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?Charge 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?Charge 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?Charge 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?Charge 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?Charge 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?Charge 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?Charge 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?Charge 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?Charge 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.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) gfactor becomes larger than gfactor of an electron in the free space (Effect 1); The bulktype Spin Hall effect due to scatterings on nonmagnetic and magnetic impurities (Effect 2)  The interfacetype Spin Hall effect due to interface scatterings (Effect 2)  spin relaxation becomes larger. Especially for delocalized electrons (conduction electrons) of p symmetry (d or f as well) (Effect 3) In a ferromagnetic material saturation magnetization becomes larger (exchange interaction is enhanced due to the spinorbit interaction) (effect 1). 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 1)  changing the magnetization and magnetization direction due to the stress. Magnetostriction (magneto elastic) effect. The stress in a metallic singlecrystal multilayer structure. (effect1)  Anomalous Hall effect (Effect 2)  Changing of the Curie temperature in a thin ferromagnetic due to the interfaceenhanced spinorbit interaction. (effect 3) 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 can not 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???A3. No!!!1) Only electrons, which occupy "spin" states, may experience the SpinOrbit interaction, because the spin of a "spin" state is 1/2. The electrons, which occupy "full" states, do not experience the SpinOrbit interaction, because the spin of a "full" state is 0. For example, the localized electrons of filled deep orbitals do not experience any spinorbit interaction. The delocalized (conductive) electrons of an energy substantially lower than the Fermi energy do not experience any spinorbit interaction. 2) An electron should have nonzero speed component perpendicular to an applied electrical field. or 3) An electron should have a nonzero orbital moment . 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.
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
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.
Two types of the SpinOrbit (SO) interaction In 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.The magnitude of the spininteraction in a solid is small!!!. Except for electron, which moves in a close vicinity of an atomic nuclear
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 phones. It is similar to the case when a supersonic 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 heavy elements 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 centro symmetric. 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 spin and induced by 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 deformation of the electron orbit
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 gfactors 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 can not be included into Magnetic susceptibility In 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 anisotropyExample 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 interaction
When a pressure applied to the film, the perpendiculartoplain magnetization may significantly increase due to the spinorbit interaction. Without a deformation the orbitals of the localized electrons is nearly spherical and the effective magnetic field 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 of the spinorbit interaction significantly increases. Note: the effective magnetic field of the exchange interaction may increase as well, because it depends on the distance between atoms.
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
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: without an external magnetic field or an exchange field there is no field of the spinorbit interaction.
Spin Hall effect
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.
for a Wikipedia explanation of the Spin Hall effect, click here (note: I do not agree with "intuitive" explanation given there)Explanation of the Spin Hall effect from the model of the spinup/spindown bands is here (my view)
The origin of the Spin Hall effect is the spindependent scatterings.
Detailed explanation of this Figure is here
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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