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.There are two distinguished classes of effects, which are originated from the spinorbit interaction:(class 1): Enhancement of external magnetic field. Effects: (1) perpendicular magnetic anisotropy; (2) magnetostriction; (3) gfactor; (4) fine structure. Localized electrons and atoms of atomic gas experience this class of effects. Timeinverse symmetry is broken by the external magnetic field.(class 2) Creation of spin polarization by an electrical current. Effects: (1) Spin Hall effect; (2) Inverse Spin Hall effect; (3) Spin relaxation. Conduction electrons experience this class of effects. Timeinverse symmetry is broken by the electrical current.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)(12a) spin relaxation. Reduction of spinpolarization of conduction electrons(13). Magnetoelastic effect(14).Voltageinduced spinorbit interaction & VCMA effect(15). Spindependent scatterings(16). Famous misunderstands and misinterpretations of the SpinOrbit (SO) interaction(17).Spinorbit interaction obtained from the Dirac equations(18). Pauli equation & SpinOrbit interaction(19).Merits and demerits of the use of the Hamiltonian approach for calculations of the SpinOrbit interaction.(20). Fine structure. Heavy and light holes..........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} Note: 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 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) Which parameter characterizes the spinorbit interaction? Energy? Spin? Orbital moment? Or any other quantum mechanical parameter?(It is important): There is only one cause of spinorbit interaction. It is the magnetic field c_{} of spinorbit interaction. All other causes are consequences of H_{SO}. For example, the electron spin align itself along H_{SO}. It change electron energy, which can be defined as an energy of the spinorbit interaction E_{SO}. However, an additional external magnetic field H_{ext} is applied, the electron spin is aligned along total magnetic field H_{SO}+ H_{ext }and E_{SO} has no physical meaning. Therefore, there is only one parameter, which is fully characterizes and describes the spinorbit interaction. It is H_{SO}. effective spinorbit magnetic field H_{SO}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 H_{SO} 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.
Is the spinorbit interaction is a quantummechanical effect, because it is only can be derived from the Dirac equations?A. It is not correct statement. The spinorbit interaction is fully relativistic effect. It is absolutely not a quantum mechanical effect, even though many quantum mechanical effects are include the features of the spin orbit interaction. See detailed answer in major misunderstandingsIn short: (1) The spinorbit interaction is a relativistic effect and can be fully described by relativistic equations. (2) The Dirac is a relativistic quantummechanical equation. The Dirac equation describes both the relativistic transformation of the electromagnetic field and the relativistic transformation of the quantum field of an electron. Calculations of the spinorbit interaction from the Dirac equation are most precise. (See SO and Dirac equation here) (3) The Schrödinger equation and Pauli equation, both describe the spinorbit interaction. Both equations describe the relativistic transformation of the electromagnetic field, but do not describe the relativistic transformation of the quantum field of an electron. However, using some corrected constants and parameters it is possible to describe the SO interaction by Schrödinger equation and Pauli equations fully precise and fully identical to description by the Dirac equation.
Energy of SO interactionThe spinorbit interaction manifests itself only by the SO magnetic field H_{SO}. The H_{SO} is a very normal magnetic field (almost). There is a precession of electron spin S around magnetic field H_{SO} of spinorbit interaction until it aligns parallel to H_{SO}. After the electron spin is aligned, the energy of SO interaction becomes μ_{B} is the Bohr magneton Note: The H_{SO} does not interact with the orbital magnetic moment of electron. It is interact only the spin magnetic momentIt is because of relativistic origin of both the SO interaction and the Lorentz force. The H_{SO} interacts only with the spin magnetic moment, but not with the orbital magnetic moment or the total magnetic moment
Why H_{SO} does not induces the Lorentz force and cannot interact with the orbital magnetic moment?The H_{SO} is the magnetic field of the relativistic origin. It appears only the coordinate system, which moves together with the electron (See here). The Lorentz force is of the relativistic origin as well and it is originated from the electrical field, which the electron experiences when moves in a magnetic field. In the moving coordinate system, where the electron experiences H_{SO} , the electron does not move and therefore has no experience any Lorentz force The interaction of the orbital moment with a magnetic field is due to the Lorentz force
Note: The total magnetic moment is a quantum mechanical sum of the orbital magnetic moment and the spinmagnetic moment. The role holds for both cases when spin is align along H_{SO} (along the orbital moment) and when the there is a spin precession around H_{SO}. Electron spin is aligned along due the spin precession damping (See Fig.6). The spin precession damping is a complex mechanism (See here), which involves an external particle with a nonzero spin (e.g. a photon, a magnon). It could take a relatively a long until full alignment of electron spin along H_{SO}.
