My Research and Inventions

### Reticle 11

Spin-Orbit Interaction

### Content

#### Introduction part:

Spin- orbit interaction is an effect describing a magnetic field, which is originated due to a movement of an electron in an electric field

## Despite its relativistic origin, the magnetic field of the spin-orbit interaction is just as real as any other magnetic field that can exist.

### (fact) The spin-orbit interaction that arises from the electric field of the nucleus and the orbital motion of the electron is widely recognized as the most prominent, influential, and robust form of spin-orbit interaction.

Magnetic anisotropy, perpendicular magnetic anisotropy (PMA), voltage- controlled magnetic anisotropy (VCMA), g- factor, spin Hall effect, Anomalous Hall effect (AHE) , Anomalous Magneto Resistance (AMR) and many other effects are due to this type of the spin orbit - interaction.

#### Main part:

 Magnetic field, which applied to electron orbital, creates additional magnetic field due to the spin- orbit interaction. As a result, the electron spin interacts with a larger magnetic field than externally applied (see details here). Magnetic anisotropy is due to this effect. See Perpendicular magnetic anisotropy Measurement of the strength of spin- orbit interaction is based on this effect (See below)

### Relativistic origin of the Spin-Orbit interaction

 Relativistic origin of the Spin-Orbit Interaction Fig.3 An object (e.g. an electron) shown in red is moving in a static electric field. In the coordinate system moving together with the object, the static electric field is relativistically transformed into the effective electric field Eeff and the effective magnetic field Heff In case if the particle has a magnetic moment (spin), there will be a spin precession around the effective magnetic field.

#### 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 Spin-Orbit 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 (spin-orbit interaction) and causes the precession of the magnetic moment around the direction of the effective magnetic field.

### Lorentz  transformation

The electromagnet field is a relativistic object and it is the Lorentz transformation rules as

where Estatic, Hstatic are the electric and magnetic field in the static coordinate system (reference frame) and Emove, Hmove 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 Hstatic, experience in own reference frame an effective electrical field EHall , which is called the Hall field (Hall voltage). Similarly, when an electron moves in a static electrical field Estatic, it experience in own reference frame an effective magnetic field HSO , which is called the effective spin-orbit magnetic field

For example, when an electron moves in the x-direction

in

The Lorentz force is a tween effect with the effect of the spin-orbit interaction
 Fig.4. The Lorentz force (Hall effect) and the spin-orbit interaction, both are relativistic effects. The both effects are originated from fact that an electrical and magnetic field are relative to the velocities of observers. A particle, which moves in a static magnetic field, experiences an effective electrical field (The Lorentz force). A particle, which moves in a static electric field, experiences an effective magnetic field. The Lorentz force (Hall effect) has a similar relativistic origin. An object (red), which is moving in a static magnetic field, experiences an effective electrical field.

#### Nonrelativistic case (v<<c),

in this case and

The Hall field can be calculated as

The spin-orbit magnetic field can be calculated as

Note: An electron should have velocity component perpendicular to a static electrical field Estatic or a static magnetic field Hstatic in order to experience the magnetic field HSO of spin-orbit interaction or the Hall field EHall

##### Note: The Hall Effect and the Spin-Orbit interaction are close cousins

the Hall effect ==== results in ====> an effective electrical field

the Spin-Orbit interaction ===results in=====> an effective magnetic field

#### An electron always moves along the electrical field (but not perpendicularly). Then, how it can experience the spin-orbit 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 nucleus. 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 spin-orbit interaction? Energy? Spin? Orbital moment? Or any other quantum mechanical parameter?

(It is important): There is only one cause of spin-orbit interaction. It is the magnetic field c of spin-orbit interaction. All other causes are consequences of HSO. For example, the electron spin align itself along HSO. It change electron energy, which can be defined as an energy of the spin-orbit interaction ESO. However, an additional external magnetic field Hext is applied, the electron spin is aligned along total magnetic field HSO+ Hext and ESO has no physical meaning. Therefore, there is only one parameter, which is fully characterizes and describes the spin-orbit interaction. It is HSO.

### effective spin-orbit magnetic field HSO

#### Since the spin-orbit magnetic field HSO is proportional to 1/c2 , the should be very small or almost negligibly small. Should we care about such a tiny effect?

it is correct. Usually, the HSO is very small except cases when the electrical field Estatic is huge. It is the case in close proximity to the nucleus. There the electrical field increase as 1/r, where r is the distance to the nucleus. The nucleus is almost "point-like" object, therefore the electrical field Estatic is huge in close proximity of the nucleus. It makes a large HSO

### Any substantial spin-orbit interaction is induced only by an electrical field of a nucleus!!!

##### Other realistic sources of the electrical field in a solid induce only a very weak spin-orbit interaction

In close vicinity of a nucleus an electron experiences a very strong electrical field of the nucleus. However, this field is very symmetric and the electron experience the opposite signs of the spin-orbit interaction on its path around nucleus. Therefore, the spin-orbit interaction cancels itself and the electron experience no spin-orbit 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 spin-orbit 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 spin-orbit interaction.

### Origin of spin-orbit interaction: the Lorentz transformation or the Dirac equation?

(fact): Both the Lorentz transformation and the Dirac equation provide equivalent and analogous descriptions of the spin-orbit interaction.

The spin-orbit interaction is a consequence of the electromagnetic field's invariance under relativistic transformations, also known as Lorentz transformations. Analogous to the electromagnetic field, the quantum field of electrons maintains its invariance under relativistic changes, a property encapsulated by the Dirac equation. Remarkably, the Dirac equation yields the exact same equation (Eq.1.7).

(reason why): This equivalence arises from the following reason: The spin-orbit interaction (SO) characterizes the interaction of a moving electron with an electric field, a phenomenon that remains consistent regardless of the chosen coordinates for calculation. It is possible to carry out calculations in a coordinate system that moves with the electron. Consequently, the electromagnetic field necessitates relativistic transformation. Similarly, computations conducted in the steady coordinate of the electric field can yield the same outcome for HSO. In this scenario, the quantum field requires a relativistic transformation, leading to the deduction of SO from the Dirac Equation.

### Confusing meaning of the historical name "spin-orbit"

#### (important fact): The orbital moment and the spin do not interact directly. They both are aligned along the direction of the broken spatial and time-inverse symmetries of the electron orbital. The orbital moment describes the degree of the broken rotational symmetry of the electron orbital. The spin is aligned along the magnetic field of the spin orbit-interaction HSO, which is also directed along the direction of the broken symmetry of the electron orbital. Even though the broken rotation symmetry and the broken orbital symmetry, which creates HSO, are very different, mostly the directions of both broken symmetries are in one direction. As a result, the spin is aligned along the orbital moment, even though there is no direct interaction between the orbital moment and the spin.

(important example 1): Existence of a strong spin- orbit interaction in absence of the orbital moment.

In a solid, the localized electrons at an interface experience the most strongest spin-orbit interaction (See the perpendicular magnetic anisotropy (PMA)). However, the orbital moment of these localized electrons are fully quenched. It literally means that their orbital moment is zero.

(important example 2): Dependence on external magnetic field

There is no spin-orbit interaction in absence of an external or internal magnetic field. An external breaking of the time- inverse symmetry is required in order for the spin-orbit interaction to manifest itself. In contrast, the strength of the orbital moment is independent of the magnetic field.

(Why the spin?)

Only one property of an electron, with which the magnetic field of the spin-orbit interaction HSO can interact, is the electron spin. The magnetic field of the spin-orbit interaction does not interact with the electron orbital moment. Only magnetic property of an electron, which remains and with which HSO can interact, is the electron spin.

(Why the orbital?)

The magnetic field of the spin-orbit interaction HSO is only substantial for orbital movement of an electron in a very strong electrical field of an atomic nucleus. In the case of an atomic nucleus, there is a magnetic field of the spin-orbit interaction only in case of an asymmetrical orbital ( a non-symmetrical orbital). The non-symmetrical orbital has a non-zero orbital moment. The larger the non-symmetry of the orbital is, the larger the magnetic field of the spin-orbit interaction and ,at the same time, the larger the orbital moment becomes. Therefore, there is such an indirect relation between HSO and the orbital moment. The only reason for this relation is that both quantities the orbital moment and the magnetic field of the spin-orbit interaction are proportional to the degree of the orbital asymmetry.

Even in absence of the orbital moment, an electron may experience a substantial spin-orbit interaction. For example, when electron is moving at a relativistic object in the space in a proximity of an electrically-charged object.

#### Q. Is the Spin-Orbit interaction a quantum-mechanical effect???

 Spin-orbit interaction in macro world Fig. 5. An imaginary case what would happen if the Sun were charged. In this case the magnetic moment of the Earth would be aligned in respect to the polarity of the Sun charge. The red arrow shows direction of the magnetic moment of the Earth. The blue arrow shows the direction of the effective magnetic field HSO due to the effect of the spin-orbit interaction, which is induced by the electrical field due to the Sun charge. When the the polarity of the charge is reversed, the direction of the effective magnetic field is reversed as well. It follows by the precession of the magnetic moment of the Earth around the effective field until it aligns itself to be antiparallel in respect to the SO magnetic field HSO.

#### A. No. The Spin-Orbit interaction affects both small objects and large objects. The spin-orbit 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 HSO of the spin-orbital interaction induced by this charge. The magnetic moment of the Earth would be aligned accordingly as it is shown in Fig.5.

#### Is the spin-orbit interaction is a quantum-mechanical effect, because it is only can be derived from the Dirac equations?

A. It is not correct statement. The spin-orbit 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 misunderstandings

In short:

(1) The spin-orbit interaction is a relativistic effect and can be fully described by relativistic equations.

(2) The Dirac is a relativistic quantum-mechanical 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 spin-orbit interaction from the Dirac equation are most precise. (See SO and Dirac equation here)

(3) The Schrödinger equation and Pauli equation, both describe the spin-orbit 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 spin- orbit interaction

Fig.6 There is a precession of electron spin S around magnetic field HSO of spin-orbit interaction until it aligns parallel to HSO. After the electron spin is aligned, the energy of SO interaction becomes

click on image to enlarge it

### Energy of SO interaction

The spin-orbit interaction manifests itself only by the SO magnetic field HSO. The HSO is a very normal magnetic field (almost). There is a precession of electron spin S around magnetic field HSO of spin-orbit interaction until it aligns parallel to HSO. After the electron spin is aligned, the energy of SO interaction becomes

μB is the Bohr magneton

#### Note: The HSO does not interact with the orbital magnetic moment of electron. It is interact only the spin magnetic moment

It is because of relativistic origin of both the SO interaction and the Lorentz force.

The HSO interacts only with the spin magnetic moment, but not with the orbital magnetic moment or the total magnetic moment

#### Why HSO does not induces the Lorentz force and cannot interact with the orbital magnetic moment?

The HSO 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 HSO , 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 spin-magnetic moment. The role holds for both cases when spin is align along HSO (along the orbital moment) and when the there is a spin precession around HSO.

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 non-zero spin (e.g. a photon, a magnon). It could take a relatively a long until full alignment of electron spin along HSO.

#### Is there any cases when the electron spin is not aligned along HSO?

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 HSO.

### Magnitude of the Spin-orbit interaction.

Except of a few weak effects, all spin-orbital effects are induced by an electrical field of an atomic nucleus and the election movement (rotation) in the close proximity of the nucleus !!

Magnitude of the spin-interaction 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 spin-orbit interaction

Except for an electron, which moves in a close vicinity of an atomic nucleus

a very strong electrical field + a moderate electron speed => result: a strong spin-orbit interaction

Example 1.

Even in the of the highest-possible electron speed in solid and largest-possible applied electrical field, the effective magnetic field of the spin-orbit interaction is small!!

### Estimation: Maximum electron speed + Maximum applied electrical field

Electron 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 spin-orbit interaction is only 0.5 Gauss

It is too small!!! For example, Earth's magnetic field at at the Earth's surface ranges from 0.25-0.65 Gauss.

# s- orbit of an atom

Check!!! electron rotates simultaneously in clockwise and counterclockwise directions Check!!! electron rotates simultaneously in clockwise and counterclockwise directions electron orbital can be represented as 2D electron rotation around a rotated axis electron orbital can be represented as 3D electron rotation around a point (nucleus position)

### Both representation of the electron orbital are equivalent.

Fig.8. s- orbital. An electron rotates around a nucleus in spherical orbital. Since at the same time the electron rotates in clockwise and anti clockwise directions, the spin-orbit interaction for opposite rotations cancels each other and the electron does not experience the spin-orbit interaction.
It is not unusual that the electron rotates in opposite directions. Just look at the animation carefully.
click on image to enlarge it

Example 2.

### An electron rotating around a nucleus.

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

Result:

#### The effective magnetic field of the spin-orbit interaction is 125 kGauss=12.5 T

It is rather large!!!. Such large magnetic field can only be obtained by a superconducting magnet.

### Spin-orbit interaction and orbital moment

#### Is the spin-orbital interaction proportional to the orbital moment?

A. Actually, not. Even though there are common tendencies between the spin-orbital 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 HSO is complex and not straightforward.

The spin-orbit interaction:

for centrosymmetric electrical field of a nucleus:

where qnucleus is the nucleus charge.

HSO is proportional to ~1/r. As a result, the main contribution to HSO is from region in proximity of the nucleus. The symmetry of electron distribution function and electron rotation symmetry in close vicinity of the nucleus mainly contribute to HSO

The orbital moment:

or in quantum-mechanical representation

L is proportional to ~r. As a result, both regions, which are close and far from the nucleus, give a substantial contributions to L.

Even though a formal relation between HSO and L is very simple:

The integration over electron distribution gives very different value of HSO and L depending on the symmetry and details of electron wavefunction.

orbital momentum vs rotation symmetry

## Points to show:

 (point 1)Orbital moment is proportional to the specific orbital symmetry (point 2) In the case of when orbital moment equals to zero, the electron orbital consists of two identical paths, in which electron moves in the opposite directions. (point 3)The spin-orbital interaction is directly proportional to the orbital symmetry and, therefore, to the orbital moment. (even though the dependency between SO and the orbital moment is indirect)

## Case 3.

orbital momentum is zero. there is no SO interaction orbital momentum is largest & SO interaction is strongest. Direction of orbital momentum is up orbital momentum is largest & SO interaction is strongest. Direction of orbital momentum is down
Full rotation only 1st-half rotation only 2d-half rotation
Look carefully on a difference of the rotation symmetry

Orbitals

Orbital moment

## Different symmetry of rotation produces a different spin-orbit magnetic field

Fig.9. click on image to enlarge it

## (Part 5a): Analyze of anisotropy field Hani. Measurement of coefficient of spin-orbit interaction kSO

(short content 1:)  the properties of spin-orbit interaction   (short content 1:)  the difference between Global, Local & Average Magnetic Fields.   (short content 1:)  how it is possible to measure  the strength of SO and strength of PMA.    (short content 1:) analyze the measurement of the anisotropy field
(short content 2:)   linear relation between the external magnetic field and Hso and about the symmetry and physics arguments, which are behind this linear relation   (short content 2:)  how to measure experimentally the internal magnetic field Hint   (short content 2:)    what holds  the magnetization along the easy magnetic axis and how to measure the strength of the holding force.
(short content 2:)  the measured oscillations of Hani and why this oscillation is an important characteristic of a nanomagnet
(short content 3:)   how the deformation of the orbital at the interface creates the perpendicular magnetic anisotropy (PMA)   (short content 3:) experimental measurements of Hint in FeB nanomagnets.   (short content 3:)   the experimental setup and main idea behind the proposed measurement   (short content 3:)   why the spin-orbit interaction depends on the polarity of the magnetic field and how to measure it.
Other parts of this video set is here
click on image to play it

### Orbital momentum vs rotation symmetry vs spin-orbit interaction

#### Q. How is it possible that an electron, while rotating around a nucleus, does not experience the Spin-Orbit 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 spin-orbit interaction are opposite, an electron does not experience any spin-orbit interaction.

#### Is HSO always equal to zero for a spherical orbital?

No, See center and right pictures

The s-orbital can be divided to the sum of two spherical orbital, for which HSO is a no zero and opposite between two orbitals.

#### Q. In case of s-orbital a half of rotations an electron experience the field of the spin-orbit 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 spin-orbit interaction at all. Therefore, the spin-orbit interaction still does affect the electron of s-orbital. Is it correct?

A. No, it is not correct. The spin-orbit interaction does not affect an electron of s-orbital 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 mean-free 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.

### Orbital rotation of conduction electrons

 Additionally to the movement along the crystal lattice, a conduction electron rotates around each atomic nucleus. Electrical field of the nucleus induces the magnetic field HSO of the spin-orbit interaction. The HSO affects the electron scattering probability and therefore creates the spin-polarized current. The effect is called the spin Hall effect. Red arrow shows the direction of electron movement. Blue circle shows the direction of orbital moment. Violet arrow shows direction of HSO induced by the orbital moment.

Yellow wavy ellipse shows the wave function of the conduction electron. Blue circle show the direction of rotation of the conduction electron around each atomic nucleus (dark spheres).Electrical field of each nucleus induces the magnetic field interaction HSO. The electron experiences the accumulative strong HSO. Even though the electrons moves along stationary nuclei,accumulative HSO remains constant.

