =3.890
notice: hcp and fcc structures are very similar (See here), BCC cell consists of 2 net atoms, The bcc unit cell has a packing factor of 0.68. FCC cell consists of 4 net atoms. The bcc unit cell has a packing factor of 0.74. hcp cell consists of 6 net atoms. The bcc unit cell has a packing factor of 0.74.
MgO lattice constant a = 4.212Å (a/2=2.106Å) Si=5.431 Å (a/2=2.7155) (along [110] 1.92 Å ) Ge=5.66 Å (a/2=2.83) (along [110] 2.001 Å ) GaAs= 5.65325 Å (a/2=2.826625 ) (along [110] 1.9987 Å )
Fe:GaAs (GaAs(110) easy axis) TiN (cubic) Young's modulus (tensile strain) & Bulk module (compressive strain)
SiO2= 68 GPa  &35 GPa Al= 69 GPa  &76 GPa Au= 79 GPa&220 GPa Ti=110 GPa &110 GPa Cu=118 GPa &140 GPa Pt=168 GPa &230 GPa Ta= 186 GPa & 200 GPa Fe=210 GPa &170 GPa Co= 209 GPa &180 GPa W=400 GPa &310 GPa Ru= 447 GPa  &220 GPa Ir=528 GPa  &320 GPa MgO= 270330 GPa &250 GPa MgO bulk elastic properties Compressive Strength 8001600 MPa Elastic Limit 80166 MPa Hardness 57 GPa Breakdown Potential= 610 MV/m=0.0060.01 V/nm Conductivities (S/m)
Strain relaxation and the critical thickness. The strain field, which acts on the filmsubstrate interface, is linearly proportional to the film thickness. The thin film has the inplane lattice parameter the same as that of the substrate. As the film thickness increases the strain field, which acts on the interface, increases. At some thickness the strain field becomes sufficient to create a dislocation at interface. This thickness is called the critical thickness. The dislocations reduce the strain in the film. The process of the creation of the dislocation is called the stain relaxation mechanism. The critical thickness depends on the crystal quality of the film and the strain relaxation mechanism. Approximately, the critical thickness h_{critical} can be calculated from relation: notice: Eq. (3) is valid only for highcrystal quality lowdefectdensity materials. Otherwise, the ratio (3) becomes smaller than 0.7. Example 1. AlGaAs (001)on GaAs(001) The lattice constant of AlGaAs (x=0.5) is 0.069 % larger than the lattice constant of GaAs. The strains are compressive. The critical thickness approximately equals to 1 um. Example 2. InGaAs(001) on GaAs(001) The lattice constant of InGaAs (x=0.5) is 3.582 % larger than the lattice constant of GaAs. The strains are compressive. The critical thickness approximately equals to 19.5 nm. Example 3. Fe(001) on MgO(001) The lattice spacing of MgO (001) in [110] direction is 3.74 % smaller than lattice spacing of Fe (001) in [100] direction. For Fe film on MgO, the strains are tensile. For MgO film on Fe, the strains are compressive. The critical thickness in both cases approximately equals to 18.7 nm. 3.74% of tensile strains in Fe correspond to mechanical tensile stress of 7.44 GPa 3.74% of strains in MgO correspond to mechanical compressive stress of 9.3 GPa. It is significantly larger than the elastic limit of MgO of 160 MPa, and compressive Strength of 1.6 GPa for more details about Fe:MgO:Fe MBE grown click to expand
bulk MgO lattice constant a = 4.212Å (a/2=2.106Å) Fe (BCC) lattice constant a = 2.870Å (along [110] 2.03 Å )
Case of MgO (1.8nm) on Fe (See Yuasa et al. Nature Material (2004)) MgO Even the is much less than the critical thickness, the 2/3 of strains is relaxed (from 3.74% to 1.2 % (2.54% of strains are relaxed)) experiment: the lattice spacing is elongated along the [001] axis by 5% and is compressed along the [100] axis by 1.2% compared with the lattice of bulk MgO (compressive stress 3 GPa . It is larger than compressive Strength of 1.6 GPa). Fe tensile strained (max 2.54%) experiment the lattice of the top Fe electrode is expanded by 1.9% along the [110] axis, which means that 0.64% is relaxed. (2.54%1.9%) tensile stress is 4 GPa
Example 4 Ta on Fe The lattice constant of Ta is 13 % larger than lattice constant of Fe. Ta is compressively strained. A thin Ta can be used with tensilestrained Fe in order to reduce the strain field and to increase the critical thickness of the tensilestrained Fe.