Is there any cases when the electron spin is not aligned along H_{SO}?There are many such cases. E.g. the conduction electrons in a metal. The size of a conduction electrons are relatively large. There are many conduction electrons, which simultaneously overlap each other. For this reason, the scattering between quantum states of conduction electrons are very frequent. The time between two consequent scatterings a conduction electron is very short (~ 1 ps). It is far not enough to finish even oven precession period and definitely it is not enough for the electron spin to align along H_{SO}.
Magnitude of the Spinorbit interaction.Except of a few weak effects, all spinorbital effects are induced by an electrical field of an atomic nuclear and the election movement (rotation) in the close proximity of the nuclear !! Magnitude of the spininteraction in is small when an conduction electron moves in any realistic extrinsic or intrinsic electrical field in a solid!!!. a moderate electrical field + a moderate electron speed => result: 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 => result: 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.Spinorbit interaction and orbital momentIs the spinorbital interaction proportional to the orbital moment?A. Actually, not. Even though there are common tendencies between the spinorbital interaction and the orbital moment. E.g. When orbital moment is zero, the SO interaction is zero. When orbital moment changes its sign, the SO interaction changes its sign as well. Even though the "orbital" is a part of name of the SO interaction, the relation between orbital moment and the H_{SO} is complex and not straightforward. The spinorbit interaction: for centrosymmetric electrical field of a nuclear:
where q_{nuclear} is the nuclear charge. H_{SO} is proportional to ~1/r. As a result, the main contribution to H_{SO} is from region in proximity of the nuclear. The symmetry of electron distribution function and electron rotation symmetry in close vicinity of the nuclear mainly contribute to H_{SO} The orbital moment: or in quantummechanical representation L is proportional to ~r. As a result, both regions, which are close and far from the nuclear, give a substantial contributions to L. Even though a formal relation between H_{SO} and L is very simple: The integration over electron distribution gives very different value of H_{SO} and L depending on the symmetry and details of electron wavefunction. Orbital momentum vs rotation symmetry vs spinorbit interactionQ. 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. Is H_{SO} always equal to zero for a spherical orbital?No, See center and right pictures The sorbital can be divided to the sum of two spherical orbital, for which H_{SO} is a no zero and opposite between two orbitals.
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.
quenched and unquenched orbital moments (See details here)) unquenched orbital: orbital moment can be freely rotated in any direction. E.g. orbital moment can align alone an external magnetic field. Localized electrons of a paramagnetic material have a nonzero unquenched orbital moment. A nonzero orbital moment is mainly feature of a paramagnetic gas. In most of ferromagnetic and paramagnetic metals,the orbital moment of localized electrons is zero and the paramagnetic properties is determined by the electron spin without influence of the orbital moment.quenched orbital: orbital moment cannot be freely rotated. Its orbital direction either is fixed or its orbital momentum is zero. An unique spacial electron distribution each orbital moment. When the orbital moment is changed, the orbital spatial distribution is changed as well. However, the bonding of atom in a solid fixes the spacial electron distribution. The orbital moment of bonding electrons cannot be rotated. Otherwise, the bonding would be destroyed. As a result, the total orbital momentum of bonding electrons is zero.