The size of a conduction electron (size of its wave function) is relatively large. A conduction electron can cover simultaneously hundreds or thousands of nuclei.
camera moves with the electron
click on image to enlarge it

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 non-zero unquenched orbital moment. A non-zero 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 non-zero and orbital can be either quenched or unquenched

## Three type of Spin-orbit interaction

#### SO interaction. Type 1. Example: the Spin Hall effect

##### The red and blue regions at wire side are regions of spin accumulation. Red and blue colors corresponds to spin-up and spin-down directions of spin accumulation. White color corresponds to regions without any spin accumulation.
Effect Origin 2:: There is a Schottky barrier at each edge of the GaAs wire and therefore an electrical field perpendicularly to the edge. Since an electrical current flows in the wire perpendicularly to that electrical field, the electrons experience the SO magnetic field HSO , which creates the spin accumulation. The direction of the electrical field is opposite on opposite sides. As a result, the polarity of and spin accumulation is opposite on opposite sides.
##### (requirement 2): The time-inverse symmetry should be broken. (The SO cannot break time-inverse symmetry by itself, but breaking ids required in order to have HSO )

(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 time-inverse 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 nucleus

(source of breaking of time-inverse symmetry): an external magnetic field

(type 3: moderate SO) creation of spin polarization by an electrical current

(source of electrical field): centrosymmetric electrical field of atomic nucleus

(source of breaking of time-inverse symmetry): an electrical current

(induced effects):Spin Hall effect & Inverse Spin Hall effect (main contributions), Spin-Orbit 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 time-inverse 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

 Enhancement of external magnetic field Fig. 14. The external magnetic field Hext induces the effective magnetic field of the spin-orbit interaction HSO, which is is in the same direction as the external magnetic field.. Therefore, the total magnetic field, which the electron experiences, becomes larger. note: Magnitude of HSO depends significantly on the direction, in which Hext is applied with orbit to orbital symmetry. click on image to enlarge it
##### only localized electrons (e.g. d- or f- electrons) experience this type of SO effect

(source of electrical field): centrosymmetric electrical field of atomic nucleus

(source of breaking of time-inverse symmetry): an external magnetic field

localized electrons experience this type of SO

Explanation of effect:

The type 2 of spin-orbit interaction is induced be an external magnetic field. E.g. in absence of a external magnetic field a localized electron does not experience any HSO. However, when external magnetic field Hext is applied, it induces strong HSO parallel to Hext and the electron experiences a stronger total magnetic field Htotal =Hext+HSO. E.g. when Hext=1 kG is applied, it induces HSO.=5 kG. Therefore, in total electron experience Htotal=6 kG.

Reason why an external magnetic field Hext induces the spin-orbit interaction HSO

(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 HSO for the CCW and ACW orbitals , therefore in total it experiences no HSO

(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, HSO becomes different for CCW and ACW orbitals and in total the electron experiences a non-zero HSO.

 Origin of SO interaction of type 2 Without external magnetic field, a localized electron does not have any orbital moment. It means the electron rotates in CCW and ACW directions on exactly the same orbital and experiences the same, but opposite HSO. The HSO is induced by the electrical field of nucleus. Therefore, in total the electron experiences no SO, HSO=0. Under external magnetic field, CCW orbital becomes slightly closer to nucleus and CCW orbital becomes more distant due to the Lorentz force FLor. As result, HSO becomes different for CCW and ACW orbitals and in total HSO becomes a non-zero click on image to enlarge it

#### 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 time-inverse 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 nucleus. The part of electron distribution, which corresponds to the electron rotation in CCW direction with respect to magnetic field, is shifted away from the nucleus.

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 s-orbital, even though the magnetic field does not change either energy or orbital moment of the s- orbital.

Effects, which are originated from Spin-orbit interaction of type 2:

#### effect 6): Magnetic anisotropy

(type 3: moderate SO) creation of spin polarization by an electrical current

## rotational moment of a conduction electron

(Fig.3a) (in absence of an electrical current) Electron scattering are different for the left and right directions. However, all electrons experience the same difference. As a result, the scattering difference does not lead to any additional current. E.g. electrons moving along +y direction are more scattered toward +x direction, but in contrast electrons moving along -y direction are more scattered toward -x direction. Since the number of electrons moving along +y and -y directions are equal. There is no any additional current along x direction (Fig.3b) (there is an electrical current along y-axis) There are more electrons moving along +y than along -y direction. As a result, there are more electrons toward +x direction than toward -x direction. The number of electrons moving along +x direction becomes larger than in -x direction. Therefore, a current flows along x-direction. This is origin 1 of the Spin Hall effect. (Fig. 3c) (Moving conduction electron experiences SO magnetic field HSO ). Red arrow shows the direction of electron movement. Blue circle shows the direction of orbital moment. Violet arrow shows direction of HSO induced by the orbital moment.

Fig. 3. Distribution of electrons with orbital moment and distribution of HSO.

The length of a vector from axis origin to the sphere is proportional to the number of electrons moving in the vector direction. Red arrow shows the direction of electron movement. Blue circle shows the direction of orbital moment. Violet arrow shows direction of HSO induced by the orbital moment. The direction of orbital moment and HSO is fixed to the electron movement direction.

## Electrical current J along y-direction creates spin-polarized currents along x- and z- directions

Each election moving in a different direction experiences magnetic field HSO in a different direction.
When a conduction electron moves along crystal lattice, it simultaneously rotates around each atomic nucleus as it passes it. The electrical field of nucleus induces the magnetic field of spin-orbit interaction. The spin of conduction electrons interacts with HSO. There is a spin precession around HSO and spin precession damping, which aligns spin to HSO.

(source of electrical field): centrosymmetric electrical field of atomic nucleus

(source of breaking of time-inverse symmetry): an electrical current

(induced effects):Spin Hall effect & Inverse Spin Hall effect (main contribution), Spin-Orbit 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 spin-polarized conduction electrons. As a result, initially spin-unpolarized gas of conduction electrons becomes spin-polarized.

(effect 3): Inverse Spin Hall effect (ISHE)

When a spin-polarized 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 spin-polarized, there are spin-polarized conduction electrons, which spins is directed in one direction. When the conduction electron has a non-zero rotational (orbital) moment, it experiences a non-zero HSO and there is a spin precession around HSO. Since the direction of HSO 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 spin-polarized electrons is misaligned from one direction and degree of the spin polarization is reduced.

there are two contributions to current-induced spin-orbit effects:

(contribution 1) band current

##### It is a feature of a metal of a high conductivity (See here)

(explanation of effect):

(step 1) The conduction electron have a non-zero rotational (orbital) moment,which created magnetic field HSO. There is a spin precession around HSO and the spin is aligning along HSO 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 HSO 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 non-zero HSO and the electron gas becomes spin-polarized.

(step 4) When there is an electrical current, the scattering probability of spin-up electrons to the left becomes different from the scattering probability to the right. As a result, e.g. the spin-up polarized current flows to the left and the spin-down 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 HSO and its spin is aligned along HSO

(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 (nucleus) of the lattice.

### Spin-orbit (SO) interaction. Facts in short.

#### (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 different

Properties 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. (HSO=0.01-1 Gauss). Facts in short.

Spin Hall effect of type 1

## due to electron movement perpendicularly to electrical field (Schottky- type) at interface

Fig. 30 . The electron current in a GaAs stripe. The view point moves together with electrons. Because of the charge depletion at the GaAs-oxide boundary, there is an electrical filed E perpendicular to the boundary (shown as the green arrows). The electrical field E induces the effective magnetic field HSO of the spin-orbit interaction (red arrows). The direction of the effective magnetic field is opposite at the opposite sides of the GaAs strip. Spins of conduction electrons is aligning along HSO. As a result, the conduction electrons becomes spin- polarized. The direction of the spin- polarization is along HSO

When the conduction electrons moved near the edge of the GaAs strip, the spins precess around the magnetic field HSO. Because the damping of the spin precession the spins are aligned along the effective magnetic field of the spin-orbit interaction.
The electrical field, HSO and spin polarization exponentially decays from the interface deep into the wire. There are no E, HSO and spin polarization at the center of the wire.
Due to the Schottky barrier at the boundary, the electrons are depleted (accumulated) at the boundary region creating the electrical field E.
click on image to enlarge it
###### note: HSO of the Weak SO interaction is comparable to Earth magnetic field 0.25-9.65 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.

#### (Origin of breaking of time-inverse 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 HSO 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 HSO becomes zero.

Effects, originated by "Weak- type" SO:

(1) Spin Hall effect (weak contribution)

##### It describes the fact that a spin-polarized 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)

#### Type 2: "non- zero orbital" SO interaction. (HSO=0.01-20 000 Gauss). Facts in short.

(Origin): It is originated from orbital movement of electron, when the orbital moment of electron is a non-zero.

(Origin of breaking of time-inverse symmetry): orbital alignment (spontaneous, local, by a magnetic field etc.)

(Source of electrical field (field is strong)): electrical field of the nucleus

(Reason why an electron moves perpendicularly to electrical field): Orbital movements,

(Source of electron movement (movement is fast)): Orbital movements

(Symmetry): (1) HSO 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. (HSO=1 000 -20 000 Gauss). Facts in short.

(Origin): It is originated by an electrical field of nucleus due to the orbital rotation of an electron around nucleus and an external magnetic field, which breaks the times-inverse 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!

#### It is only type of SO interaction, which exists without an electrical current

(Origin of breaking of time-inverse symmetry): external magnetic field

(Source of electrical field (field is strong)): electrical field of the nucleus

.(Reason why an electron moves perpendicularly to electrical field): Orbital movements,

(Source of electron movement (movement is fast)): Orbital movements

(Symmetry): (1) HSO is linearly proportional to the external magnetic field. (2)

Effects, originated by "Strong- type" SO:

#### Type 4: "Moderate- type" SO interaction. (HSO=1 0 - 500 Gauss). Facts in short.

(Origin): It is originated by an electrical field of nucleus due to the orbital rotation of an electron around nucleus and an electrical current, which breaks the times-inverse symmetry of the orbital

###### note: The moderate-type SO effect cannot break the time-inverse symmetry by itself. It requires an electrical current flow to break it.

(Origin of breaking of time-inverse symmetry): electrical current

(Source of electrical field (field is strong)): electrical field of the nucleus

(Reason why an electron moves perpendicularly to electrical field): Orbital movements

(Source of electron movement (movement is fast)): Orbital movements

(Symmetry): (1) HSO is linearly proportional to the current (2)

Effects, originated by the "Moderate- type" SO:

(1)

### Key properties of the spin-orbit interaction

.....

#### Effect 1:

-- Enhancement (magnification) of the applied magnetic field.
Due to the spin-orbit interaction, an electron experiences the effective magnetic field, which is larger than the actual applied magnetic field.
where it is the case: (1) changing of g-factor; (2) perpendicular magnetic anisotropy; (3) magnetostriction

#### Effect 2:

-- Spin-dependent scatterings.
Due to the spin-orbit interaction, the scattering probability for electrons with opposite spins becomes different.
where it is the case: (1) Anomalous Hall effect; (2) Spin Hall effect

#### Effect 3:

-- Spin precession. Spin relaxation.

When electron moves across a strong electrical field, the effective magnetic field of the spin-orbit 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.

#### In a non-magnetic material (paramagnetic or diamagnetic)

(Effect 1) -g-factor becomes larger than g-factor of an electron in the free space ;

(Effect 2)-The bulk-type Spin Hall effect due to scatterings on non-magnetic and magnetic impurities

(Effect 3)- The interface-type 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 spin-orbit interaction)

(Effect 6)-interface-induced perpendicular anisotropy (for example, Co/Pt). It is due to a large difference in the spin-orbit 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 single-crystal multilayer structure.

(Effect 8)- Anomalous Hall effect (AHE)

#### Q1. The Spin-Orbit interaction. What it is ??

The Spin-Orbit interaction describes the fact that an electron experiences an effective magnetic field when it moves in an electrical field.

#### Q2. The Spin-Orbit interaction. How does it affect an electron??

The effective magnetic field HSO of the Spin-Orbit interaction affects only the electron spin. Interaction of HSO 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 spin-orbit interaction

3) Electron transport can become spin-dependent

4) The electron energy becomes spin-dependent.

#### Example to understand how the spin-orbit interaction works in a solid

 Increase of spin-orbit interaction due to crystal deformation. click on image to enlarge it Fig. 20. Due to the crystal deformation, the orbital are deformed and the nuclei are shifted out of the center of the orbital. It makes HSO (green arrow) larger and the magnetic energy larger. Green arrows show the effective magnetic field HSO of the spin-orbit interaction. It is large only when the orbitals are deformed. Blue arrows show the intrinsic magnetic field Hinside. It the total magnetic field, which electron experience: external magnetic field, magnetic field from neighbor orbitals. The total magnetic field except HSO White spheres show the orbitals of the localized d-electrons The magnetic energy and HSO increase only when the magnetization and Hinside (blue arrow) are perpendicular to the deformation and film surface. There is no enhancement of HSO when the magnetization is in-plane. This effect is called the perpendicular magnetic anisotropy (PMA)

Q. Both localized (-d,-f) and delocalized (-s,-p) electrons are rotating around nuclei (atoms), is it sufficient for them to experience a strong spin-orbit interaction?

A. No. It is far not sufficient. There are several other conditions the electron should satisfy in order to experience the spin-orbit interaction:
1) The orbital moment (symmetry) should be non-zero. Only electrons, which wave function has the -p,-d,-f- like spacial symmetry may experience the spin-orbit interaction.
2) The time-inverse symmetry should be broken!!

The following describes the reasons why an electron does not experience the spin-orbit interaction when the electron orbit is spherical and why it does experience the spin-orbit interaction for other shapes of the orbital.

## Spin-orbit interaction. Type 2. Magneto-elastic effect

#### This effect described facts that the elastic stress may enhance the spin-orbit interaction and increase the energy of the perpendicular magnetic anisotropy (PMA).

##### see details here

When 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): nuclei are shifted out of the center of the orbital. Both deformation makes the effective magnetic field HSO of the spin-orbit interaction larger.

In a ferromagnetic material the localized electrons have a non-compensated spin, which creates a magnetic field Hmag

At an interface between a magnetic and non-magnetic material, the demagnetization field Hdemag is created due to uncompensated magnetic moment at the interface. The direction of Hdemag is perpendicular to the interface and opposite to Hmag.

The magnetic field Hinside inside of the ferromagnetic field equals Hmag- Hdemag. The Hinside is the total magnetic field except HSO. It includes the external magnetic field

Important fact: Additionally, the electron experience the effective magnetic field HSO of the spin-orbit interaction, which is always directed along Hinside. The magnitude of HSO is proportional to Hinside and the degree of the orbital deformation.

Without a deformation the orbitals of the localized electrons is nearly spherical and the effective magnetic field HSO of the spin-orbit interaction is small.

When the pressure applied, the orbitals are deforms in the direction of the applied pressure and the effective magnetic field HSO of the spin-orbit interaction increases.

The magnetic energy of an electron equals to a product of the electron spin and Hinside+HSO.

When magnetization is perpendicular to the film, the orbital deformation is larger, HSO is larger and the magnetic energy is larger.

When magnetization is in-plane, the orbital deformation is smaller, HSO 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)

# Spin-orbit interaction in the centrosymmetric electric field of atomic nucleus.

 Animated figure 6. Spin-Orbit Interaction in the center-symmetrical electrical field of atom nucleus. It is an imaginary case when electron moving in a circular orbit around nucleus (for s- symmetry electron, the orbit is spherical). Red arrow indicates the spin direction of electron. Blue arrow indicates the effective magnetic field of the spin-orbit interaction. The effective field appears only when the electron moving. The directions of the effective field is opposite for the opposite direction of the electron movement. There is a spin of precession around the effective magnetic field of the spin-orbit interaction. The electrons spin slowly aligns itself along the effective magnetic field because of the precession damping. click here to enlarge

#### Distribution of conduction electrons with orbital moment

Fig.

click on image to enlarge it

Spin-orbit interaction (type 3: moderate SO) creation of spin polarization by an electrical current

(source of electrical field): centrosymmetric electrical field of atomic nucleus

(source of breaking of time-inverse symmetry): an electrical current

(induced effects):Spin Hall effect & Inverse Spin Hall effect (main contribution), Spin-Orbit Torque

 Movement of a conduction electron in a metal Simultaneously with movement along the crystal a conduction electron rotates around each nucleus. The rotation and the linear movent cause two different types of the spin-orbit interaction.

Q. How to make an electron to rotate in one direction more than in the opposite direction??? How to make the spin-orbit 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 effect

The orbital can be distorted by an electrical field. In this case, the electron experiences the effective magnetic field due to the spin-orbit interaction.

When the orbital is distorted by an external electrical field, the existence of the effective magnetic field due to the spin-orbit 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 spin-orbit 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 time-inverse symmetry, which is a key condition for SO to occur (See below).

orbital momentum in

click on image to enlarge it

### Direct (weak) and indirect (strong) spin-orbit interaction (SO)

There are two kinds of the spin-orbit interaction in a crystal lattice. In both cases an electron experiences an effective magnetic field of the spin-orbit 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 externally-applied 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 spin-orbit interaction is induced not by an external electrical field, but by the electrical field of a nucleus. The external electrical field just deforms the electron orbital making the spin-orbit 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 spin-orbit interaction may experience localized electrons, delocalized (conduction) electrons and standing-wave electrons.