Magnetostriction The mechanical stress σ can be calculated where ε is the total strain, E is the Young’s modulus at magnetic saturation and λ is the magneto elastic strain
Magnetostriction. Click to expend
Magnetostrictionwiki page is hereThe effect describes the change of shape of a ferromagnetic material when its magnetization changes. The origin of the effect Magnetostatic interaction between domains in the ferromagnetic materials. When shape, size, magnetization inside domains changes, the strength of the magnetostatic interaction between domains changes and the lattice contracts or expands. Note: in a singledomain nanomagnet the magnetostriction of this type does not exists. Materials TerfenolD (Tb_{x}Dy_{1x}Fe_{2}) The magnetostriction of the TerfenolD generates strains 100 times greater than traditional magnetostrictive, and 25 times greater than traditional piezoceramics. For typical transducer and actuator applications, TerfenolD is the most commonly used engineering magnetostrictive material. Elastic properties (Tb_{0.3}Dy_{0.7}Fe_{1.92}) Young's Modulus=2535 GPa
Voltageinduced spinorbit interaction & VCMA effectdetails about VCMA effect are here
The external magnetic field induces the magnetic field along its direction. In the case of nearspherical orbit (Fig. 17), the enhancement is small and the magnetic field of the spinorbit interaction is small.
In the external electrical field the positivelycharged nuclear moves a little toward the direction of the electrical field. The negativelycharged electrons move in the opposite direction. Without the electrical field the charge was symmetrically distributed (Fig. 17 left). When the electrical field is applied there is more positive charge at right side and there is more negative charge at the left side. Therefore, the electrical field induces a dipole polarization in the material. The dipole polarization is described by permittivity of the material. Also, the magnetic field of the spinorbit interaction becomes larger. Under the electrical field the electron orbit is deformed so that at the left side the electron distribution becomes denser in the close vicinity of the nuclear. Therefore, at the left side of the nuclear the electron experiences a larger electrical field and a larger corresponded magnetic field of the spinorbit interaction. Even at the right side of the nuclear the spinorbit interaction is reduced, in total the spinorbit interaction becomes larger in the electrical field. It is because the electrical field of nuclear decays as 1/r^2 and at left side it increases sharply, but at the right the decrease is small. It is important: In absence of an external magnetic field, there is no spinorbit interaction and H_{SO}=0.The spinorbit interaction by itself cannot break the time inversesymmetry!
Spindependent scatteringsDue to the spinorbit interaction the electron scattering may become the spindependent. The total spin of the electron gas is not conserved after each spindependent scattering.An electron scattering becomes spindependent when the spinorbital interaction connected the spin of the electron gas to a spin particle outside of the equilibrium of the electron gas
The Spin Hall effect is the effect describing accumulation of the spins at a surface of a metallic wire, when an electrical current flows through the wire due to the SpinOrbit interaction.
Detailed explanation of this Figure is here
Famous misunderstands and misinterpretations of the SpinOrbit (SO) interactionmisinterpretations (1): The SpinOrbit interaction is the QuantumMechanical effect, because it can be derived from the Dirac equations. The SO is only the feature of very small quantummechanical objects. A larger object does not experience any SO interaction.The SO interaction is important for a large objects as well. For example, the influence of SO interaction is very strong in case of a Giant object (like a neutron star or a black hole). In the vicinity of these giant objects the particles move at near the speed of light and there is a substantial electrical field. As a result, the SO magnetic field is very strong in the vicinity of a neutron star or a black hole. The SpinOrbit interaction is a relativistic effect, but not a QuantumMechanical effect. The Dirac equations are relativistic quantummechanical equations describing the quantum field of electrons. As any relativistic equations, they contain the information about the SO interaction. However, any relativistic equations, which describe the photonelectron interaction, should contain a description of the SO interaction. For example, the Maxwell's equations contain the description of the SO interaction as well. e.g. See Landau, Lifshitz. The Classical Theory of Fieldsmisinterpretations (2): There is a "special" "quantummechanical" field of the SpinOrbit interactionOnly a field of the SO interaction is the magnetic field. The SO magnetic field H_{SO} is a very "normal" magnetic field, which is undistinguished from any other magnetic fields (e.g. the magnetic field created by an electrical current) misinterpretations (3): Since the SO interaction can "interact" only with the electron spin, which is a quantum mechanical object, but not with the orbital moment, the SO interaction has quantummechanical origin. Additionally, the SO magnetic field does not induce the Lorentz force. It is an additional proof of the quantummechanical origin of the SO interaction.An electron experiences the SO magnetic field only in the coordinate system, which moves together with the electron. In this coordinate system, the electron does not move. It stay still. Therefore, in this coordinate system the electron does have any orbital moment or any movementrelative property and the SO magnetic field can only interact with the electron spin. The spin is only one magnetic property remaining for a motionless still object. For the same reason, the SO magnetic field does not create the Lorentz force. The Lorentz force is created when an electron moves in a magnetic field. In the coordinate system where the SO magnetic field H_{SO} exists, the electron does not move. misinterpretations (4): Since the strength of the SO interaction is proportional to 1/c^{2}, the influence of SO interaction always can be calculated as a tiny perturbation.The SO interaction is not small at all. In a ferromagnetic metal with perpendicular magnetic anisotropy (PMA), H_{SO}_{} may reach tens of kGauss. For example, in a thin Fe film it can override the demagnetization field of 2030 kGauss and align the magnetization perpendicularly to the film surface. misinterpretations (5): The spinorbit is proportional to the orbital moment of an electron.The SO interaction is not directly related to the orbital moment. Even though in same specific cases, such relation can be established. As was shown above, the strength of the spin orbit interaction substantially depends on the breaking on the orbital spatial symmetry and the timeinverse symmetry. The orbital symmetry is different for an electron of a different orbital moment. As was shown above, the dependence of the SO interaction on the orbital spatial symmetry is more rich and complex than just its dependence on its orbital moment. It is the case (the most common case) of substantial strength of the SO interaction, when the SO interaction is induced by an electrical field of atomic nuclear.