orbital moment of electrons in a metal (common case) localized electrons: the orbital moment is zero and orbital is quenched conduction electrons: the orbital moment is a nonzero and orbital can be either quenched or unquenched
Three type of Spinorbit interactionIt is convenient to divide effects related to the SO into 3 classes depending on a source of electrical field and a source of breaking of the timeinverse symmetryThree types of the SpinOrbit (SO) interaction
Key requirements for the SO to exist:(requirement 1): electrons should move perpendicularly to the electrical field(requirement 2): The timeinverse symmetry should be broken. (The SO cannot break timeinverse symmetry by itself, but breaking ids required in order to have H_{SO} )(type 1: weak SO) SO interaction induced by an external electrical field , which is perpendicular to electron current (source of electrical field): electrical field at interface; electrical field of a Schottky barrier: (source of breaking of timeinverse symmetry): an electrical current flowing along interface and perpendicularly to the interface electrical field: (induced effects): Spin Hall effect & Inverse Spin Hall effect (weak contributions) (type 2: strong SO) Enhancement of an external magnetic field (source of electrical field): centrosymmetric electrical field of atomic nuclear (source of breaking of timeinverse symmetry): an external magnetic field (induced effects): Perpendicular magnetic anisotropy (PMA), voltagecontrolled magnetic anisotropy (VCMA) (type 3: moderate SO) creation of spin polarization by an electrical current (source of electrical field): centrosymmetric electrical field of atomic nuclear (source of breaking of timeinverse symmetry): an electrical current (induced effects):Spin Hall effect & Inverse Spin Hall effect (main contributions), SpinOrbit Torque
(type 1: weak SO) SO interaction induced by an external electrical field , which is perpendicular to current only conduction electrons experiences this type of SO effect(source of electrical field): electrical field at interface; electrical field of a Schottky barrier: (source of breaking of timeinverse symmetry): an electrical current flowing along interface and perpendicularly to the interface electrical field: (induced effects): Spin Hall effect & Inverse Spin Hall effect (weak contribution) It is weakest, but easiest to understand SO type.Explanation of effect: An electrical field, which exists at interface, or an external electrical field is applied (exists) perpendicularly to the electron current. The conduction electrons are confined in a 2D structure (e.g. a quantum well). Therefore, they do not flow in the perpendicular direction along the perpendicular magnetic field. The geometry of this type of SO effect is nearly the same as the geometry explaining the relativistic origin of the SO interaction, when electrons move perpendicularly
The reasons why the type of SO interaction is small, see here
(type 2: strong SO) Effect of Enhancement of external magnetic field
only localized electrons (e.g. d or f electrons) experience this type of SO effect(source of electrical field): centrosymmetric electrical field of atomic nuclear (source of breaking of timeinverse symmetry): an external magnetic field (induced effects): Perpendicular magnetic anisotropy (PMA), voltagecontrolled magnetic anisotropy (VCMA) localized electrons experience this type of SO Explanation of effect: The type 2 of spinorbit interaction is induced be an external magnetic field. E.g. in absence of a external magnetic field a localized electron does not experience any H_{SO}. However, when external magnetic field H_{ext} is applied, it induces strong H_{SO} parallel to H_{ext} and the electron experiences a stronger total magnetic field H_{total} =H_{ext}+H_{SO}. E.g. when H_{ext}=1 kG is applied, it induces H_{SO}.=5 kG. Therefore, in total electron experience H_{total}=6 kG. Reason why an external magnetic field H_{ext} induces the spinorbit interaction H_{SO}_{} (without external magnetic field): The orbital moment of the localized electrons is zero (or unquenched (See details here)). Any 3D orbital can be divided as a sum of two 2D orbitals of CCW and ACW electron rotation. Since the total moment of the localized electrons is zero, the CCW and ACW orbitals are identical. As a result, the electron experience the same but opposite H_{SO}_{} for the CCW and ACW orbitals , therefore in total it experiences no H_{SO}_{} (with external magnetic field):Since electron rotation in the CCW and ACW orbitals is opposite, the Lorentz force is opposite for CCW and ACW orbitals. As a result, the CCW and ACW orbitals are deformed differently in an external magnetic field, H_{SO} becomes different for CCW and ACW orbitals and in total the electron experiences a nonzero H_{SO}_{}.
How an external magnetic field affect a localized electron and electron orbital?(influence 1) Electron energy is changed. (less important for SO) The electron energy changes in a magnetic field according to its orbital moment. The orbital moment is aligned along magnetic field minimizing magnetic energy. The energy of s electrons (orbital moment L=0) does not change. The energy of p, d, f electrons (orbital moment L=1,2,3) changes. (influence 2) Time inverse symmetry is broken. (very important for SO) The magnetic field changes the spacial distribution of an electron orbital, which breaks the timeinverse symmetry for the orbital. The part of electron distribution, which corresponds to the electron rotation in ACW direction with respect to magnetic field, is becomes closer to the nuclear. The part of electron distribution, which corresponds to the electron rotation in CCW direction with respect to magnetic field, is shifted away from the nuclear.
Note: The breaking of the time  inverse symmetry of the orbital does not depend whether the electron energy or electron orbital moment is changed or not. For example, a magnetic field breaks the time inverse symmetry even for the sorbital, even though the magnetic field does not change either energy or orbital moment of the s orbital.