In contrast to direct SO, the indirect SO can occurs only when the time-inverse symmetry is broken. It can be broken by an external magnetic field or a local magnetic field. (See below)

### The spin-orbit 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 spin-orbit interaction. This is reason, for example, why the spin-orbit interaction is significantly stronger in GaAs than in Si.

In an ionic crystal the covalent electrons are nearly-fully transformed from a cation to a anion and the electron orbital becomes again more center-symmetrical with a weak spin-orbit interaction. This is reason, for example, why the spin-orbit 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 spin-orbit interaction is directly proportional to the electric field of the nucleus. The nucleus charge is larger for an element of a larger atomic number. Therefore, the electrical field of the nucleus and the spin-orbit 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 spin-orbit interactions by inner electrons.

Because of the screening of an electrical field of a nucleus by inner electrons , the strength of spin-orbit interaction reduces.

The effects of screening:

(effect 1) The spin-orbit 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 nucleus field of anion is larger. Therefore, the spin-orbit interaction induced by the nucleus of anion is smaller.

(effect 2) In atoms of unfilled inner shells the spin-orbit interaction is stronger.

In the case when the inner shell of atom is not fully filled, the screening of the nucleus by the electrons of the inner shell is not centrosymmetric. It makes the spin-orbit interaction stronger.

### It is good to know.

#### (fact) The effective spin-orbit magnetic field HSO 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.

# Main part:

## Time- inverse symmetry and the spin-orbit interaction

Time-inverse symmetry is not broken

Fig. 12. Electron orbital. The green arrows show the direction of the effective magnetic field of the spin-orbit interaction HSO

Time-inverse symmetry: not broken

Average effective magnetic field of the spin-orbit interaction: zero

When the time-inverse symmetry in the material is not broken, there is an equal probability that electron circulating around the nucleus in the clockwise and anti clockwise directions. Since the electron experiences equal and opposite effective magnetic field of the spin-orbit interaction, in the average the electron does not experiences any effective magnetic field of the spin-orbit interaction. (See Fig. above)

Even in the case when the orbital moment of the electron is not zero, when the time-inverse 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 spin-orbit interaction: zero

note: in this case the spin-orbit interaction affects the spin relaxation

---------------------------------------------

#### 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 spin-orbit 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 spin-orbit interaction. The magnification is different for the orbital of a different symmetry. As a result, the orbitals experience a different HSO and the orbital energies become different. (See more details Here)

#### (fact) The spin-orbit interaction cannot break the time inverse symmetry

How the time-inverse symmetry is broken

It results: Average effective magnetic field of the spin-orbit interaction becomes non-zero

An external magnetic field breaks the time-inverse symmetry and it causes a non-zero average effective magnetic field of the spin-orbit interaction in the direction of the external magnetic field.

Since the electron moves around the nucleus, it experiences the Lorentz force in the magnetic field. The Lorentz force is in opposite directions for electron moving in the clockwise and counterclockwise directions around the magnetic field. The Lorentz force modifies the orbital of electrons. When an electron moves in the counterclockwise direction, it moves closer to the nucleus and it experiences the larger electrical field from the nucleus and the larger corresponded effective magnetic field of the the spin-orbit interaction. When an electron moves in the clockwise direction, it moves more distant from the nucleus and it experiences the smaller electrical field from the nucleus and the smaller corresponded effective magnetic field of the the spin-orbit interaction. In the average, the average the electron experiences a non-zero effective magnetic field of the the spin-orbit interaction in the direction of the external magnetic field.

##### note: The effective electrical field of the Lorentz force can not induce the spin-orbit 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 spin-orbit 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 spin-orbit interaction.

# Enhancement of magnetic field due to the spin-orbit interaction

 Magnetic field, which applied to electron orbital, creates additional magnetic field due to the spin- orbit interaction. As a result, the electron spin interacts with a larger magnetic field than externally applied. Magnetic anisotropy is due to this effect. See Perpendicular magnetic anisotropy Measurement of the strength of spin- orbit interaction is based on this effect (See below)

## This is the most important property of the spin-orbit interaction !!!.

#### This property determines how the spin-orbit interaction affects electrons in a solid

In fact, it is the joint work of two relativistic effects: 1) the Lorentz force 2) the spin-orbit interaction

- The Lorentz force, which is induced by an external magnetic field, deforms the electron orbital and breaks the time-inverse symmetry;

- Because of the broken time-inverse symmetry, the strong effective magnetic field is induced by the spin-orbit interaction.

## (key property of spin- orbit interaction): The magnetic field of the spin- orbit interaction HSO is proportional to an externally- applied magnetic fieldHext

In absence of external magnetic field   In presence of external magnetic field

### strong SO magnetic field. HSO ≠0

The electron experiences the magnetic field HSO of the spin-orbit interaction, when it is orbiting the nucleus. However, the magnetic field HSO is opposite, when the electrons moves in clockwise and counterclockwise directions. As a result, in total the electron experiences no spin- orbit interaction.   In the presence of an external magnetic field Hext, the electron experience the Lorentz force FLor, which is opposite for the clockwise and anti- clockwise paths of the electron orbital. The opposite Lorentz forth makes clockwise and anti- clockwise paths of electron orbital to become different. Since during the counter-clockwise movement, the electron path becomes closer to the nucleus, the electron experiences a larger the electrical field of nucleus and, as result, a larger magnetic field HSO of the spin-orbit interaction. In contrast, during the clockwise movement, the electron path becomes further from the nucleus, the electron experiences a smaller the electrical field of nucleus and, as result, a smaller magnetic field HSO of the spin-orbit interaction. Since the HSO becomes unequal for opposite rotation directions, in total, the electron experiences a non-zero magnetic field HSO of the spin-orbit interaction.
Any orbital of a localized electron consists of orbital paths of clockwise and counter-clockwise rotations. The time-reversal symmetry requires that the path of the clockwise rotation to be exactly equal to the path of counter- clockwise rotation.   (proportionality of HSO to the external magnetic field Hext) When the external magnetic field Hext becomes larger, the Lorentz force FLor, becomes larger, which leads to larger difference between clockwise and counter- clockwise orbitals and, as a consequence, a larger magnetic field HSO of the spin-orbit interaction.
This feature of the spin-orbit interaction originates the perpendicular magnetic anisotropy (PMA). See here for details.
click on image to enlarge it

### (origin of the effect):

The part of the orbital that rotates clockwise around the magnetic field contracts due to the Lorentz force, bringing it closer to the nucleus, where the electric field is stronger. Consequently, this portion of the orbital experiences a greater spin-orbit magnetic field.

Conversely, the counterclockwise rotating portion of the orbital expands and moves away from the nucleus, resulting in a smaller spin-orbit magnetic field.

Since the electric field diminishes with increasing distance from the nucleus following a 1/r decay, the gain from the clockwise rotating component surpasses the loss from the counterclockwise rotating component. As a result, the electron experiences an overall amplified magnetic field due to the spin-orbit interaction.

:

Time-inverse symmetry is broken

Fig. 13. Electron orbital in the presence of external magnetic field (blue arrow). The green arrows show the direction of the effective magnetic field of the spin-orbit interaction HSO The red arrows show the direction the Lorentz force

## Broken time inverse symmetry & Enhancement of magnetic field due to the spin-orbit interaction

 The increase of the effective magnetic field due to the spin orbit interaction Fig. 14. The external magnetic field Hext induces the effective magnetic field of the spin-orbit interaction HSO, which is is in the same direction as the external magnetic field.. Therefore, the total magnetic field, which the electron experiences, becomes larger. click on image to enlarge it

When a magnetic applied to the material, it breaks the time inverse symmetry. As result, the electron starts to experience non-zero effective field of the spin-orbit interaction.

The effective magnetic field HSO of the spin-orbit 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 spin-orbit 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-Orbit interaction due to the orbital deformation

The increase of the spin-orbit interaction due to deformation of the electron orbital

Fig. 9. Electron orbital. When the orbital is spherical the effective magnetic field of the spin-orbit interaction HSO is zero. Only when the orbital is deformed, there is the magnetic field of the spin-orbit interaction. The effective magnetic field is the largest in the case of a circle or elliptical orbital.

##### The effective magnetic field of the spin-orbit interaction for localized electrons due to a deformation of electron orbit may be very large. It may reach 1-30 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 spin-orbit interaction

(1) The electron orbit should be deformed along one direction

(2) The electron orbit should be deformed asymmetrically in respect to its nucleus

The orbital is significantly deformed in compound materials with covalent bonding (like GaAs). Therefore, they have a larger spin-orbit interaction.

In materials with ionic bonding, the orbital is less deformed and they have a smaller spin-orbit interaction (like ZnO).

The p- , d- and f- orbitals are inherently asymmetrical. For each individual p- , d- and f- orbital, the spin-orbit interaction may be strong.

For each individual p- or d- or f- orbital, the time-inverse symmetry is broken. However, in a non-magnetic metal or a semiconductor, where the time-inverse 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 spin-orbit interaction of electrons of p- or d- or f- symmetry may be not as strong as in the case of a separated atom.

## Orbital symmetry vs strength of the spin-orbit interaction

### not- deformed orbital. HSO is small

When electron orbital is deformed, the part of the electron path becomes very close to the atomic nucleus, where the electrical field of the nucleus is very large. Since the magnetic field of spin-orbit interaction HSO is proportional to the electrical field, the HSO becomes substantially larger. Even though, at same time the HSO reduces from to be small to become negligibly small at the orbital part, which become far from the nucleus, in average the HSO substantially increases for the deformed orbital   When electron orbital is centrosymmetric, the magnetic field of spin-orbit interaction HSO is equally small at all points of the orbital. In average, the HSO is small.
(note: simplified schematic): the figures shown above are simplified schematic representation of the electron orbital aiming to demonstrate the operation principle of the effect.
(note about PMA) In the case of the PMA is only important how the HSO changes under applied magnetic field, but not the absolute value of. At parts closer to nucleus the influence of the the Lorentz force FLor may be weaker. Therefore, for the PMA a complex consideration on an increase of HSO and on a possible decrease of FLor should be considered to analyze the deformed orbital.
(note about atomic neighbors) In the case of a crystal, a deformation of the orbital and its influence on the HSO depend on neighbor atoms. Proximity of a neighbor can either push out or pull closer the electron orbital to the atomic nucleus. It can influence both the HSO and the orbital deformation due to the Lorentz force FLor.
(note interface and bulk) at an interface the neighborhood of the atom is different from that of the bulk and the orbital reconstruction (deformation) always occurs at the interface. As a result, the HSO and, therefore, the PMA is very different for the interface and the bulk of the material.
click on image to enlarge it

## Spin relaxation in gas of conduction electrons

Spin relaxation due to non-zero orbital moment of a conduction electron

## Distribution (pink sphere) of speeds, movement directions, orbitals moments and magnetic field HSO for conduction electrons

Fig.33a. The spin precession of spin of spin-polarized electrons around magnetic field of the spin- orbit interaction for different movement direction of conduction electrons. The direction of vector from axis origin to the pink sphere corresponds to the electron movement direction. The paused frame corresponds initial spin direction of spin polarized electrons, at which all spins are directed in one direction (along x- axis). Since electrons, which move in different directions, experience HSO in different directions, the axis of the spin precession is different for spin- polarized electrons moving in different directions. It results to a misalignment of spins of spin- polarized electrons and therefore to the spin relaxation Fig. 33b. Distribution of electrons with orbital moment and distribution of HSO. The length of a vector from axis origin to sphere is proportional to the number of electrons moving in the vector direction. The direction of orbital moment and HSO is fixed to the electron movement direction k (red arrow). The direction of HSO is different for conduction electrons moving in different directions.
Note: the scatterings in electron are rather frequent. As a result, the electron is scattered out much earlier before even one period spin precession is finished. In fact, the precision angle is rather small (less than 1 degree). The scatterings are constantly re-aligning spins into spin- direction distributions of two groups of spin-polarized and spin- unpolarized electrons (See details here)

## Alignment of HSO with respect to the spins of spin-polarized conduction electrons

Fig. 33c. Spin precession of a conduction electron. The pink circle shows the orbital (rotational) moment of a conduction electron. Blue arrows shows HSO, which is along the orbital moment. Red arrow shows movement direction of conduction electron. The direction of HSO is fixed to the movement direction of the conduction electron. There is a precession of spin of conduction electrons around HSO. The precession direction different for a different movement direction of a conduction electron. It results of misalignment of spin- polarized electrons and therefore the spin relaxation.
When rotational moment of conduction electron is non-zero, it experiences a magnetic field HSO of spin- orbit interaction. There are a precession of electron spin around HSO. The direction of and consequently direction of the spin precession is fixed to movement direction of the conduction electron E.g. spins of spin- polarized conduction electrons are aligned along the same direction (along x-axis). The alignment of HSO and electron spin are different for different movement direction of the conduction electrons. It results that the precession axis is different for electrons moving in different directions.
click on image to enlarge it

#### 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 HSO

The spins of all spin- polarized electrons are directed in one direction. In contrast, the direction of HSO is different for electrons moving in different directions. As a result, the angle between electron spin and HSO is different for the spin- polarized electrons moving in different directions. There is a spin precession around HSO. Since the directions of HSO 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.

#### Since the spin precession is fast (~0.1-1 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 spin-polarized 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.

Spin relaxation

## Spin relaxation is a joint work of (1) spin misalignment due to spin precession around HSO and (2) scatterings

 (initial state): all spins of spin- polarized conduction electrons are aligned along the same direction (along x- direction) (spin misalignment): There is a spin precession around magnetic field HSO of spin- orbit interaction. Direction of HSO is different for electrons moving in different direction. As a result, the axis direction for spin precession is different for electrons moving in different directions and spin directions of spin- polarized electrons becomes misaligned. (spin alignment & reduction of number of spin- polarized electrons): The electron scattering constantly separate electrons into two groups: spin- polarized and spin- unpolarized electrons (see here for details). As a result, misaligned spins of the scattered electrons are aligned again along the x- axis. However, the number of spin- polarized electrons becomes smaller, because of the conservation of the total spin of the electron gas.
The direction of vector from axis origin to the pink sphere corresponds to the electron movement direction. The length of a vector from axis origin to sphere is proportional to the number of electrons moving in the vector direction. The blue arrows show the direction of . The direction of is different for electron moving in different directions. The green balls shows the spin direction of spin- polarized conduction electrons. The size of the green balls is proportional to the number of the spin- polarized electrons
click on image to enlarge it

## g-factor

wiki page about g-factor is here

The g-factor 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 g-factor is used and it can be measured:

1) Ferromagnetic resonance and electron paramagnetic resonance. The g-factor 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 g-factor describes the energy difference for electrons, which spins are along and opposite to the direction of the magnetic field.

-------------------

Important notice: The g-factor, which is measured from the ferromagnetic or paramagnetic resonance, is not always same as the g-factor, 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 g-factor

In atoms when the spin is compensated and the magnetic moment is only due to the orbital moment, the g-factor equals to 1. The g-factor 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 spin-orbit interaction and the g-factor

non-magnetic 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 spin-orbit interaction.

Even though in the reality the effective electron g-factor does not change and only the effective magnetic field changes due to the spin-orbit interaction, it is convenient to assume the g-factor of the material is changed, but the magnetic field remains unchanged. Therefore, the Larmor frequency can be calculated as

where kSO is coefficient, which described the enhancement of the magnetic field due to the spin-orbit interaction. From Eq. (g4), the Larmor frequency is calculated as

where the g-factor is

Often the g-factor is defined and measured for the external magnetic field strength H instead of the magnetic induction B. In this case the effective g-factor can be used

g-factor and Specific magnetic susceptibility for non-magnetic materials

Content

g-factor

conduction band (bulk):

GaAs : -0.3 (300 K) -0.45 (50 K)

InAs: -15

InP: 1.5

GaSb=-8

InSb=-51.3

n-Si: =1.9985

p-Si=2

Cu=

Specific magnetic susceptibility (CGS-emu=Si-unit/4pi)

 Ge Si InAs GaAs InSb GaSb Al Ag Cu -1.15 -1.08 -1.2 -1.25 -1.25 -1.35 1.75×10−6 −1.84×10−6 −0.083×10−6

Paramagnetic (Si unit)

 FeO Pt Al W Cr Ti 720×10−5 26×10−5 2.2×10−5 6.8×10−5 3.13×10−4 1.81×10−4

Diamagnetic (Si unit)

 Ag Cu Au Si Al2O3 -2.6×10−5 -1×10−5 -3.44×10−5 -0.41×10−5 -1.81×10−5

EPR for Ge=2

9.3882 GHz-> 3.35 kG

ferromagnetic metals

In ferromagnetic metals

g-factor, saturated magnetization and width of FMR peak in ferromagnetic metals. Click to extend

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
g factor
free electron
g=2.0023
Fe g=2.088
Co g=2.18
Ni g=2.2
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
4piM=
Ni= 6.2 kG
Py=17.1 kG
Co=17.8 kG
Fe=21.4 kG
;;;;;;;;;;;;;;;;;;

;;;;;FMR

;;;;;;;;;;;;;;;;;;;;;;;;;
FMR magnetic field for 9.8 GHz
Ni= 1600 G
Py=700 G
Co=690 G
Fe=600 G
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;

relaxation parameters
Fe 57 MHz ( alfa= 0.002)
Ni 220 MHZ
Co 170 MHz
Py 114 MHz
;;;;;;;;;;;;;;;;;;;;;;;;;

for YIG width of FMR resonance
0.15 Oe -0.2 Oe

#### Q. Why the spin-orbital enhancement of the magnetic field cannot be included into Magnetic susceptibility

In the of paramagnetic metals, the spin-orbit 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 spin-orbit 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 spin-orbit 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 g-factor, the enhancement due to the spin-orbit interaction is 1-10 % for the most of materials. In the case of III-V semiconductors (GaAs,InAs), the enhancement may be more than 100 %.