Can the Schrödinger equation be used to describe the spinorbit interaction?A. Yes. Even though the Schrödinger equation is not a relativistic equation, its combination with the Maxwell's equations, which are relativistic equations, gives a correct description of the SO interaction. The spin properties should be included into the solution. The spinorbit interaction is the effect of the electromagnetic field, but not of the quantum field of the electrons. The relativistic description of the electromagnetic field is enough to describe the SO interaction. It does not matter whether the quantum field of the electrons is described relativistically (Dirac equations) or nonrelativistically (Schrödinger equation with inclusion of the spin). It is matter of convenience whether to use the Dirac equations or Schrödinger equation or Pauli equation for a description of the SO interaction. Note: evaluation of the spinorbit interaction from the Dirac equations is a very beautiful and educational method. Spinorbit interaction obtained from the Dirac equationsThe Dirac equations include the full and correct description of the SO interaction.The Einstein's relativistic equation for the energy is which should describe the electron field as well. The quantummechanical for the energy and the momentum are Substituting Eq.(3.2) into Eq.(3.2) gives KleinGordon equation as Considering limitation on the possible symmetries of the wave function, the wave equation should be 1st order differential equation with respect of time and space. Dirac has found that the KleinGordon equation can be represented as a product of 1st order differential equation and its conjugate. Therefore, such the wave equation fully describes the electron field. The Dirac equation (classical form) iswhere the gamma matrices (2 × 2 submatrices taken from the Pauli matrices) The Dirac equation, which includes the gauge potential, is Does KleinGordon equation include information about the conservation of the timeinverse symmetry and spin?Probably not. The Dirac equation and the Pauli equation, both do include the conservation of the timeinverse symmetry and spin. It is difficult to answer about the KleinGordon equation.
Nonrelativistic form of Dirac equationcase v<<cIn this case wave function can be represented as a sum of a large "electron" part and a small "positron" part. a perturbation method using 1/c^{2} as a small parameter.
Pauli equation & SpinOrbit interactionThe Pauli equation is the wave equation describing interaction of electron spin with the electromagnetic field. Wiki page is hereDoes the Pauli equation describe the spinorbit interaction?Yes, the effective magnetic field H_{SO} should be used in Eq.(3.2) additionally to the applied magnetic field. In the Pauli equation the gauge invariant A is used, which is the invariant for the Lorentz transformation, therefore the Pauli equation (Eq.(3.1))should automatically include the spinorbit interaction without usage of H_{SO}?The gauge invariant A is the invariant for the Lorentz transformation, but the used wave function. Therefore, in contrast to the Dirac equation the Pauli equation is not an invariant for the Lorentz transformation. The Pauli equation is the equation valid in only one static coordinate system. When an electron moves in this static coordinate system, it experience the effective magnetic field of the spinorbit interaction, which should be included into Pauli equation The Pauli equation is the extended Schrödinger equation, where electron spin properties are included The Pauli equation is between simpler, but approximate Schrödinger equation and full, but more complex Dirac equation. The Pauli equation can be obtained from the Dirac equation. The Pauli equation (classical form) iswhere A is the magnetic vector potential and is φ the electric scalar potential. σ is the Pauli matrices The magnetic field can be calculated as Substituting Eq.(4.2) into Eq.(4.2) gives The Pauli equation, which include the Spinorbit interaction, isMerits and demerits of the use of the Hamiltonian approach for calculations of the SpinOrbit interactionThis method has limitations and restrictions, which are discussed below
merit (1): The SpinOrbit interaction can be easily included into a more general Hamiltonian as an additional termmerit (2): It allows to use a calculation based on a minimization of Hamiltonian. It does not require any assumption about orbital symmetry.
demerits: demerit (1): It often approximates that the spinorbit interaction is small. It is a very rough approximation and it is often incorrect.
demerit (2): a small calculation error may lead to a substantial error in the final result. It does not use the important features of the orbital symmetry for the calculation of SO.
Content of this page represents my personal view and it is reflected my own finding. It may slightly different from the "classical" view on PMA, which is described in following references M. T.Johnson et. al. Reports on Progress in Physics(1996) ; P.Bruno PRB (1989);

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