Effects, which are originated from Spinorbit interaction of type 2: (effect 1): Perpendicular magnetic anisotropy (PMA) (also see here)(effect 2): gfactor(effect 3):. Magnetoelastic effect(effect 4): VCMA effect ? (one possible origin)(effect 5): Interface sensingeffect 6): Magnetic anisotropy
(type 3: moderate SO) creation of spin polarization by an electrical current
(source of electrical field): centrosymmetric electrical field of atomic nuclear (source of breaking of timeinverse symmetry): an electrical current (induced effects):Spin Hall effect & Inverse Spin Hall effect (main contribution), SpinOrbit Torque Only conduction electrons experience this type of SO Explanation of the effect: (effect 1): Spin Hall effect When electrical current flows in a metallic wire, it generates a spin current flowing perpendicularly to the electrical current (effect 2): Spin pumping When electrical current flows in a metallic wire, it creates spinpolarized conduction electrons. As a result, initially spinunpolarized gas of conduction electrons becomes spinpolarized. (effect 3): Inverse Spin Hall effect (ISHE) When a spinpolarized current flows in a metallic wire, it generates a charge current (conventional current)flowing perpendicularly to the spin current. (effect 4): Spin damping When the electron gas is spinpolarized, there are spinpolarized conduction electrons, which spins is directed in one direction. When the conduction electron has a nonzero rotational (orbital) moment, it experiences a nonzero H_{SO} and there is a spin precession around H_{SO}. Since the direction of H_{SO} is different for electrons moving in different directions, the spin precession is along different directions for electrons moving in different directions. As a result, the spins of spinpolarized electrons is misaligned from one direction and degree of the spin polarization is reduced.
there are two contributions to currentinduced spinorbit effects: (contribution 1) band current It occurs only when conduction electrons have nonzero rotational moment.It is a feature of a metal of a high conductivity (See here)(explanation of effect): (step 1) The conduction electron have a nonzero rotational (orbital) moment,which created magnetic field H_{SO}. There is a spin precession around H_{SO} and the spin is aligning along H_{SO} due to the damping of the spin precession. (step 2) When there is no electrical current, there are equal numbers of electrons moving in any two opposite directions. Since the rotational (orbital) moment and H_{SO} are equal and opposite for electrons moving in opposite direction, both the total rotational (orbital) moment and total are zero for the electron gas and scattering probabilities are independent on electron movent direction and electron spin (step 3) When there is an electrical current, the number of conduction electrons moving along current is larger than number of electrons moving in the opposite direction. As a result, the rotational (orbital) moment of electrons moving along current is not fully balanced by the opposite moment of electrons moving in the opposite direction and the total the electron gas experience a nonzero H_{SO} and the electron gas becomes spinpolarized. (step 4) When there is an electrical current, the scattering probability of spinup electrons to the left becomes different from the scattering probability to the right. As a result, e.g. the spinup polarized current flows to the left and the spindown polarized current flows to the right. (contribution 2) scattering current It is originated from electrical field of defects and from direction dependence of scattering probability in the vicinity of an interface.It is a feature of a metal of a low conductivity (See here) (explanation of effect): (step 1) There is an electrical field in close vicinity of a defect in a metal and an interface between two metal or at edge of a metal wire. The conduction electrons are screening any electrical field in a metal. However, in close proximity of a defect or interface the electrical field is not fully screened. Especially it is the case of a metal of a low conductivity (step 2) When a conduction electron moves along the electrical field of the defect or interface, it experience H_{SO} and its spin is aligned along H_{SO} (step 3) Since direction of the electrical field is opposite from left and right sides
The conduction electrons move simultaneously in the forward direction along lattice and around each atom (nuclear) of the lattice.
Spinorbit (SO) interaction. Facts in short.(fact 1) SO describes the fact that an electron moving perpendicularly to electrical field experiences a magnetic field H_{SO}. The magnetic field H_{SO} affects the electron spin. As a result, the electron properties become spindependent.(fact 2)There are three types of SO interaction: the "weak type" SO , "non zero orbital" type, "strong type" and "moderate  type" SO. Their properties and features are completely differentProperties distinguish each type of SO interaction What is the origin of electrical field?  Direction of electron movent  What (an electrical current or an external magnetic field) breaks the time inverse symmetry.
Type 1: "Weak type"SO interaction. (H_{SO}=0.011 Gauss). Facts in short.