### Perpendicular-to-plane magnetic anisotropy (PMA)

##### Detailed description of the PMA is here

Perpendicular-to-plane magnetic anisotropy at a Fe/Pt interface

### Magnetization is along the interface

Fig.15. Schematic diagram of Pt/Fe interface. The blue and red spheres show the electron orbitals in Fe and Co, respectively. Blue arrows shows the magnetization (spin of localized d-electrons). The green arrows show the effective magnetic field HSO of the spin-orbit interaction. The orbital of the localized electrons are shown.

The magnetization of a single-material 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 multi-layered films are Co/Pt, Fe(fcc)/Pt, Co/Tb, Fe(fcc)/Tb.

Since the strength of the spin-orbit interaction depends of the shape of the electron orbit in a crystal, the perpendicular-to-plane magnetic anisotropy only a feature of a specific crystal orientation and only a specific crystal orientations of the interfaces. For example, in all above-mentioned cases the perpendicular-to-plane magnetic anisotropy is feature of only fcc(111) interfaces or very similar hcp interfaces

Perpendicular-to-plane magnetic anisotropy occurs due to a strong effective magnetic field of the spin-orbit interaction at the interface . The enhancement of the effective field of the spin-orbit interaction occurs because of a deformation of the orbital of the ferromagnetic and non-magnetic 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 blue-colored spheres). In the vicinity of the contact, the orbitals are deformed.

Thickness-dependence of PMA

### Thinner metal

#### Magnetization is perpendicular to plane

Fig.16. Magnetization of CoFeB film grown on MgO. The magnetization of a thick film (thickness >1.5 nm) is in-plane , but the magnetization of a thinner film is perpendicular-to-plane

##### 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 spin-orbit interaction depends of the shape of the electron orbit in a crystal, the perpendicular-to-plane 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 in-plane.

## Magneto-elastic effect (Villari effect) and the Spin-orbit interaction

#### This effect described facts that the elastic stress may enhance the spin-orbit interaction and increase the energy of the perpendicular magnetic anisotropy (PMA)

 Increase of spin-orbit interaction due to crystal deformation. click on image to enlarge it Fig. 20. Due to the crystal deformation, the orbital are deformed and the nuclei are shifted out of the center of the orbital. It makes HSO (green arrow) larger and the magnetic energy larger. Green arrows show the effective magnetic field HSO of the spin-orbit interaction Blue arrows show the intrinsic magnetic field Hinside White spheres show the orbitals of the localized d-electrons The magnetic energy and HSO increase only when the magnetization and Hinside (blue arrow) are perpendicular to the deformation and film surface. There is no enhancement of HSO when the magnetization is in-plane. This effect is called the perpendicular magnetic anisotropy (PMA)

When 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): nuclei are shifted out of the center of the orbital. Both deformation makes the effective magnetic field HSO of the spin-orbit interaction larger.

Without a deformation the orbitals of the localized electrons is nearly spherical and the effective magnetic field HSO of the spin-orbit interaction is small.

When the pressure applied, the orbitals are deforms in the direction of the applied pressure and the effective magnetic field HSO of the spin-orbit interaction increases.

In a ferromagnetic material the localized electrons have a non-compensated spin, which creates a magnetic field Hmag

At an interface between a magnetic and non-magnetic material, the demagnetization field Hdemag is created due to uncompensated magnetic moment at the interface. The direction of Hdemag is perpendicular to the interface and opposite to Hmag.

The magnetic field Hinside inside of the ferromagnetic field equals Hmag- Hdemag

Additionally, the electron experience the effective magnetic field HSO of the spin-orbit interaction, which is always directed along Hinside. The magnitude of HSO is proportional to Hinside and the degree of the orbital deformation.

The magnetic energy of an electron equals to a product of the electron spin and Hinside+HSO.

When magnetization is perpendicular to the film, the orbital deformation is larger, HSO is larger and the magnetic energy is larger.

When magnetization is in-plane, the orbital deformation is smaller, HSO 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 perpendicular-to-plain 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 spin-orbit 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 spin-orbit 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= 270-330 GPa-- &--250 GPa

MgO bulk elastic properties

Compressive Strength 800-1600 MPa

Elastic Limit 80-166 MPa

Hardness 5-7 GPa

Breakdown Potential= 6-10 MV/m=0.006-0.01 V/nm

Conductivities (S/m)

 Silver 6.30E+07 Copper 5.96E+07 Gold 4.10E+07 Aluminium 3.50E+07 Tungsten 1.79E+07 Co 1.66E+07 Nickel 1.43E+07 Ru 1.40E+07 Iron 1.00E+07 Platinum 9.43E+06 Tin 9.17E+06 Cr 7.87E+06 Ta 7.40E+06 Carbon steel (1010) 6.99E+06 Lead 4.55E+06 Titanium 2.38E+06 Stainless steel 1.45E+06 titanium Nitride 1.42-3.33E6

Strain relaxation and the critical thickness.

The strain field, which acts on the film-substrate interface, is linearly proportional to the film thickness. The thin film has the in-plane 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 hcritical can be calculated from relation:

notice: Eq. (3) is valid only for high-crystal quality low-defect-density 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 tensile-strained Fe in order to reduce the strain field and to increase the critical thickness of the tensile-strained 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

## Magnetostriction

##### wiki page is here

The 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 single-domain nanomagnet the magnetostriction of this type does not exists.

Materials

Terfenol-D (TbxDy1-xFe2)

The magnetostriction of the Terfenol-D generates strains 100 times greater than traditional magnetostrictive, and 2-5 times greater than traditional piezoceramics.

For typical transducer and actuator applications, Terfenol-D is the most commonly used engineering magnetostrictive material.

Elastic properties (Tb0.3Dy0.7Fe1.92)

Young's Modulus=25-35 GPa

### Voltage-induced spin-orbit interaction & VCMA effect

##### details about VCMA effect are here

Enhancement of the spin-orbit interaction due to electrically induced orbital polarization

### When electrical field is applied

Fig.17. The yellow mesh shows the electron orbital; the blue arrow shows direction and magnitude of external magnetic field; the green arrow shows direction and magnitude of the effective field of the spin-orbit interaction; the red arrows shows direction and magnitude of the applied electrical field; the oval at left- bottom corner shows the induced dipole polarization of the orbital.

The external magnetic field induces the magnetic field along its direction. In the case of near-spherical orbit (Fig. 17), the enhancement is small and the magnetic field of the spin-orbit interaction is small.

In the external electrical field the positively-charged nucleus moves a little toward the direction of the electrical field. The negatively-charged 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 spin-orbit 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 nucleus. Therefore, at the left side of the nucleus the electron experiences a larger electrical field and a larger corresponded magnetic field of the spin-orbit interaction. Even at the right side of the nucleus the spin-orbit interaction is reduced, in total the spin-orbit interaction becomes larger in the electrical field. It is because the electrical field of nucleus 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 spin-orbit interaction and HSO=0.

The spin-orbit interaction by itself cannot break the time inverse-symmetry!

Enhancement of the spin-orbit interaction due to electrically induced orbital polarization

Fig.

### Spin-dependent scatterings

Due to the spin-orbit interaction the electron scattering may become the spin-dependent.

### Famous misunderstands and misinterpretations of the Spin-Orbit (SO) interaction

#### misinterpretations (1): The Spin-Orbit interaction is the Quantum-Mechanical effect, because it can be derived from the Dirac equations. The SO is only the feature of very small quantum-mechanical 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 Spin-Orbit interaction is a relativistic effect, but not a Quantum-Mechanical effect. The Dirac equations are relativistic quantum-mechanical 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 photon-electron interaction, should contain a description of the SO interaction. For example, the Maxwell's equations contain the description of the SO interaction as well.

#### misinterpretations (2): There is a "special" "quantum-mechanical" field of the Spin-Orbit interaction

Only a field of the SO interaction is the magnetic field. The SO magnetic field HSO 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 HSO and "normal" magnetic field, HSO cannot induce the Lorentz force

#### misinterpretations (3): Since the SO interaction can "interact" only with the electron spin, which is a quantum mechanicalobject, but not with the orbital moment, the SO interaction has quantum-mechanical origin. Additionally, the SO magnetic field does not induce the Lorentz force. It is an additional proof of the quantum-mechanical 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 movement-relative 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 HSO 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/c2, 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), HSO may reach tens of kGauss. For example, in a thin Fe film it can override the demagnetization field of 20-30 kGauss and align the magnetization perpendicularly to the film surface.

#### misinterpretations (5):The spin-orbit 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 time-inverse 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 nucleus.

Note: the formulas for orbital momentum and the spin-orbit interaction are very similar. The difference between them is only coefficient 1/r2. In close vicinity of the nucleus, the 1/r2 is huge and it makes a huge difference.

#### Can the Schrödinger equation be used to describe the spin-orbit 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 spin-orbit interaction from the Dirac equations or Schrödinger equation

Dirac equations calculates both contributions to the spin-orbit 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.

### Spin-orbit interaction obtained from the Dirac equations

##### The 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 quantum-mechanical for the energy and the momentum are

Substituting Eq.(3.2) into Eq.(3.2) gives Klein-Gordon 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 Klein-Gordon 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) is

where the gamma matrices (2 × 2 sub-matrices taken from the Pauli matrices)

The Dirac equation, which includes the gauge potential, is

#### Does Klein-Gordon equation include information about the conservation of the time-inverse symmetry and spin?

Probably not. The Dirac equation and the Pauli equation, both do include the conservation of the time-inverse symmetry and spin. It is difficult to answer about the Klein-Gordon equation.

### Non-relativistic form of Dirac equation

##### case v<<c

In this case wave function can be represented as a sum of a large "electron" part and a small "positron" part.

### Pauli equation & Spin-Orbit interaction

#### Does the Pauli equation describe the spin-orbit interaction?

Yes, the effective magnetic field HSO 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 spin-orbit interaction without usage of HSO?

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 HSO 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 HSO 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) is

where 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

demerits:

## Correct and incorrect Hamiltonian for Spin- orbit interaction

 Approximated Hamiltonian for Spin- orbit interaction (note) This Hamiltonian captures only a very general tendency that the strength and the direction of the magnetic field of the spin-orbit interaction are similar to the strength and the direction of the spin-orbit interaction. The orbital moment and the magnetic field of the spin-orbit interaction do not interact directly. Instead, they align themselves in accordance with the directional and strength patterns resulting from the broken spatial and time-inverse symmetries of the electron orbital. Consequently, they exhibit similar tendencies in terms of their directions and strengths. (example violation of this Hamiltonian) At an interface, the localized electrons undergo a notably potent spin-orbit interaction (See the perpendicular magnetic anisotropy (PMA)). Despite this strong SO interaction, the orbital moment of these localized electrons are fully quenched. It literally means that their orbital moment is zero. click on image to enlarge it

The classic incorrect Hamiltonian HSO of the spin-orbit interaction is just a product of the spin and the orbital of a n electron

Incorrect Hamiltonian, which only shows a general tendency:

where L is the orbital moment, S is spin, λ is the spin-orbit coupling constant

Correct Hamiltonian of Spin-orbit interaction

The effect of spin-orbit interaction describes the relativistic magnetic field HSO of a moving electrons and nothing else. As a consequence, the correct Hamiltonian is a product of the electron spin and the magnetic field of spin orbit interaction:

The spin-orbit interaction cannot break the time-inverse symmetry by itself. It requires an external magnetic to break the time-inverse symmetry. Only then the spin- orbit interaction manifests itself. As a result, the strength of the spin-orbit interaction is linearly proportional to the the total magnetic field Htotal (internal magnetic field + external magnetic field), which is applied to the electron orbital.

where kSO is the coefficient of spin-orbit interaction, which can be measured experimentally with a very high precision.

##### (note) In an atomic gas and the electron gas of conduction electrons, the time- inverse symmetry can be locally broken. As a result, magnetic field HSO of spin- orbit interaction can exist even in absence of an external magnetic field.

Substitution of Eq (11.4) into (11.3) gives the Hamiltonian of Spin- Orbit interaction as

where S is spin, kSO is the coefficient of spin-orbit interaction and Htotal is the total magnetic field (internal magnetic field + external magnetic field), which is applied to the electron orbital.

### HSO and external magnetic field

 Fine structure The fine spliting of the energy levels of a hydrogen atom. The fine structure correction predicts that the Lyman-alpha line (emitted in a transition from n=2 to n=1) must split into a doublet. Image taken from here. click on image to enlarge it

#### (fact) HSO is always proportional to an external external magnetic field

Two 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 spin-orbit interaction. Why?

### Fine structure. Heavy and light holes.

#### (possible contradiction) The spin-orbit interaction cannot break the time-inversion symmetry. As a result, the spin-orbit interaction does not manifest itself until the time-inverse 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 spin-orbit 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 non-zero, locally the time-inverse-symmetry is broken for each atom and each atom experiences a magnetic field HSO of the spin-orbit interaction. Since the directions of the orbital moments are equally distributed in all directions, the gas is non-magnetic and globally there is no breaking of the time-inverse 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 time-inverse 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 spin-orbit interaction is not involved. Let us consider a gas of atoms. Each atom has a magnetic moment and a non-zero 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 sufficiently-strong 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 time-inverse 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 time-inverse 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.

 Spin-orbit interaction makes properties of light and heavy holes different The the light and heavy holes experience a different magnitude of the spin-orbit interaction, due to the different symmetry of their the spacial distribution. The spin-orbit interaction causes the difference of properties between the light and heavy holes. click on image to enlarge it

It describes the fact that in the hydrogen gas the p- energy level is splits in two levels: the lower-energy level (J=1/2) and the higher-energy level (J=3/2).(More details see here)

(The reason of the energy splitting:) The orbital moment of a p- electron is non-zero. As a result, the p- electron experiences the additional spin-orbit magnetic field HSO and corresponding additional energy ESO=HSO*S, where S is the electron spin. HSO 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 HSO and have a different energies.

In a molecular or atomic gas, all values of the orbital moment, HSO and electron spin are non-zero. The orbital moment, HSO 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, HSO and spins are equally distributed in all direction. Still HSO 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 HSO on the orbital moment (orbital symmetry) causes the fine energy splitting.

Light and heavy holes

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 spin-orbit interaction, due to their different orbital symmetry (orbital moment). The spin-orbit interaction causes the difference of energies (properties) between the light and heavy holes.

The orbital moment of a hole is non-zero. As a result, the hole experiences the spin-orbit magnetic field HSO and corresponding additional energy ESO=HSO*S, where S is the electron spin. HSO 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 HSO and have a different energies.

## 3 types of magnetic field

Type 1: Conventional magnetic field. Global field Type 2: Magnetic field of Spin-orbit interaction Type 3: effective magnetic field of the exchange interaction
This is the conventional magnetic filed. This magnetic field "fills" all the space and all electrons experience equally this magnetic field. For this reason, it is called the global magnetic field

The magnetic field HSO of the spin- orbit interaction is the component of the electromagnetic field similar to the conventional magnetic field. However, each electron experiences an individual HSO of different magnitude and direction. The HSO of each electron does not influence the spin any neighbor electron. Even electrons, which rotate around the same nucleus, may have different HSO when their orbital symmetry is different.

In the exchange field the electron spin is aligned either parallel or anti parallel to the spin direction of its neighbor electrons. The exchange field is not a component of the electromagnetic field. However, in a solid the spin properties of the electron in a exchange field are exactly the same as in a conventional magnetic field. Therefore, the exchange field can be assigned as an effective magnetic field.
The HSO is originated from electric field of a nucleus and depends on the orbital symmetry of the electron. That is why the HSO is individual for each electron. Details about the spin-orbit interaction are here. Details about the exchange interaction are here
The spin properties of electrons are exactly the same for each type of the magnetic field. In an equilibrium the electron spin is aligned along the total magnetic field, which is a vector sum of all three types of the magnetic field. There is a spin precession before the alignment.
click on image to enlarge it

### Similarity between the spin- orbit interaction and the dynamo effect

#### Dynamo effect

In contrast, the production of fakes in a larger numbers is much easier

Click on image to enlarge it

#### Similarity: The result of both effects is absolutely identical

The result of the both effect is absolutely identical. Both effects, the spin- orbit interaction and the dynamo effect, amplify an externally applied magnetic field. When a weak magnetic field is applied, the dynamo effect or the spin-orbit interaction creates much stronger magnetic field (10 times stronger or more) parallel to the the external magnetic field.

#### Dynamo effect creates:

(result 1 of dynamo effect): Earth magnetic field ~0.5 Gauss

(result 2 of dynamo effect): Sun magnetic field ~1 Gauss

(result 3 of dynamo effect): magnetic field of a Sun dark spot ~4000 Gauss

(result 4 of dynamo effect): magnetic field of a neutron star millions of Gauss

## Coefficient of spin-orbit interaction kSO is a measure of strength of the spin- orbit interaction

Only parameter, which describes the strength of spin- orbit interaction, is the magnetic field Hso of spin- orbit interaction. The strength of Hso is proportional to the total applied magnetic field Htotal. The proportionality coefficient is called the coefficient of spin- orbit interaction and which gives the strength of the spin- orbit interaction.