note: H_{SO} of the Weak SO interaction is comparable to Earth magnetic field 0.259.65 Gauss (See here)(Origin): It is originated by an electrical current flowing in quantum well (QW) or in the vicinity of the interface. An external magnetic field is not required to create the weak SO effect. note: In the case of weak SO effect the timeinverse symmetry is broken by the electrical current(Origin of breaking of timeinverse symmetry): electrical current(Source of electrical field (field is weak)): (1) electrical field of charge accumulation at interface (Schottky type or due to a difference of work functions of materials at sides of the interface). (2) electrical field of defects (Reason why an electron moves perpendicularly to electrical field) Electrical current in the perpendicular direction is blocked by the interface or 2D confinement. (Source of electron movement (movement is slow)): electron current along a 2D QW or along an interface (Symmetry): (1)Polarity of H_{SO} is reversed when is the electrical current is reversed. (2) Asymmetrical structure perpendicularly to current direction is often required. Otherwise, SO interaction compensate itself at two opposite interfaces and H_{SO} becomes zero. Effects, originated by "Weak type" SO: (1) Spin Hall effect (weak contribution) It describes the fact that a spinpolarized current is generated perpendicularly to a flow of electrical current in a nonmagnetic material.(2) Anomalous Hall effect (weak contribution) It describes the fact that a Hall voltage is generated at side of a ferromagnetic wire when the wire magnetization is perpendicular to the electrical current(3) Effect of Anomalous Magneto  Resistance (AMR) (weak contribution) It describes the fact that(4) Rashba effect (weak contribution) It describes the fact that there is a spin precession for a conduction electron when there is an electrical current in a quantum well (QW)
Type 2: "non zero orbital" SO interaction. (H_{SO}=0.0120 000 Gauss). Facts in short.(Origin): It is originated from orbital movement of electron, when the orbital moment of electron is a nonzero. (Origin of breaking of timeinverse symmetry): orbital alignment (spontaneous, local, by a magnetic field etc.) (Source of electrical field (field is strong)): electrical field of the nuclear (Reason why an electron moves perpendicularly to electrical field): Orbital movements, (Source of electron movement (movement is fast)): Orbital movements (Symmetry): (1) H_{SO} is linearly proportional to the orbital moment of electron. (2) Effects, originated by "non zero orbital" SO: (1) Fine structure Type 3: "Strong type" SO interaction. (H_{SO}=1 000 20 000 Gauss). Facts in short.(Origin): It is originated by an electrical field of nuclear due to the orbital rotation of an electron around nuclear and an external magnetic field, which breaks the timesinverse symmetry of the orbital An external magnetic field is required to create the strong SO effect. Without an external magnetic the effect does not exists! note: The strongtype SO effect cannot break the timeinverse symmetry by itself. It requires an external magnetic field to break it.It is only type of SO interaction, which exists without an electrical current(Origin of breaking of timeinverse symmetry): external magnetic field (Source of electrical field (field is strong)): electrical field of the nuclear .(Reason why an electron moves perpendicularly to electrical field): Orbital movements, (Source of electron movement (movement is fast)): Orbital movements (Symmetry): (1) H_{SO} is linearly proportional to the external magnetic field. (2) Effects, originated by "Strong type" SO: (1) Perpendicular magnetic anisotropy (PMA) It describes the fact that a spinpolarized current isType 4: "Moderate type" SO interaction. (H_{SO}=1 0  500 Gauss). Facts in short.(Origin): It is originated by an electrical field of nuclear due to the orbital rotation of an electron around nuclear and an electrical current, which breaks the timesinverse symmetry of the orbital note: The moderatetype SO effect cannot break the timeinverse symmetry by itself. It requires an electrical current flow to break it.(Origin of breaking of timeinverse symmetry): electrical current (Source of electrical field (field is strong)): electrical field of the nuclear (Reason why an electron moves perpendicularly to electrical field): Orbital movements (Source of electron movement (movement is fast)): Orbital movements (Symmetry): (1) H_{SO} is linearly proportional to the current (2) Effects, originated by the "Moderate type" SO: (1) It describes the fact that a spinpolarized current is
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 (AHE) 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 H_{SO} of the SpinOrbit interaction affects only the electron spin. Interaction of H_{SO }with electron spin leads to 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 3) Electron transport can become spindependent 4) The electron energy becomes spindependent. Important Note 1: The effective magnetic field H_{SO} of the SpinOrbit interaction cannot induce the Lorentz force or the Hall effect.Important Note 2: The effective magnetic field H_{SO} of the SpinOrbit interaction does not interact with the magnetic moment induced by the orbital moment of the electron.Example to understand how the spinorbit interaction works in a solid
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. Type 2. 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)
Spinorbit interaction in the centrosymmetric electric field of atomic nuclear.
Spinorbit interaction (type 3: moderate SO) creation of spin polarization by an electrical current (source of electrical field): centrosymmetric electrical field of atomic nuclear (source of breaking of timeinverse symmetry): an electrical current (induced effects):Spin Hall effect & Inverse Spin Hall effect (main contribution), SpinOrbit Torque
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.(fact) The effective spinorbit magnetic field H_{SO} acts only on spin and it does not effect the magnetic moment due to the orbital moment.The magnetic moment of an electron is a quantum mechanical sum of magnetic moments induced by the spin and induced by 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  ( important fact:) The spinorbit interaction cannot break time inverse symmetry by itself. E.g. the object cannot be magnetized due to spinorbit interaction only. The spinorbit interaction enhances alreadyexisted magnetic field, but it cannot create its own magnetic field. The effective spinorbit magnetic field H_{SO} is always proportional to the external magnetic field. It does not exist when there is no external magnetic field. This condition is the consequence of the conservation law of the timeinverse symmetry.This fact contradicts with the existence of a fine structure in the hydrogen gas and the existence of the heavy and light holes in a semiconductor? In both case, the spinorbit interaction causes a spliting of the energy levels without any external magnetic field.There is no contradiction. Globally, there is no magnetic field in both cases. However, locally each atom experience a magnetic field, which is magnified by the spinorbit interaction. The magnification is different for the orbital of a different symmetry. As a result, the orbitals experience a different H_{SO} and the orbital energies become different. (See more details Here) (fact) H_{SO} does not exist when there is no external magnetic field(fact) H_{SO} is always proportional to an external external magnetic field(fact) The spinorbit interaction cannot break the time inverse symmetryHow the timeinverse symmetry is broken:
It results: Average effective magnetic field of the spinorbit interaction becomes nonzero
An external magnetic 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 H_{SO} 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.
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).
Spin relaxation in gas of conduction electrons
Spin relaxation describes reduction of the number of spin polarized electrons. Or the same, the conversion of electrons from group of spin polarized electrons to the group of spin unpolarized electrons. (See here about spin polarization of conduction electrons)(Mechanism): The incoherent spin precession around H_{SO} The spins of all spin polarized electrons are directed in one direction. In contrast, the direction of H_{SO} is different for electrons moving in different directions. As a result, the angle between electron spin and H_{SO} is different for the spin polarized electrons moving in different directions. There is a spin precession around H_{SO}. Since the directions of H_{SO} are different for conduction electrons moving in different directions, their precession directions are different as well. The precession in different directions misaligns spins of spin polarized electrons, which causes the spin relaxation.
The spin relaxation reduces the amount of spinpolarized conduction electrons in electron gas (See here)
Since the spin precession is fast (~0.11 GHz) and therefore all spin polarized electrons should be misaligned very quickly within one period of Ferromagnetic resonance (FMR), which is about 1 nanosecond? Therefore, all spins should be misaligned and spinpolarized electrons should not exists?A. Additionally to the mechanism of the spin misalignment (mechanisms of the spin relaxation), there are mechanisms, which align all spins in one direction (mechanism of the spin pumping). The simple electron scatterings are most efficient as a spin alignment mechanisms. The symmetry and spin feature of electron scatterings is that they redistribute randomly spin misaligned group of conduction electrons into a group of spin polarized electrons, in which all spins are aligned in one direction, and the group of spin unpolarized electrons, in which spins are distributed equally in all directions. See here more details about scatterings and spin distributions. (fact 1) Spin relaxation is a balance between two mechanisms: the mechanism 1 of spin misalignment due to the incoherent precession around H_{SO} and mechanism 2 of spin alignment due to scatterings.
(fact 2) Spins of all spin polarized electrons are aligned constantly along one direction due to electron scatterings. However, the number of spin polarized electrons becomes smaller after each such alignment, because of the conservation of the total spin of the electron gas.
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 here
The magnetization of a singlematerial ferromagnetic film is in  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 at the interface . 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.
Detailed description of the dependence of PMA on film thickness is here
It should be noticed that the magnetizations of a thin Fe(bcc)(001) on Cu(bcc)(001), on Ta (001), on W(100) is also is perpendicular to plane.
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 on Cu is inplane.
See features of the PMA for different interfaces hereM. T.Johnson et. al. Reports on Progress in Physics(1996) ; P.Bruno PRB (1989);
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.
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. 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). There is only one difference between H_{SO} and "normal" magnetic field, H_{SO} cannot induce the Lorentz force 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. This property is related to the Quantum Mechanics. 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 (see here). 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.Note: the formulas for orbital momentum and the spinorbit interaction are very similar. The difference between them is only coefficient 1/r^{2}. In close vicinity of the nuclear, the 1/r^{2} is huge and it makes a huge difference. 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. Such description includes the relativistic features of the electromagnetic field, but does not include the relativistic features of the quantum field of an electrons. However, for a description of effects in a solid state these features can be included by adjusting some parameters and constants.