The total applied magnetic field is the sum of intrinsic and external magnetic field.

#### Which parameter defines the strength of the spin- orbit interaction?

The coefficient of spin-orbit interaction kSO, which gives the strength of the spin- orbit interaction. kSO is the proportionality coefficient between the magnetic field Hso of spin- orbit interaction and the total applied magnetic field Htotal (internal plus external magnetic fields).

## Measurement of strength of spin-orbit interaction

Eq. 45. Anisotropy field Hani is linearly increases when an external magnetic field Hz is applied along the magnetic easy axis. The proportionality coefficient between Hani and Hz is the coefficient of the spin orbit interaction kSO.

(measurement method) The anisotropy field Hani is measured as a function of an external magnetic field Hz applied along the magnetic easy axis. The proportionality coefficient gives kSO
(math) The detailed math of how to obtain this equation is here

The measurement of the strength of the spin- orbit interaction is relatively simple and straightforward.

The magnetic field of spin-orbit interaction Hso holds the magnetization along the magnetic easy axis. A measurement of how strongly the magnetization is held along its easy axis evaluates the strength of the spin- orbit interaction.

For this purpose, an external magnetic field is applied along the hard axis and, therefore, perpendicular to the easy axis forcing the magnetization to tilt out from the easy axis. The stronger the spin- orbit interaction is, the harder it is to tilt the magnetization.

The anisotropy field is a magnetic field, at which the magnetization is fully tilted along the hard axis, and is a good measure of the spin-orbit interaction.

If additionally an external magnetic field is applied along the easy axis, the spin- orbit interaction becomes stronger and, correspondingly, the anisotropy field becomes larger. As you can see, the anisotropy field is linearly proportional to the strength of the spin-orbit interaction. Therefore, a measurement of anisotropy field versus the external field,gives the strength of the spin- orbit interaction.

## Measurement of strength of spin-orbit interaction

Two magnetic fields are independently controlled in this measurement. The in-plane magnetic field Hx is applied in-plane and, therefore, perpendicularly to the easy axis. Hz is applied along the easy axis.

(measurement method) For each fixed Hz, Hx is scanned and the magnetization tilting angle is measured. From this dependence, the value of Hani is evaluated for each Hz. From linear dependence Hani vs. Hz, kSO is evaluated

(key feature of the measurement): Two magnetic fields (along Hz and perpendicular Hx to the easy axis) are applied and independently controlled

(measurement method)

(step 1): For each fixed Hz, Hx is scanned and the magnetization tilting angle is measured.

(step 2): From the dependence of Hani vs. Hx, the value of Hani is evaluated for each Hz.

(step 3):From linear dependence Hani vs. Hz, kSO is evaluated

Fig.41. Examples of experimental measurements of strength spin-orbit interaction
Nanomagnet with a single ferromagnetic layer Nanomagnet with a single ferromagnetic layer Slope is small and negative and, therefore, kSO is negative and small. Slope is large and positive and, therefore, kSO is positive and large. There are only 2 interfaces: FeB/Ta & FeB/MgO. The positive contribution to SO strength of interface is not sufficient to overcome the negative contribution from the bulk. As a result, kSO is negative and small There are a total of 10 interfaces: FeB/W, FeB/Ta, and FeB/MgO. Each of these interfaces contributes to the enhancement of the spin-orbit interaction strength. Consequently, the value of kSO becomes significantly large and positive.
(interface & bulk contributions): The strongest and positive contribution to the strength of spin- orbit interaction is from interfaces. The bulk contribution is small and negative. As a consequence, kSO is small in a single- ferromagnetic- layer nanomagnet, which has only two interfaces and in which the interface contribution is comparable to the bulk contribution, and is large in a multi- layer nanomagnet, which has many interfaces and in which the interface contribution dominates.
(oscillations): There are clear oscillations on top of the linear dependence. The oscillation is due to a periodic dependence of spin- orbit interaction on the magnetic field. The oscillation is a feature of an interface and is stronger when interface contribution is larger and the bulk contribution is weaker (See below)
(two measured parameters): A linear fitting of this dependence gives two parameters: (parameter 1) slope of the line, which gives coefficient of spin- orbit interaction kSO. (parameter 2): offset of the line, which gives the anisotropy field in absence of an external field or simply the anisotropy field Hani.
click on image to enlarge it

## Relation between spin-orbit interaction and anisotropy field

When some conditions at a nanomagnet interface are changing, it affects both the strength of spin orbit interaction kSO and demagnetization field Hdemag. Since the anisotropy field Hani depends on both kSO and Hdemag, the anisotropy field changes as well. However, the polarity change of Hani may be different according to whether the kSO or Hdemag contribution to anisotropy field is dominated for the specific nanomagnet.

When polarity of change of Hani is the same as that of kSO, the spin- orbit contribution dominates.
When polarity of change of Hani is opposite to that of kSO, the demagnetization contribution dominates.
click on image to enlarge it

There are two parameters, which are evaluated from the measured linear dependence of Fig.41: (parameter 1): coefficient kSO of spin orbit interaction (the slope of the line) and (parameter 2): anisotropy field in absence of external field or simply the anisotropy field Hani ( the offset of the line)

Both kSO and Hani depend on a variety of nanomagnet parameters such as interface roughness, nanomagnet thickness, current, gate voltage etc.

It is important, the dependencies of kSO and Hani are not independent.

(reason):

The anisotropy field is proportional to the strength of the spin- orbit interaction and the internal magnetic field Hint:

The internal magnetic field is the field along the magnetization M minus the demagnetization field Hdemag and plus the magnetic field of spin- orbit interaction HSO:

Therefore, the anisotropy field is proportional to the coefficient of spin-orbit interaction and the demagnetization field:

(note) Usually kSO and Hdemag increase or decrease simultaneously. As a consequence, the anisotropy field Hani can either increase or decrease depending which contribution kSO or Hdemag to Hani is larger

Factors, which affect kSO and Hdemag:

(factor 1): interface roughness (see below)

(rougher interface) → (kSO is smaller) & (Hdemag is smaller) ⇒ (Hani is either smaller or larger)

The dependence of on the interface roughness is different for a nanomagnet containing one or several ferromagnetic layers.

a single-layer nanomagnet: (rougher interface) →(Hani is smaller)

a multi-layer nanomagnet: (rougher interface) →(Hani is larger)

(factor 2): nanomagnet thickness (see below)

##### when kSO contribution is positive for interface and negative for the bulk

(thicker nanomagnet) → (kSO is smaller) & (Hdemag is the same) ⇒ (Hani is smaller )

(factor 3): polarity of applied magnetic field (see below)

##### only in the case of asymmetrical nanomagnet

(larger change with reversal of H) → ( ΔkSO is larger) & (ΔHdemag is the same) ⇒ (ΔHani is larger )

(factor 4): Electrical current. SOT effect (see below)

(a larger current) → (kSO is ) & (Hdemag is ) ⇒ (Hani is )

(factor 5): Gate voltage. VCMA effect (see below)

(a larger positive gate voltage) → (kSO is larger) & (Hdemag is smaller) ⇒ (Hani is smaller)

## Measurement of internal magnetic field

Extending measured linear dependence of anisotropy Hani vs. external magnetic field Hz till value Hani=0, gives the value of the internal magnetic field Hint

The case Hani=0 corresponds to the case when there is no magnetic anisotropy and, therefore, when the anisotropy field equals zero and when there is no internal field.
click on image to enlarge it

The magnetic field, which holds the magnetization along its easy axis, is called the internal magnetic field.

(fact) The measured internal magnetic field in a FeCoB nanomagnet is in range between 2.5-5 kG. However it can be as large as 15 kG in a single- layer FeCoB nanomagnet and can be less than 1 kG in a multilayer nanomagnet.

(how to measure the internal magnetic field)

##### Extending measured linear dependence of anisotropy Hani vs. external magnetic field Hz till value Hani =0, gives the value of the internal magnetic field Hint

The anisotropy filed depends linearly on the external magnetic field Hz, which is applied along easy axis

In absence of external magnetic field Hz, only the internal magnetic field Hint holds the magnetization along the magnetic easy axis and the anisotropy field is linearly proportional to the internal magnetic field Hint. Since both the external and internal magnetic field are just the magnetic field and, therefore, should have the similar force on the spin, Eq. (41.1) can be written as

Similar to the external magnetic field Hz, the internal magnetic field Hint also has the bulk contribution, which should be considered. Then, correct form of Eq. (41.2) would be

Comparison of Eqs (41.1) and (41.2) gives the internal magnetic field Hint as

In the case when there is no external magnetic field Hz =0 and no internal magnetic field Hint=0, the anisotropy field eqauls zero (See Eq. 41.2a) and there is no magnetic anisotropy

(fact) In vacuum, there is no magnetic anisotropy. Therefore, the anisotropy field equals zero and there is no internal field.

Measured internal magnetic field in FeCoB nanomagnets Samples. Thickness in nm shown in blankets
Fig 44 The internal magnetic field vs. coefficient kSO of spin- orbit interaction systematically measured in FeCoB nanomagnets of different thickness, size, structure, composition. Dots of the same color and shape correspond to nanomagnet fabricated at different places of the same wafer. Stars show multilayer nanomagnets, which contain several ferromagnetic layers.
The internal magnetic field is stronger for single- layer nanomagnets and weaker for multi-layer nanomagnets (shown by stars). The nanomagnet of a different width and length in range from 50 x 50 nm to 3000 nm x 3000 nm are shown.

#### Three contributions to the internal magnetic field Hint:

(contribution 1) : Magnetic field along magnetization M

(contribution 2) : Demagnetization field

(contribution 3) : Magnetic field HSO of the spin-orbit interaction

(fact) The electrons in the bulk of a nanomagnet and at the interface experience a very different internal magnetic field Hint, because they experience a very different magnetic field HSO of the spin-orbit interaction

#### Factors, which affects the strength the internal magnetic field Hint:

(factor 1) : Magnetization M

A nanomagnet, which has a larger magnetization, often has a larger Hint, because of larger contribution 1. However, other factors can reverse this tendency.

(factor 2) : Magnetic anisotropy. Anisotropy field Hani

A harder nanomagnet has a larger Hint. The anisotropy field Hani is linearly proportional to the internal magnetic field Hint

(factor 3) : Roughness and sharpness of an interface

Both the demagnetization field and the magnetic field of Spin-orbit interaction substantially depend on the perfection of the interface.

(factor 4) : Thickness of a nanomagnet

The strength of spin- orbit interaction is substantially different for the electrons in the bulk of a nanomagnet and at the interface. The bulk contribution is stronger for a thicker nanomagnet and the interface contribution is stronger for a thinner nanomagnet

(factor 5) : Structure of a nanomagnet

The demagnetization field is substantially in a multilayer nanomagnet than in a single layer nanomagnet. As a consequence, the internal magnetic field in a multilayer nanomagnet is smaller than in a single layer nanomagnet. This is because the effect of interfacial imperfections, which reduces the demagnetization field, is less prominent in a multilayer nanomagnet.

Why is the internal magnetic field in a multilayer nanomagnet substantially smaller than in a multi- layer nanomagnet?

It is because of a larger demagnetization field. The larger the number of interfaces is, the more efficiently the demagnetization field is created. In an ideal non- existed case of 100% efficiency of creation of the demagnetization (e.g. ideally- smooth ideally- planar ideally- sharp interface), the demagnetization field exactly equals the magnetization field and, as a consequence, the internal magnetic field becomes zero.

## free-space contribution to Hani

(left) In vacuum, the spin is always aligned precisely along the external magnetic field. In a nanomagnet, the magnetic field HSO of spin- orbit interaction tilts the spin out of the direction of the external magnetic field Hext
(center) When the coefficient kSO of spin-orbit interaction is positive, the spin is tilted towards the surface normal. (right) When the coefficient kSO of spin-orbit interaction is positive, the spin is tilted towards the in-plane direction.
click on image to enlarge it

The origin of the free-space contribution to Hani is the simple and obvious fact that in vacuum the spins always align perfectly along the magnetic field.

(fact): Because of the free-space contribution, the dependence of Hani -Hz vs. Hz is more informative than the dependence of Hani vs. Hz

In vacuum, the spin is always aligned exactly along the external magnetic field.

In a nanomagnet, the external field also creates the SO magnetic field.

When this field is directed along an external field, kSO is positive.

When SO field is directed opposite to external field, kSO is negative

# Excluding free-space contribution

### with free- space contribution

sample 1 sample 2 wafer VOLT 58A Ta(5 nm)/ (FeCoB 1 nm, x=0.3) (see here) nanomagnet L43C SOT wafer Volt46B W(3):[FeB(0.68)/W(0.25)]x4/ FeB(0.68) nanomagnet free18.
The presence of non-informative free-space contribution masks crucial properties embedded within this measured dependence. These properties include the slope of the line, which directly relates to the strength of the spin-orbit interaction, as well as the oscillations and disparities observed during magnetization reversal.

### without free space contribution

sample 1 sample 2 wafer VOLT 58A Ta(5 nm)/ (FeCoB 1 nm, x=0.3) (see here) nanomagnet L43C SOT wafer Volt46B W(3):[FeB(0.68)/W(0.25)]x4/ FeB(0.68) nanomagnet free18.
Upon eliminating the non-informative free-space contribution, the polarity and strength of the spin-orbit interaction (reflected by line slopes), the amplitude and period of oscillation, and the distinctions arising from magnetization reversal become distinctly visible and unequivocally clear.
click on image to enlarge it

### Oscillations of the strength of spin- orbit interaction under an external magnetic field

The strength of the spin-orbit interaction exhibits oscillations as the external magnetic field increases. The oscillation is due to a periodic dependence of spin- orbit interaction on the magnetic field.

The oscillations of the strength of spin- orbit interaction are very clear in experimental measurements and are observed for any nanomagnet, which I have measured.

Oscillations of the strength of spin- orbit interaction
Nanomagnet with a single ferromagnetic layer Nanomagnet with a single ferromagnetic layer Slope is small and negative and, therefore, kSO is negative and small. Measured data (blue line) are oscillating on top of linear dependence. (red line) Slope is large and positive and, therefore, kSO is positive and large.
(oscillations): There are clear oscillations on top of the linear dependence. The oscillation is due to a periodic dependence of spin- orbit interaction on the magnetic field. The oscillation is a feature of an interface and is stronger when interface contribution is larger and the bulk contribution is weaker
click on image to enlarge it

(How to measure?)

Measured dependence of anisotropy field Hani vs. external magnetic field Hz has two components: (component 1): linear increase; (component 2) oscillating component

Origin of Oscillations of the strength of spin- orbit interaction under an external magnetic field
The oscillations are very clear in experimental measurements and are observed for any nanomagnet, which I have measured.
(close guess): The strength of spin- orbit interactions increases because orbital deformation under the Lorentz force in the magnetic field is different for parts of orbitals corresponding to the electron rotation in the clockwise and counterclockwise directions. The deformations are influenced by the other orbitals. The influence by each orbitals might be optimum at some value of the magnetic field making the maximum of the strength of the spin- orbit interaction. Each orbital has a different maximum. As a result, the dependence looks periodical.
As 2023.06, there are no enough experimental measurements to clarify the Origin of observed Oscillations

Two components of dependence Hani vs. Hz

(component 1 of Hani vs Hz) (general (main) dependency): linear dependency

The strength of spin-orbit interaction is linearly proportional to the external magnetic field Hz. As a result, the anisotropy field Hani can be expressed as

where Hz are external magnetic field, Hani (Hz =0) is the anisotropy field in absence of the external magnetic field and kSO is the coefficient of spin- orbit interaction.

(component 2 of Hani vs Hz) (minor dependency): oscillating dependency

The strength of the spin-orbit interaction exhibits oscillations as the external magnetic field increases. These oscillations serve as a minor contribution to the primary linear dependence of Hani vs. Hz, which slope is determined by the strength of the spin-orbit interactions.

where Aosc is amplitude of oscillation, Hperiod is the period of oscillations (the magnetic field, after which the oscillations repeats itself), Hphase is the phase of oscillations, Hdecay is field for the decay of oscillations.

#### In total, the measured dependence Hani vs. Hz is fitted by formula:

(fact) This formula give a a perfect fitting of measured data for all nanomagnets I have measured so far.

The observed oscillations correspond to variations in the strength of the spin-orbit interaction as the external magnetic field increases.

There are clear oscillations on top of the linear dependence. The oscillation is due to a periodic dependence of spin- orbit interaction on the magnetic field. The oscillation is a feature of an interface and is stronger when interface contribution is larger and the bulk contribution is weaker

(interfacial origin of oscillations): The oscillations are weaker for a thicker nanomagnet, where the bulk contribution dominates, and stronger for a thinner nanomagnet, in which the interfacial contribution dominates. It clearly indicates the interfacial origin of the oscillations.

(linear proportionality of oscillation amplitude to the strength of spin- orbit interaction): The measured amplitude Aosc of oscillation is linear proportional to the strength of the spin-orbit interaction or , the same, to the coefficient of SO (kso). It is true for nanomagnets fabricated on a single wafer (See Fig.19) and for the nanomagnets of a different size, material, composition, thickness, structure fabricated on different wafer (See Figs. 22b,22c below). The slope of dependence Aosc vs. kSO is positive. Since the positive contribution to kSO is from the interface, the positive means again the interfacial origin of the oscillations.

Distribution of oscillation amplitude vs. strength of spin orbit interaction Raw data
Fig. 19 Measured amplitude of oscillation of the strength of spin- orbit interaction vs. measured coefficient kso of spin-orbit interaction. Each dot corresponds to a measurement of one individual nanomagnet fabricated at a different place of one wafer. The variations of amplitude and kso are due to variations of roughness and thickness over the wafer. Arrow shows the tendency. Fig.19a. Measured anisotropy field Hani vs. external magnetic field Hz for two nanomagnets (R12C & L43C) fabricated on the same wafer Volt57B. Dots are measured data, lines are fitting. The fitting is perfect in both cases. Nanomagnet R12C has a stronger spin-orbit (kso= + 0.06) and a larger oscillation amplitude of 0.75 kG. Nanomagnet L43C has a weaker spin-orbit (kso= - 0.17)(slope is negative) and a smaller oscillation amplitude of 0.46 kG.

### There is a clear correspondence between slope and oscillation amplitude!!!

wafer Volt57B :Ta(5 nm)/ FeCoB( x=0.5 1.1 nm) (click here fore more details) click on image to enlarge it

(period of oscillation): The period Hperiod of oscillation is nearly identical for all nanomagnets and equals approximately 8 kGauss (See Fig. 21c below).

##### (note) When amplitude of oscillation becomes, there is some ambiguities of oscillation fitting. This is the reason why some nanomagnets show the period smaller than 8 kGauss

(phase of oscillations): The non-zero measured phase Hphase of oscillation means that the 1st maximum of oscillation is not at Hz=0, but is slightly shifted. For all measured nanomagnets, the phase Hphase is in range 0.1-0.2 kGauss.

##### (note) As 2023.06, I have not analyzed systematically the measured data of Hphase or its dependence or correlation with other measured parameters. The data of measured Hphase can be found in this origin file: AllSampleHani.opj

(decay of oscillations): The amplitude of oscillation decreases when the external magnetic field increases. Typically, at a half period of oscillation the oscillation amplitude decreases for about 40%. Some nanomagnets do not show any decrease of the oscillation amplitude and some nanomagnets show even an increase of the oscillation amplitude. The decrease or increase of the oscillation amplitude substantially depends on size, material, composition, thickness, and structure of the nanomagnet.

##### (note) As 2023.06, I have not analyzed systematically the measured data of Hdecay or its dependence or correlation with other measured parameters. The data of measured oscillation amplitude at Hz=0 and at Hz= Hperiod/2 can be found in this origin file: AllSampleHani.opj

The data of measured Hani, kSO , Hperiod, , Hphase, Hdecay for all nanomagnets, which I have measured so far, can be found in this origin file: AllSampleHani.opj

(systematic error due to oscillations): The amplitude of oscillation in a nanomagnet with a strong spin- orbit interaction (kso>0.2) becomes very large of about 1 kGauss and larger. It creates ambiguity of the fitting of the experimental data. The oscillating contribution may become dominating and there may be an ambiguity to distinguish the linear contribution. This can create some systematic error in measurement of kSO and Hani. See, for example, Fig. 19a above.

### (fact) The oscillations of the strength of the spin- orbit interaction are large!!!

Especially, the oscillations are large in nanomagnets with a strong interfacial anisotropy and with a large spin-orbit interaction

Systematic measurements in FeCoB nanomagnets

#### period of oscillations vs. kso

Samples. Thickness in nm shown in blankets

Fig.22b The amplitude of oscillations linearly depends on the strength of the spin- orbit interaction for both the single- layer and multi-layer nanomagnets. The linear dependence indicates the interfacial origin of the oscillations.   Fig.22c It is same figure as the left figure (Fig.22b) but in a different scale. For all single- layer nanomagnets, the dependence is linear with the same positive slope.   Fig.21c. The period is about 8 kGauss   The nanomagnet of a different width and length in range from 50 x 50 nm to 3000 nm x 3000 nm are shown.
Dots of the same color and shape correspond to nanomagnet fabricated at different places of the same wafer. Stars show multilayer nanomagnets, which contain several ferromagnetic layers.
click on image to enlarge it

### Strength of spin- orbit interaction vs. nanomagnet thickness.

There are small variations of thickness and interface roughness from a point to a point at the same wafer. Parameters of nanomagnets, which are fabricated at different places( e.g. at center or at edge of wafer or just at two close neighbor points), vary due to a variation of the interface roughness and due to a variation of the nanomagnet thickness.

The strength of spin orbit interaction and the demagnetization field are two parameters, which are directly affected by variations of thickness and roughness. Because of the variations of these two parameters, the anisotropy field Hani and the intrinsic magnetic field are affected by variations of thickness and roughness as well.

#### (effect of interface roughness):

(kSO & Hdemag): A rougher interface reduces both the strength of the spin- orbit interaction kSO and the demagnetization field Hdemag.

(anisotropy field Hani): A rougher interface may either decrease or increase the anisotropy field.

When the contribution of kSO to Hani dominates, a rougher interface causes a decrease of Hani. Hani is flowing the change of kSO. E.g. it is a case of a single- layer nanomagnet.

When the contribution of Hdemag to Hani dominates, a rougher interface causes an increase of Hani. Hani is flowing the change of Hdemag. E.g. it is a case of a multi- layer nanomagnet.

(intrinsic magnetic field Hint): Hint is affected strongly by the roughness. The internal magnetic field Hint is just a difference between the magnetic field along magnetization and the demagnetization field. For an ideally- flat surface (a non-existed subject), the demagnetization field exactly equal to the magnetization and, therefore, the becomes zero. A rougher interface reduces the demagnetization field and, therefore, increases the internal magnetic filed.

#### (effect of nanomagnet thickness):

(SO strength kSO): There are different contributions to the total strength of SO from the bulk and the interface. As a nanomagnet become thinner .

(anisotropy field Hani): A rougher interface may either decrease or increase the anisotropy field.

(intrinsic magnetic field Hint): Hint is not affected strongly by the nanomagnet thickness when the interface roughness and other parameters remains the same.

(Why there is a thickness variation): In MRAM applications, the FeCoB nanomagnet typically has a thickness of 1 nm. Considering that the interatomic distance in FeCoB is approximately 0.1 nm, this implies that there are merely around 10-15 atomic layers spanning the thickness. Consequently, the absence of even a single atom can lead to a significant variation of 5-10% in the thickness of the nanomagnet.

## Distribution of strength of spin- orbit interaction kSO and anisotropy field Hani due to variations of thickness and roughness is not random, but linear.

### fact for nanomagnets fabricated at different places of one wafer

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Two independent parameters, which are affect by variations of the nanomagnet thickness and the interface roughness: strength of spin-orbit interaction and demagnetization field.

(fact for nanomagnets fabricated at different places of one wafer) Distribution of strength of spin- orbit interaction kSO, anisotropy field Hani, intrinsic magnetic field and demagnetization field due to variations of thickness and roughness is not random, but linear.

(reason why:)Tendencies of how the strength of spin- orbit interaction and the demagnetization field are similar. A rougher interface makes the spin- orbit interaction weaker and the demagnetization field smaller. In a thicker nanomagnet, the bulk contribution is larger and, as consequence, the average spin- orbit interaction becomes smaller. The demagnetization field is not affected by a variation of the nanomagnet thickness.

(fact about distribution of Hani and kSO due to roughness/ thickness variation) The slope of distribution of Hani vs. kSO can be either positive (more common) or negative (less common). The slope is negative when the dogmatization field is larger and when the contribution due to the variation of the demagnetization field is dominated.

(slope polarity for distribution of Hint vs. kSO) The slope is positive for single- layer nanomagnets which have a small coefficient kSO of spin- orbit interaction and large internal magnetic field Hint. The slope is negative for multi- layer nanomagnets , which have a large kSO and small Hint.

(slope polarity for distribution of Hint vs. kSO) The slope is always negative for distribution of the internal magnetic field Hint vs. kSO for both single- layer and multi- layer nanomagnets. (See Fig.44 above). It means that a smother interface always results in a larger coefficient kSO of spin- orbit interaction and a smaller the internal magnetic field Hint. The Hint is smaller because the demagnetization field becomes larger for a smoother interface.

Measured anisotropy field vs. measured strength of spin-orbit interaction. Systematic study.
##### Comparison of data of many different nanomagnets of a different size, material, composition, thickness, structure.
Full scale Small scale Samples. Thickness in nm shown in blankets
Measured anisotropy field vs. measured coefficient of spin-orbit interaction for nanomagnets of a different size, composition, structure. Arrows show the tendency for single-layer nanomagnets (blue arrow) and multi-layer- nanomagnets (brown arrow) It is the same graph as the left one, but in a different scale Dots of the same color and shape correspond to nanomagnet fabricated at different places of the same wafer. Stars show multilayer nanomagnets, which contain several ferromagnetic layers.
The slope is positive for single- layer nanomagnets having a small kSO. The slope is negative for multi- layer nanomagnets (shown by stars) having a large kSO.   The nanomagnet of a different width and length in range from 50 x 50 nm to 3000 nm x 3000 nm are shown.
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Why is the distribution of Hani and kSO measured for nanomagnets fabricated on one wafer important?

It is because in one wafer there are not so many variations of parameters. There is only a weak variation of interface roughness and a weak variation of the nanomagnet thickness. Other parameters remain the same. Therefore, it is easier to trace the effect of the roughness and thickness variations on parameters of a nanomagnet.

### Different slope polarities in distribution of Hani vs. kSO in a single-layer and a multi-layer nanomagnet.

##### .
slope of Hani vs kSO. Single-layer vs. multi-layer nanomagnets.
single- layer nanomagnet vs. multi- layer nanomagnet multi-layer nanomagnets with different number of layers

Anisotropy field Hani vs. coefficient kSO of SO interaction. Comparison of a single- ferromagnetic -layer and a multilayer- ferromagnetic -layer nanomagnet.

Anisotropy field Hani vs. coefficient kSO of SO interaction for a multilayer- ferromagnetic -layer nanomagnet. Dependence on number of layers.
The slope is positive for a single- ferromagnetic- layer nanomagnet. The slope is negative for a multi- ferromagnetic- layer nanomagnet. When number of layer is small (3 layer) the slope is still positive, but it becomes negative and its absolute value increases, when the number of ferromagnetic layers increases. However, when the number of layers reaches some critical value (6 layers in this case), the absolute value of the slope starts decreasing.
single-layer nanomagnets: 53B Ta(2.5)FeBCo0.3(1); 55 Ta(5)FeB(0.9); 57ATa(5)FeBCo0.5(1.1); multi-layer nanomagnets: 45B W(3):[FeB(0.55) W(0.5)]5 FeB(0.55); 48A W(3): [FeB(0.45)/W(0.2)]7 FeB(0.45) (3 layers) 44B Ta(2):FeB(0.8):W(0.5):FeB(0.8):W(0.8):FeB(0.8); (5 layers) 46B W(3):[FeB(0.68)/W(0.25)]x4/ FeB(0.68); (6 layers) 45B W(3):[FeB(0.55) W(0.5)]5 FeB(0.55); (8 layers) 48A W(3): [FeB(0.45)/W(0.2)]7 FeB(0.45)
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( vs. ) Why is the slope of dependence Hani vs. kSO positive for a single- ferromagnetic- layer nanomagnet, but negative for a a multi- ferromagnetic- layer nanomagnet?

Both the demagnetization field Hdemag and the magnetic field HSO of spin- orbit interaction are originated at an interface and, therefore, become larger when the number of interfaces increases. In a multilayer nanomagnet, the number of interfaces is larger. For this reason, both kSO and Hdemag are larger in a multilayer nanomagnet

The internal magnetic field Hint is smaller when Hdemag is larger. For a smoother interface, kSO becomes larger but Hint becomes smaller. Therefore, a positive change ΔkSO corresponds to a negative change ΔHint.

Hint is small and kSO is large in a multilayer nanomagnet. As a result, ΔHint contribution to ΔHani is large, ΔkSO contribution is small and the slope is negative (See Eq. 42.3)

Hint is large and kSO is small in a single- layer nanomagnet. As a result, ΔHint contribution to ΔHani is small, ΔkSO contribution is large and the slope is positive (See Eq. 42.3)

( vs. vs. ) Why is the absolute value of slope of dependence Hani vs. kSO increases at first, when number of layers increases, but starts to decreases when the number of layer exceeds some critical number?

When the number of ferromagnetic layers increases, the interface roughness often increases as well. As a consequence, an increase of kSO and Hdemag due to the increase of the number of interface is overturned by the decrease of kSO and Hdemag due to the increase of the interface roughness.

Effect of increase of nanomagnet thickness or interface roughness on strength of spin-orbit interaction (kSO), demagnetization field (Hdemag) , anisotropy field (Hani) and internal magnetic field (Hint)
increase of interface roughness increase of nanomagnet thickness A rougher surface makes the strength of spin- orbit interaction weaker and the demagnetization field smaller. The internal magnetic field is a difference between magnetic field M along magnetization and demagnetization field. As the demagnetization decreases and, therefore, becomes more different from M, the internal magnetic field increases. The anisotropy field is linearly proportional to kSO and Hint. Therefore, Hani decreases due to the decrease of kSO, but increases due to the increase of Hint. In a thicker nanomagnet, the negative bulk contribution to the strength of spin- orbit interaction becomes larger and the positive interface contribution becomes smaller. As a result, kSO decreases. The demagnetization field originates at the interface and is affected by the nanomagnet thickness. The anisotropy field is linearly proportional to kSO and , therefore, decreases with the decrease of kso.
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## Perfection of fabrication technology

 (good fabrication technology: wafer 54A): all data points cluster tightly within a small circle with a small radius. (bad fabrication technology: wafer 55): the measured data points are noticeably sparse, dispersed over a larger area.

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In any technological application, it is crucial to ensure uniformity among the parameters of all devices fabricated on a single wafer. This holds particularly true for nanomagnets employed in memory or sensor applications, where consistency in magnetic parameters is of utmost importance. Among these parameters, the anisotropy field and the strength of the spin-orbit interaction stand out as key factors. By measuring the distribution of the anisotropy field against the coefficient of the spin-orbit interaction, one can obtain a clear indication of the quality and precision of the underlying technology being utilized.

The measured distribution of the anisotropy field (Hani) against the coefficient (kSO) of the spin-orbit interaction serves as an indicator of the fabrication technology's quality and precision:

(perfect fabrication technology): all data points cluster tightly within a small circle with a minimal radius.

(moderate fabrication technology): although there may be some slight scattering of data points, they still align along a straight line.

(bad fabrication technology): the measured data points are noticeably sparse, dispersed over a larger area.

## Evaluation of a nanomagnet fabrication technology.

A measured distribution of Hani vs. kso is a good evaluation of the fabrication technology. The case, when all data points cluster tightly within a small circle with a minimal radius, is a clear indication of perfection of the fabrication technology.

How can I determine whether the variation in roughness or the variation in thickness is responsible for the changes in magnetic properties observed in nanomagnets fabricated on a single wafer?

The demagnetization field does not depend on the nanomagnet thickness. As a consequence, a variation of thickness does not affect the internal magnetic field Hint, but does affect the strength of spin-orbit interaction. The distribution Hint vs. kSO along a straight horizontal line is a good indication that the variation of thickness is minimal.

It should be noted that the slope of distribution Hint vs. kSO is increases for a smaller Hint and larger kSO (See Fig 44 above)

Is the distribution of the coercive field Hc for nanomagnets fabricated on the same wafer related to the distribution Hani vs. kSO?

Yes. The coercive field Hc is related to Hani and kSO. A sparse distribution of Hani vs. kSO usually corresponds to a sparse distribution of Hc. However, there is one parameter, which may make the Hc distribution even more sparse. It is the size of the nucleation domain (see here).

The size of the nucleation domain depends very much on the smoothness, perfections and sharpness of the nanomagnet boundary, which may be very different from a nanomagnet to nanomagnet. As a consequence, a variation of size of the nucleation domain may be substantial even in the case of a reasonably good nanofabrication technology.

## asymmetrical nanomagnet

In a symmetrical nanomagnet, the strength of spin- orbit interaction and the anisotropy field are the same for opposite magnetization directions.   In a asymmetrical nanomagnet, the strength of spin- orbit interaction and the anisotropy field are deferent for opposite magnetization directions.

magnetization up = magnetization down

magnetization up magnetization down
magnetization up: Ta/FeB+ FeB/Ta   magnetization up: Ta/FeB+ FeB/MgO
magnetization down: FeB/Ta + Ta/FeB   magnetization down: FeB/Ta + MgO/FeB

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There are two stable opposite magnetization directions along the nanomagnet easy axis. Both the strength of the spin- orbit interaction and the anisotropy field are different for those opposite magnetization directions.

The reason for the observed disparity is the correlation between the strength of the spin-orbit interaction and the orientation of the magnetic field relative to the orbital deformation taking place at the interface.