Difference between description of the spinorbit interaction from the Dirac equations or Schrödinger equationDirac equations calculates both contributions to the spinorbit interaction (contribution 1) due to relativistic nature of the electromagnetic field. (contribution 2) due to relativistic nature of the quantum field of an electron. Schrödinger equation calculates only contribution 1 due to relativistic nature of the electromagnetic field. It does not calculate contribution 2 due to relativistic nature of the quantum field of an electron. 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 external magnetic field. Then, the Pauli equation correctly describes the SO interaction 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 wave function of the Pauli equation is not a invariant. Therefore, in contrast to the Dirac equation the Pauli equation is not an invariant for the Lorentz transformation. The reason why H_{SO} is not included into the Pauli equation, but should be input as additional magnetic field, is following. The Pauli equation is the equation, which is valid in only one static coordinate system. When an electron moves in this static coordinate system, the Pauli equation becomes not valid. The relativistic transformation of the quantum field of an electron are missing in Pauli equation. However, the adding of H_{SO} fixes the problem and the Pauli equation becomes valid again. The Pauli equation is the extension of Schrödinger equation, where electron spin properties are included The Pauli equation can be considered as a semi relativistic equation. They place 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.H_{SO} and external magnetic field
(important fact:) The spinorbit interaction cannot break time inverse symmetry by itself. E.g. the object cannot be magnetized due to spinorbit interaction only. The spinorbit interaction enhances alreadyexisted magnetic field, but it cannot create its own magnetic field. The effective spinorbit magnetic field H_{SO} is always proportional to the external magnetic field. It does not exist when there is no external magnetic field. This condition is the consequence of the conservation law of the timeinverse symmetry.(fact) H_{SO} does not exist when there is no external magnetic field(fact) H_{SO} is always proportional to an external external magnetic fieldTwo effects: (1) the fine structure and (2) the difference between energies of heavy and light holes) exists due to the spin orbit interaction. In both cases there is no external magnetic field, but there is a substantial spinorbit interaction. Why? Fine structure. Heavy and light holes.(possible contradiction) The spinorbit interaction cannot break the timeinversion symmetry. As a result, the spinorbit interaction does not manifest itself until the timeinverse symmetry is broken by another mean. E.g. the external magnetic field is applied or an electrical current flows. This fact contradicts with the existence of a fine structure in the hydrogen gas and the existence of the heavy and light holes in a semiconductor? In both case, the spinorbit interaction causes a spliting of the energy levels without any external magnetic field.There is no contradiction. Globally, there is no magnetic field in both cases. However, when the orbital moment of an atom is a nonzero, locally the timeinversesymmetry is broken for each atom and each atom experiences a magnetic field H_{SO} of the spinorbit interaction. Since the directions of the orbital moments are equally distributed in all directions, the gas is nonmagnetic and globally there is no breaking of the timeinverse symmetry. Still changes the electron energy and there is a difference of electron energies of electrons of different orbital moments. Difference between Global and Local breaking of the timeinverse symmetry (TIS) It can be understood from the following example, which describes the Zeeman splitting of a gas of atoms in a magnetic field and in which the more complex effect of the spinorbit interaction is not involved. Let us consider a gas of atoms. Each atom has a magnetic moment and a nonzero spin. Both the local and global breaking of TIS lead to the energy splitting. However, only the global breaking creates the directional dependency of gas or material properties. (global symmetry breaking): The symmetry is broken globally, when a sufficientlystrong external magnetic field is applied. As a result, the magnetic moment of all atoms is aligned along the magnetic field. The magnetic field field breaks the timeinverse symmetry. It results in two changes. The first change the energy of electrons with spins along and opposite the magnetic field becomes different. As a result, one energy level splits into two levels. The second result of the timeinverse symmetry breaking is that the properties of the atomic gas become direction dependent. E.g. left and right circular polarized light is absorbed differently, when light propagates along the magnetic field and the absorption is the same when the light propagates perpendicularly to the field. (local symmetry breaking): The symmetry is broken locally, but not globally, when there is a magnetic field in each atom along the atom magnetic moment, but the magnetic moments of all atoms are not aligned. It is the case when there is no external magnetic field and the magnetic moments of atoms of the gas are distributed equally in all directions. In this case there is no any directional asymmetry, but the energy splitting still remains. Each atom has the equal energy splitting independently on direction of its magnetic moment. Even though globally there is no magnetic field, the atoms of the gas experiences the Zeeman splitting.