(demagnetization field vs. spin- orbit interaction): The demagnetization field remains unaffected by the magnetization polarity as it is solely a geometric phenomenon independent of polarity. On the other hand, the strength of the spin-orbit interaction does depend on magnetization polarity due to its reliance on the polarity of the orbital deformation with respect to the interface. Specifically, it is influenced by the direction of the orbital center's shift in relation to the nuclear position. The strength of the spin-orbit interaction varies depending on whether the magnetic field is applied in the same direction as the shift or in the opposite direction.

(how to measure) There is a substantial difference in measured dependence of anisotropy field vs. an external magnetic field Hz, which is applied along the easy axis, for two opposite directions of Hz. Both the offset of the dependence (Hani) and the slope (kSO) are different for opposite directions of the magnetic field.

Measured dependences of Hani and kSO on polarity of magnetic field & magnetization
 There is a difference of offset (Hani) and slope (kSO) of the lines between measurements when the magnetization direction is up (shown in blue) and the magnetization is down (shown in red) The same graph as shown in the left, but versus absolute value of Hz. It allows to compare the difference for opposite direction of the magnetic field Measurement of 0 deg corresponds to a scan of in-plane magnetic field along current. Measurement of 90 deg corresponds to a scan of in-plane magnetic field perpendicularly to the electrical current. Independent measurements give the same difference of the offset (Hani) and slope (kSO)
Anisotropy field Hani vs. external magnetic field. Dots are measured data. Lines are fit.
wafer VOLT 58A Ta(5 nm)/ (FeCoB 1 nm, x=0.3) (see here) nanomagnet L43C SOT
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(fact): The variation in the strength of the spin-orbit interaction in relation to magnetization reversal is a characteristic commonly found in nearly all interfaces.

What is the reason behind the lack of variation in the strength of the spin-orbit interaction during magnetization reversal for a symmetrical nanomagnet?

For a symmetrical nanomagnet, the absence of variation in the strength of the spin-orbit interaction with respect to magnetization reversal can be attributed to its balanced and uniform structure. There are two equal but opposite interfaces in a symmetrical nanomagnet. For example, in Ta/FeB/Ta nanomagnet there are two opposite interfaces: Ta/FeB and FeB\Ta. A variation in the strength of spin- orbit is the same, but opposite at each interface. In total, there is no variation for the symmetrical nanomagnet.

When the magnetization direction is up, the magnetic field penetrates from Ta to Fe at the lower interface and from Fe to Ta at the upper interface. When the magnetic field is reversed, there is still one interface (the upper one), where the magnetic field penetrates from Ta to Fe, there is one interface (the lower one), where the magnetic field penetrates from Fe to Ta. Even though the strength of the spin-orbit interaction is different between cases when the magnetic field penetrates from Ta to Fe and from Fe to Ta, there is no difference for the whole nanomagnet.

Symmetrical vs. Asymmetrical nanomagnets
Symmetrical Symmetrical Asymmetrical
wafer VOLT 58A Ta(5 nm)/ (FeCoB 1 nm, x=0.3) (see here) nanomagnet L43C SOT wafer Volt 59A Ta(8 nm)/ FeB(0.9 nm) (see here) nanomagnet L75C SOT wafer Volt 46B W(3):[FeB(0.68)/W(0.25)]x4/ FeB(0.68) nanomagnet free18.
Anisotropy field Hani vs. external magnetic field. Dots are measured data. Lines are fit.
wafer VOLT 58A Ta(5 nm)/ (FeCoB 1 nm, x=0.3) (see here) nanomagnet L43C SOT
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#### Origin for dependence of the strength of spin-orbit interaction on the polarity of magnetization

Why does the strength of the spin- orbit interaction depend on the magnetization polarity?

(reason why the strength of the spin-orbit interaction increases under an external magnetic field):

The increase in the strength of the spin-orbit interaction with an increase in the external magnetic field can be attributed to the difference of orbital modifications under the influence of the Lorentz force.

The part of the orbital that rotates clockwise around the magnetic field contracts due to the Lorentz force, bringing it closer to the nucleus, where the electric field is stronger. Consequently, this portion of the orbital experiences a greater spin-orbit magnetic field.

Conversely, the counterclockwise rotating portion of the orbital expands and moves away from the nucleus, resulting in a smaller spin-orbit magnetic field.

Since the electric field diminishes with increasing distance from the nucleus following a 1/r decay, the gain from the clockwise rotating component surpasses the loss from the counterclockwise rotating component. As a result, the electron experiences an overall amplified magnetic field due to the spin-orbit interaction.

(reason why the the strength of the spin-orbit interaction depends on the polarity of an external magnetic field):

When the magnetic field is reversed, the clockwise component expands and the counterclockwise component contracts. However, due to the asymmetric nature of the orbital at the interface, the gain and loss of the spin-orbit interaction differ from the previous case. Consequently, the total gain in the magnetic field from the spin-orbit interaction varies for opposite polarities of the external magnetic field.

(orbital quenching) Given that the orbital moment of localized electrons is completely quenched (see here) , resulting in a net orbital moment of zero, it raises the question of why an asymmetry exists between the clockwise and counterclockwise components of the orbital. Furthermore, what causes the interaction between the clockwise component and the up-magnetic field to differ from the interaction between the counterclockwise component and the down-magnetic field?

Indeed, you are correct. The orbital moment of localized electrons is effectively quenched, resulting in no net orbital moment. This is achieved by the compensating contributions from the orbital components associated with electrons rotating in clockwise and counterclockwise directions. However, it is important to note that this compensation occurs for the overall orbital as a whole.

On a local scale, there can still be subtle differences. For instance, the clockwise component may be slightly shifted to the left, while the counterclockwise component may be shifted to the right due to variations in the surrounding atoms on the left and right sides of the orbital. As a result, there can be localized differences in the interaction of the clockwise and counterclockwise rotating components with an external magnetic field.

(bulk vs. interface) What is the underlying reason for the significant dependence of the strength of the spin-orbit interaction on the magnetization polarity specifically at the interface, while such dependence is absent in the bulk of a ferromagnet?

For the spin-orbit interaction to exhibit a dependence on magnetization polarity, a distinct spatial symmetry of the electron orbital must be disrupted. Typically, this symmetry is broken at the interface but remains intact in the bulk of a ferromagnetic material. While local symmetry breaking can occur, on average, the symmetry is maintained within the bulk.

The bulk of the ferromagnetic material resembles a multilayer nanomagnet, where each interface breaks the symmetry, but subsequent interfaces exhibit equal and opposite symmetry breaking. As a result, the overall symmetry remains unbroken.

(neighboring orbitals) Does the contraction or expansion of an electron's orbital under the influence of the Lorentz force take into account the neighboring orbitals of other localized electrons surrounding it?

Indeed, the orbital of a localized electron forms strong bonds with the orbitals of neighboring atoms. Because of the strong bonding, the strength of these bonds remains unaltered under the influence of an external magnetic field. However, within the context of unchanged overall bonding strength, there is a redistribution of the clockwise and counterclockwise rotation components of the orbital, which helps to break the required spatial symmetry.

The presence of neighboring orbitals plays a significant role in breaking the required spatial symmetry. For instance, when different atoms are situated on each side of the orbital, the center of the orbital becomes shifted away from the nucleus position. Furthermore, such neighboring orbitals make the clockwise and counterclockwise rotation components of the orbital experience shifts in different directions, thereby breaking the symmetry between them. As a result, the interaction of the clockwise and counterclockwise rotating components with an external magnetic field becomes different.

#### Distribution of a change of Hani vs. a change of kso under magnetization reversal

The anisotropy field (Hani) and the coefficient of spin-orbit interaction (kSO) both exhibit a dependence on the magnetization polarity. These dependencies are not independent but are correlated with each other. Specifically, a larger change in kSO corresponds to a smaller change in Hani when the magnetization is reversed.

Change of anisotropy field and strength of spin- orbit interaction under magnetization reversal.
Each dots corresponds to a measurement of one nanomagnet. Dots of the same shape and color correspond to nanomagnets fabricated on the same wafer.
A negative slope is very clear. A nanomagnet, which have a larger step of kso, has a smaller step of Hani.
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What is the explanation for the negative slope observed in the distribution of the change in anisotropy field versus the change in the strength of the spin-orbit interaction under a reversal of magnetization?
The distinction in the strength of the spin-orbit interaction during magnetization reversal is evident, but it is expected that the demagnetization field remains unaffected as it is purely a geometrical phenomenon. Consequently, one would anticipate a positive slope in the distribution. However, experimental observations reveal a negative slope. The question then arises: why is this the case?
As 2023.06, the reason for the experimentally- observed negative slope of distribution ΔHani vs. ΔkSO is not understood.

This correlation becomes evident when comparing measurement data from nanomagnets fabricated on the same wafer. Within a single wafer, there are slight variations in thickness and interface roughness from point to point. Consequently, nanomagnets fabricated at different locations, such as the center or edge of the wafer, or even neighboring points, will exhibit variations in their parameters due to differences in interface roughness and nanomagnet thickness.

(fact) The strength of spin-orbit interaction and, therefore, kSO does depend on the magnetization polarity

It is because the position of the center of the electron orbital with respect to the nucleus position is shifted towards (outwards) the interface. The magnetic field opposite or along that shift induces a different strength of spin- orbit interaction.

(fact) The demagnetization field does not depend on the magnetization polarity

It is The demagnetization is a geometrical effect. There is any reason why the demagnetization filed should depend on the magnetization polarity.

(unexplained experimental fact): negative slope between a change of kSO vs change of Hani under magnetization reversal

Hani is proportional to kSO and the internal magnetic field Hint:

Since the demagnetization field remains unaffected by changes in magnetization polarity, the internal magnetic field Hint: should be independent of the magnetization polarity as well. Therefore, one would expect the change in anisotropy field with magnetization reversal to have the same polarity as the change in kso. However, contrary to these expectations, the observed polarity in experiments is opposite.

(A possible reason): dependence of demagnetization field on the magnetization polarity

But why and how? What is the mechanism?

The negative slope is a systematic feature, which is observed in all studied wafer

Change of anisotropy field and strength of spin- orbit interaction under magnetization reversal. Systematic study.

##### Comparison of data of many different nanomagnets of a different size, material, composition, thickness, structure.
Full scale small scale Samples. Thickness in nm shown in blankets

Measured step of anisotropy field vs. measured step of coefficient of spin-orbit interaction as magnetization is reversed to opposite direction measured for nanomagnets of a different size, composition, structure. Arrows show the tendency.

It is the same graph as the left one, but in a different scale Dots of the same color and shape correspond to nanomagnet fabricated at different places of the same wafer. Stars show multilayer nanomagnets, which contain several ferromagnetic layers.
The slope is negative for both single-layer and multi- layer nanomagnets.   The nanomagnet of a different width and length in range from 50 x 50 nm to 3000 nm x 3000 nm are shown.
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#### Examples. Measured dependences of Hani and kSO on polarity of magnetic field & magnetization

Measured dependences of Hani and kSO on polarity of magnetic field & magnetization
large Hani~9 kG, kSO>0 small Hani~3 kG, kSO<0 moderate Hani~7 kG , kSO<0
wafer Volt48A nanomagnet L74C g wafer Volt54A nanomagnet R65C wafer VOLT 58A Ta(5 nm)/ (FeCoB 1 nm, x=0.3) (see here) nanomagnet L43C
Anisotropy field Hani vs. external magnetic field. Dots are measured data. Lines are fit.
Measurement of 0 deg corresponds to a scan of in-plane magnetic field along current. Measurement of 90 deg corresponds to a scan of in-plane magnetic field perpendicularly to the electrical current. Independent measurements give the same difference of the offset (Hani) and slope (kSO)
(note) In all shown nanomagnets, the split due to magnetization reversal is identical for the 0-deg and 90-deg scans. It is not always the case.
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#### Dependence of strength of spin- orbit interaction on direction of magnetic field

The strength of spin-orbit interaction relies on the relative angle between the magnetization and the magnetic easy axis or the surface normal for a nanomagnet with PMA. At particular angles, there exists a peak or a trough in the strength of the spin-orbit interaction. This specific angle corresponds to characteristics related to the distortion of electron orbitals resulting from bonding at the nanomagnet's interface. Across all nanomagnets examined, a consistent and systematic correlation was observed between the magnetization angle and the spin-orbit interaction.

#### Considering that the standard method for measuring the strength of spin-orbit interaction involves tilting the magnetization under an external magnetic field, how can one effectively determine the relationship between the spin-orbit interaction strength and the angle of magnetization tilt?

In addition to the prominent linear relationship between Mx and Hx, which serves as the basis for assessing the strength of spin-orbit interaction , there exists a secondary weak oscillatory dependence on the tilting angle. This weak oscillating pattern allows for the evaluation of the angle dependency of the spin-orbit interaction.

Major prominent linear relationship between Mx and Hx, from which the strength of the spin orbit interaction is evaluated,:

where Mx is in-plane component of the magnetization, Hx is in-plane external magnetic field and Hani is the anisotropy field , which is evaluated from the linear fitting of measured data of Mx vs. Hx.

Real measured dependency between Mx and Hx,

where osc is a very weak oscillations.

measured dependency Mx vs. Hx, & its linear fit

Deviation of measured Mx vs. Hx from linear Relationships

## Mx-Hx/Hani

wafer Volt40A nanomagnet L43C

 wafer Volt54A nanomagnet R65C wafer Volt58A nanomagnet L23C wafer Volt40A nanomagnet L43C

### voltage-controlled magnetic anisotropy. VCMA effect.

The VCMA effect describes the modulation of the magnetic properties of ferromagnetic metal by a gate voltage
 Under a gate voltage, magnetic properties of ferromagnetic metal (Fe) are changed. For example, the magnetization direction changes. Gate voltage is not applied to bulk to Fe, but only to Fe/oxide. Still the gate voltage greatly affects magnetic properties of Fe
The gate voltage modulates the coercive field, anisotropy field, spin polarization, the absolute value of the magnetization and magnetization switching time
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There is a screening of the gate voltage in the bulk of the nanomagnet. Even though the gate voltage is applied to the nanomagnet, in fact, due to the screening the gate voltage is only applied to the interface and, therefore, modulation of magnetic properties of the interface by the gate voltage is the key effect here.

Change of strength of spin- orbit interaction and anisotropy field under a gate voltage
 Anisotropy field Measured Hani vs perpendicular- to- plane external magnetic field Hz. Lines of different color correspond to a measurement under a different gate voltage. Measured anisotropy filed Hani and coefficient of spin- orbit interaction kSO as a function of the gate voltage. Measured change of anisotropy filed ΔHani and the coefficient of spin- orbit interaction ΔkSO as the gate voltage changes from 0 V to 1 V. Each dots corresponds to a measurement of a different nanomagnet fabricated at a different part of the same wafer. The dependence on the gate voltage is substantial. Both the offset (Hani) and the slope (kSO) of the line depend on the gate voltage. The dependencies are opposite. Hani is largest at a negative gate voltage. kSO is largest at a positive gate voltage. Even though the change is slightly different from a nanomagnet to a nanomagnet, ΔHani is always positive and ΔkSO is always negative Even in the absence of an external magnetic field the Hani depends on gate voltage. It decreases as the gate voltage increases. Additionally, the slope of lines depends on the gate voltage. Due to a different slope, the gap between lines decreases at a larger field.
wafer Volt 54a (see here), nanomagnet L25C
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### Possible reason why an increase of anisotropy field sometimes does lead to an increase of anisotropy field

The magnetic anisotropy itself is originated by the spin-orbit interaction. In absence of the spin- orbit interaction there is no magnetic anisotropy and the anisotropy field Hani equals zero. It is plausible to assume that an increase of the strength of the spin- orbit interaction always leads to an increase of the anisotropy field. Often it is true. However, sometimes the dependency is opposite, an increase of the strength of the spin- orbit interaction is accompanied by an decrease of the anisotropy field, when some of nanomagnet parameters changes. It is because additionally there are changes of the demagnetization field and the internal magnetic field, which can be opposite to the change of the spin-orbit interaction and which can reverse the change of Hani.

.

### dependence on anisotropy field Hani

strength of spin-orbit interaction kSO amplitude of oscillations of strength of spin-orbit interaction vs external magnetic field period of oscillations of strength of spin-orbit interaction vs external magnetic field Samples. Thickness in nm shown in blankets Fig.21a. Strength of spin-orbit interaction is largest in multilayer nanomagnets, which are shown by stars and which contain several ferromagnetic layers. There is a linear dependence between kSO vs. Hani. In case of a single- layer nanomagnet, the slope of the linear dependence is positive. In case of a multi- layer nanomagnet, the slope is negative. Fig.21b. The dependency is sparse, because it is indirect.. In case of a single- layer nanomagnet, the slope is approximately positive. In case of a multi- layer nanomagnet, the slope is approximately negative. Oscillation amplitude depends directly on kSO (see graph below) and kSO depends on Hani (see left graph). Fig.21c. The period is about 8 kGauss and becomes smaller for nanomagnets with small oscillation amplitude.

### dependence on strength of spin-orbit interaction kSO

internal magnetic field, Hint amplitude of oscillations of strength of spin-orbit interaction vs external magnetic field amplitude of oscillations of strength of spin-orbit interaction vs external magnetic field Samples. Thickness in nm shown in blankets Fig.22a. It is the internal magnetic field, which holds the magnetization of the nanomagnet along its easy magnetic axis. The internal magnetic field is stronger for single- layer nanomagnets and weaker for multi-layer nanomagnets (shown by stars). Fig.22b The amplitude of oscillations linearly depends on the strength of the spin- orbit interaction for both the single- layer and multi-layer nanomagnets. The linear dependence indicates the interfacial origin of the oscillations. Fig.22c It is same figure as the left figure (Fig.22b) but in a different scale. For all single- layer nanomagnets, the dependence is linear with the same positive slope.