It describes the fact that in the hydrogen gas the p energy level is splits in two levels: the lowerenergy level (J=1/2) and the higherenergy level (J=3/2).(More details see here) (The reason of the energy splitting:) The orbital moment of a p electron is nonzero. As a result, the p electron experiences the additional spinorbit magnetic field H_{SO} and corresponding additional energy E_{SO}=H_{SO}*S, where S is the electron spin. H_{SO} depends on the spacial symmetry of the orbital. The spacial symmetry of the orbitals of J=1/2 and J=3/2 are different. As a result, the electrons experience a different H_{SO} and have a different energies. In a molecular or atomic gas, all values of the orbital moment, H_{SO} and electron spin are nonzero. The orbital moment, H_{SO} and electron spin are directed in one direction specific for each individual atom. Each atoms creates a dipole magnetic field around itself and theretofore it can be considered as a tiny magnet. However, the atomic gas in total is not magnetic. It is because the directions of the orbital moments, H_{SO} and spins are equally distributed in all direction. Still H_{SO} changes the energy of each atom, which is independent on the direction of magnetic moment of each atoms and remains as a feature of the whole gas. The dependence of H_{SO} on the orbital moment (orbital symmetry) causes the fine energy splitting. In a semiconductor, the holes have p spacial symmetry. The holes are divided into two classes: the light (J=1/2) and heavy (J=3/2) holes. (reason of difference between light and heavy holes) The the light and heavy holes experience a different magnitude of the spinorbit interaction, due to their different orbital symmetry (orbital moment). The spinorbit interaction causes the difference of energies (properties) between the light and heavy holes. The orbital moment of a hole is nonzero. As a result, the hole experiences the spinorbit magnetic field H_{SO} and corresponding additional energy E_{SO}=H_{SO}*S, where S is the electron spin. H_{SO} depends on the spacial symmetry of the orbital. The spacial symmetry of a light hole (J=1/2) and a heavy hole (J=3/2 ) are different. As a result, the the light and heavy holes experience a different H_{SO} and have a different energies. Questions & Comments about SpinOrbit interactionQ. Why does the Spinorbit interaction increase when the electronic orbital is deformed and becomes less symmetrical?A. There are several reasons why: (reason 1): In the case of a deformed orbital, the electron distribution become more close to the atomic nuclear (at least some of the distribution). In order for the SO be large, the electron should move in a very large electrical field, which is only exists in a very close proximity of the nuclear. Only this region mainly contributes to SO. If the ionic or atomic radius (see here or here) is about 50100 pm, the region with radius only 12 pm contributes about 99% to SO (number might be slight different from case to case due to the symmetry and frequent cancelation of SO in the proximity of the nuclear).Usually a deformation of orbital moves electron (electron distribution) closer to the nuclear, which makes the SO larger. (reason 2): In the case when the orbital moment is zero, the SO is zero, because it has two opposite contributions, which of compensate and cancel each other(see here). The deformation makes the orbital moment larger. There is no balance between two opposite contributions and the SO is enlarged. The localized electrons and the conduction electrons of the ssymmetry (e.g. ntype electrons in a semiconductor) have zero orbital moment. (reason 3): The SO is enhanced by an external magnetic field, because of the symmetry breaking due to the Lorentz force induced by the external magnetic field. This enhancement is more efficient for less symmetrical orbital. Q. Why an external magnetic field does not break symmetry for a spherical orbital? (Jan asked): So i guess i understand the Point why an external field is making a net spin orbit interaction magnetic field, but i DONT see why this is NOT the case for a spherical Orbit.A. You are right. An external magnetic field enhances the SO in the case of the spherical orbital as well. However, the enhancement is more efficient for a deformed orbital than for a spherical orbital. This can be understood as follows. A deformed orbital has some part of electron distribution, which is very close to the nuclear. Q. You always depict an electron as a pointlike particle. Should you use a wave function instead and full quantummechanical description of the SO interaction?A. The spinorbit interaction is relativistic effect (it is not a quantum mechanical effect). All effects due to the spin orbit interaction exist for both a small object (which should be described by a wave function) and a large object ( which can be approximated as a point like classical object). Q. Why the spin orbit interaction cannot break the time symmetry by itself?A. The spin orbit interaction is a relativistic effect, which just describes the transformation of the electro  magnetic field between coordinate systems moving with different speeds (Also, it describes the transformation of the quantum field of electron between different coordinate systems (See Dirac Eq.)). If the time inverse symmetry is not broken in one coordinate system, it is not broken in any other coordinate system. It doesn't matter whether the coordinate system is moving or not. It is the reason why the spin orbit interaction cannot break the time inverse symmetry. In order to manifest itself, the spin orbit interaction always requires an external breaking of the time inverse symmetry. Q. I have a question on breaking the timereversal symmetry, which you mentioned as a key ingredient to induce spinorbit interaction.

I will try to answer your questions as soon as possible