Dots of the same color and shape correspond to nanomagnet fabricated at different places of the same wafer. Stars show multilayer nanomagnets, which contain several ferromagnetic layers.
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Strength of spin-orbit interaction is largest in multilayer nanomagnets, which are shown by stars and which contain several ferromagnetic layers. There is a linear dependence between kSO vs. Hani. In case of a single- layer nanomagnet, the slope of the linear dependence is positive. In case of a multi- layer nanomagnet, the slope is negative.

#### Q. Why does the Spin-orbit 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 nucleus (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 nucleus. Only this region mainly contributes to SO. If the ionic or atomic radius (see here or here) is about 50-100 pm, the region with radius only 1-2 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 nucleus).Usually a deformation of orbital moves electron (electron distribution) closer to the nucleus, 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 s- symmetry (e.g. n-type 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 nucleus.

#### Q. You always depict an electron as a point-like particle. Should you use a wave function instead and full quantum-mechanical description of the SO interaction?

A. The spin-orbit 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 electromagnetic 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 time-reversal symmetry, which you mentioned as a key ingredient to induce spin-orbit interaction. I am not sure how electrical current flowing along with the interface and perpendicularly to the interface electrical field (type 1, weak SO) and electrical current (type 3, moderate SO) can break the TRS. Thank you for your answer in advance!?

A.(time- inverse symmetry & electrical current) An electrical current is always breaking the time inverse symmetry. The reversal of time makes the electrons to move in the opposite direction. The time -inverse symmetry is broken for any material property, which depends on the direction of electron movements or the polarity of the current.

(weak SO & breaking of time- inverse symmetry) It is the basic relativistic SO, which describes the fact that an electron is experiences a magnetic field when it moves perpendicularly to an electrical field. This effect can be understood as following. The electrical field is only one specific case of the more general electromagnetic (EM) field. The electron can experience only electrical field without magnetic field only in the case when both the electron and the electrical field do not move. When electron moves perpendicularly to the electrical field, additionally to the electrical- field component, the electron experiences the magnetic component of the electromagnetic field. This magnetic component is called the magnetic field HSO of spin- orbit interaction.
(breaking of time inverse symmetry for weak (or basic) SO) The direction of magnetic field HSO is fixed relatively to the direction of the electrical field and the direction of electron movement. In the case of the forward movement and the direction of the electrical field to the right hand, the direction of HSO is up (See page top fig.). When the movement direction of electron is reversed (or the time direction is reversed), the electrical field becomes toward the left hand and therefore the direction of HSO becomes down. To summarize the origin of the weak SO contains the breaking of time- symmetry

(moderate SO & breaking of time- inverse symmetry) The example of the moderate SO is the Inverse Spin Hall effect (ISHE) (See my page on the Spin Hall effect). The ISHE describes the fact that an electrical current is created perpendicularly to the flow of the spin- polarized current (spin current) due to spin- dependent scatterings (and SO). E.g. it occurs when the probability of electron scattered to the left and right direction are slightly different with respect to the electron movement direction. This asymmetry of scattering can occurs in a spin- polarized electron gas (e.g. in a ferromagnetic metal) and its origin is the SO.
The reason why the breaking of the time- inverse symmetry is critically important for this effect can be understood as follows. Let us start from the case when there is no current and the time- inverse symmetry is not broken. From a view of a static observer, if for a forward- moving electron the scattered probability to the left is higher than to the right, for a backward- moving electron the probability is higher to the right than to the left. When there is no electrical current and the time- inverse symmetry is not broken, there are equal amounts of electrons moving in the forward and backward directions. In total, the number of electrons scattered into the left and into the right are equal and there is no electron current from the left to right. When there is an electron current in the forward direction and the time- inverse symmetry is broken, the number of electrons moving in the forward direction is larger than in the backward direction. As a result, there are more electrons are scattered to the left than to the right, which results an electron current from the left to the right. To summarize, the breaking the time- inverse symmetry by electrical current creates a perpendicular electron current.

#### Q. If I am correctly understood from the contents, you said that HSO alone cannot break the time-reversal symmetry. But in the comments on weak SO, it looks like Hso can break the symmetry.

A.(breaking of time- inverse symmetry ) Magnetic field, spin, electrical current, orbital moment, they are all objects for which the time inverse- symmetry is already broken. They all reverse their own direction when the time is reversed. Additionally, when the time- asymmetrical object is interacting with a time- symmetrical object, the time- symmetrical object may become time- asymmetrical. For example, an external magnetic field may induce a magnetic moment for electron orbital, which initially has no magnetic moment. It is the origin of the PMA effect (strong SO).

(conservation of time- inverse symmetry ) Some objects can spontaneously break the time- inverse symmetry. For example, assembly of spins + exchange interaction can do it. The spins can spontaneously be aligned along one direction. However, it is a very exceptional case. The time-inverse symmetry is a well- conserved quantity and cannot be broken.

(HSO & time- inverse symmetry) Now I am approaching to the answer to your question. It is very important that the spin- orbit interaction cannot break the time- inverse symmetry by itself. It needs some external breaking (like an external magnetic field or an electrical current) in order to manifest itself. It is a very important feature of SO.

It should be noted that the SO can still influence the electron energy without breaking the time- inverse symmetry. For example, in an atomic gas the SO aligns electron spin along the orbital moment for each individual atom. However, in total for whole atomic gas is not broken due to random orientation of orbital moments in the gas.

In the case of the weak and moderate SO, the electrical current breaks the time- inverse symmetry. In the case of the strong SO (PMA), the magnetic field (magnetization) breaks the time- inverse symmetry.

#### Q. What is the difference between SOC and SOI? or are they same? What I know is: Spin orbit coupling (SOC) is interaction of spin of electron and its orbital motion around nucleus. Spin orbit interaction (SOI) is interaction between associated magnetic field of moving electron in electric potential with its spin. However, some refer spin orbit coupling and spin orbit interaction as a same phenomena. Is am I right?

The Spin-orbit Interaction (SOT) is a general physical effect, which describes a relativistic property of the electromagnetic field. The SOT describes the magnetic field, which an electron experiences, when moving in an electrical field.

The Spin-orbit coupling (SOC) is a perturbation calculation method for the SOT for a specific case when the electric field is the electric field of a nucleus. The SOC assumes that the SOT is small and calculates it as a tiny perturbation. It is an absolutely incorrect assumption. In close proximity of the nucleus the electrical field is huge and the SOT induced by a nucleus is not small at all. For example, in a sample having PMA (See my page on PMA) the SOT magnetic field may be huge (about 10 kGauss).

The SOC perturbation method was developed about 100 years ago, when there were no computers and there was no another option.

The SOC perturbation method often diverges and gives an incorrect result. Therefore, its usage should be minimized.

(about dependency of the spin-orbit interaction of external magnetic field)

##### (from Xian-Peng Zhang) Q. I have a question about the spin-orbit magnetic field. Why it is proportional to the total magnetic field? (see below for details) May you provide me some references?

Your question is why the spin-orbit interaction depends on an external magnetic field and in which reference it was described for the first time.

This fact is known for a very long time and therefore it is more about the history of Science, in which I am not expert. I am sorry if I refer something incorrectly.

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(SO fact 1) The fact about the relativistic origin of the magnetic field and the existence of the spin-orbit interaction (the existence of the magnetic field of the spin-orbit interaction). This fact is one of basic facts, on which the Special Theory of Relativity was build. Therefore, I think Einstein and Lorentz knew about this fact. Myself, I have read all about this fact (all details and many nice explanations) in Landau textbook "The Classical Theory of Fields"

(SO fact 2) The fact that the strength of spin- orbit interaction (the SO magnetic field) is proportional to the strength of an externally applied magnetic field. This fact are direct consequence of the another important fact about SO:

(SO fact 3) The spin-orbit interaction cannot break the time- inverse symmetry. As a result, in order to manifest itself, the SO needs an external breaking of the time- inverse symmetry (E.g. by an orbital moment or by an external magnetic field  or an electrical current etc.).

Fact 2 follows very directly from the fact 3. The strength of SO is zero , when the time inverse- symmetry is not broken (case when the orbital moment is zero, there is no external magnetic field etc.) . The SO magnetic field becomes a non-zero only when the time- inverse symmetry is broken. The strength of  the SO ( the strength of the SO magnetic field)  becomes larger when the degree of the breaking of the time inverse symmetry becomes larger (e.g. the orbital moment becomes larger or the external magnetic field becomes larger etc.) .

Once again

zero magnetic field -> time-inverse symmetry is not broken -> spin-orbit is zero

external magnetic field becomes larger -> degree of breaking of time-inverse symmetry becomes larger -> spin-orbit becomes larger.

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(fact 3 explanation)  Why the spin-orbit interaction cannot break the time- inverse symmetry. This feature is the feature of the time-space symmetry and can be obtained from he Special Theory of Relativity (e.g. see Landau textbook " The Classical Theory of Fields  " ). A simplified understanding of the fact 3  can be as follows.  The spin-orbit effect is a relativistic effect and therefore can only occur when the object moves or the field, with which the object interacts, moves. The relativistic effect requires a movement.  The close movement speed is to the speed of the light, the stronger any relativistic effect is.  Any movement means a breaking of the time inverse symmetry. This is why the SO occurs only when the time-inverse symmetry is broken.

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Even though the direct relation between the spin- orbit interaction and the breaking of the time- inverse was understood  a long time ago  as well as the proportionality of the SO strength to the degree of TIS breaking (e.g., to the strength of the orbital moment or  to strength of the external magnetic field), all details and specifics have not been fully understood.

For example, the SO strength  depends on the orbital symmetry. Usually  the orbital is more symmetric in the bulk of a material and the SO is weaker, but at an interface the orbital becomes less symmetrical and the SO becomes larger.

You can read some details  here

This effect is used for molecular-recognition sensor, See here

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The dependence of PMA and therefore SO on an external magnetic field is very strong. The experimental measurement of this dependence is rather simple  and the measured data well fit to the theory. The anisotropy field is a measurable parameter, which characterizes the strength of PMA and SO.

Near for all nanomagnet, I have studied, I measure the dependence of anisotropy field on a perpendicular magnetic field. Such data contains a lot of important information.

Introductory presentation on this measurement method I gave at MMM 2020 conference. You can watch it here

(about validity of representation of electron orbital as an electron rotating around a nucleus. Electron rotation vs. the orbital spacial distribution vs. breaking time-inverse symmetry)

##### (from Ke Chen) Q. You cannot think of orbital angular momentum (e.g., L =1) as the angular momentum generated by an electron rotating around a nucleus. In fact, it corresponds to a probability distribution of the electron that can be described in terms of spherical harmonics. Specifically l=1, its spherical harmonic function (electron distribution) is two equally tangent balls. So in clockwise and counterclockwise the orbital angular momentum has the same magnitude and the same sign. The orbital moment, on the other hand, occurs in the magnetic field and has a value of plus or minus.

(Answer). In fact you can and you must. Let me explain by steps

(step 1: about probability) The electron is a wave particle. The probability describes how this wave interacts with external objects. E.g. if you put a point-like electron detector in some point of space, the probability describes efficiency of the detector to detect this electron.
Still as a wave, the electron can move, be reflected and have all properties of a wave.
A good example is a photon, which reflected back and forth between two mirrors. It consists of two opposite moving wave, but the probability does not change in time.

(step 2: about angular momentum): In general, the angular momentum (e.g. L=1) describes the degree of breaking of the time-inverse symmetry. The spin describes the internal breaking of the time-inverse symmetry and the orbital moment describes the spatial breaking of the time-inverse symmetry. Any spatial breaking of the time-inverse symmetry corresponds to some spatial rotation of the object. It is the mathematical fact. Even though the electron orbital might be complex, it can be divided into components, along which the electron rotates into clockwise or/and counterclockwise direction with respect to a fixed axis (axis of the orbital moment). Arbitrary 3D is rather complex (See animation above), but still when electron has a non-zero orbital moment, the rotation in one direction slightly larger than in the opposite direction. It is strictly determined by the degree of the breaking of the time-inverse symmetry.

(step 3: about a complex orbital): The shape of the orbital is not directly related to the breaking of the time- inverse symmetry. These are two different properties, even though they often influence each other and might to be fixed to each other by some specific interaction.
In a hydrogen atom, the orbital of a larger orbital moment have a more complex orbital, but it is not always the case. For example, in a crystal, the orbital moment of localized electrons is usually quenched. It means it is zero, even though the shape of the orbital could be very complex.

(step 4: rotation and symmetry):

Any 1D wave function can be represented as a wave moving forward, a wave moving backward or a standing wave as a sum of waves moving forward and backward.

Any 2D wave function can be represented as waves moving forward or backward or a standing wave. Additionally to that, the wavefunction can be represented as a wave rotating clockwise or a wave rotating counterclockwise or a standing wave as a sum of waves rotating rotating clockwise and counterclockwise.

Any 3D wavefunction is the same as the 2D wavefunction but the rotation is the 2D.

What I want to say that the spacial distribution and the breaking of the time- inverse symmetry are two separate properties of the wavefunction. It does not matter what is the spacial distribution of the wavefunction, the time- inversion symmetry can be either broken or not. Even though sometimes the combination of the other eigen wavefunctions is required to make a specific time- inversion symmetry breaking (e.g. for p- or d- symmetries)

(about rotation speed of electron around atom)

##### Q. What is rotation speed of an electron around atom. Is it fast enough for a relativistic effect (such as the Spin-Orbit interaction) to be relevant?

(Answer). It is fast enough. For example, the ratio of the velocity to the speed of light c is equal to the fine-structure constant α (α =1/137) for the first orbit of the hydrogen atom.

velectron=α · c=c/137 (q1)

c is speed of light.

In fact, Eq.(q1) is one of definitions of the fine-structure constant α (See here).

(about rotation of an electron around a nucleus)

(from Ekta Yadav) "A conduction electron rotates around each nucleus. " Can you give me some reference for this statement?

yes, the electron does rotate around a nucleus. Literally. It is not a rotation when one dot is rotating around another dot. For example, as the Earth is rotating (orbiting) around the Sun. The rotation of an electron around a nucleus is more complex, because the electron is a wave. Nevertheless, it is as a real rotation as any rotation could possibly be.

(Rotation in Quantum mechanics. Rotation & electron orbital) In order to understand it, it is better to start from a 1D structure. Let us look at a wave, which is reflected back and forward between two mirrors. This quantum state is static. It does not move in space.. The field distribution is static as well. However, there are waves, which move forward and backward. Next,Let us look at a similar 2D structure. A wave can move around a circle. In this case, the quantum state has a positive and negative orbital moment for a clockwise and counterclockwise rotation, respectively. Additionally, a quantum state may have zero orbital moment. In this case, the state consists of two waves moving or rotating in the opposite directions. This 2d case is similar to the 1D case of a wave reflecting between mirrors. In the 3D case, the electron rotation can be along any 3D vector (x,y,z). Correspondingly, the direction of the orbital moment can be along any 3D vector (x,y,z).

(Rotation & Orbital symmetry) The fact of rotation of the electron around the nucleus has an even more fundamental origin. In nature, all conserved parameters of an object have corresponding broken symmetries. The spin describes the broken time-inverse symmetry. The rotation (as well as the orbital moment) describes more complex breaking of the symmetry. The rotation means breaking of the space symmetry along with breaking of the time- inverse symmetry. The electron of an atom orbital breaks just this specific symmetry corresponding to the rotation. Meaning that, judging from the most fundamental definition of the rotation, the electron of an atomic orbital is truly rotating around the nucleus.

(Rotation & Bonding between neighbor atoms) The electron rotation or, the same, the electron having a non-zero orbital moment also means that the spatial distribution of the electron wavefunction is changing in time. When an electron state participates in a bonding between atoms, the spatial distribution of the electron wavefunction is fixed, cannot change in time and, therefore, cannot be rotated. For this reason, the electron moment is quenched meaning the orbital moment is zero and there is no electron rotation around the nucleus (See here).

## Video of Conference presentations.

67th Annual Conference on Magnetism and Magnetic Materials (MMM 2022)

Intermag 2023. Sendai, Japan

click on image to see YouTube Video   click on image to see YouTube Video
(title): Measurement of strength of spin-orbit interaction   (title): Systematic study of the strength of VCMA effect in nanomagnets of small and large strength of spin-orbit interaction.

Explanation Video

## (Part 5a): Analyze of anisotropy field Hani. Measurement of coefficient of spin-orbit interaction kSO

(short content 1:)  the properties of spin-orbit interaction   (short content 1:)  the difference between Global, Local & Average Magnetic Fields.   (short content 1:)  how it is possible to measure  the strength of SO and strength of PMA.    (short content 1:) analyze the measurement of the anisotropy field
(short content 2:)   linear relation between the external magnetic field and Hso and about the symmetry and physics arguments, which are behind this linear relation   (short content 2:)  how to measure experimentally the internal magnetic field Hint   (short content 2:)    what holds  the magnetization along the easy magnetic axis and how to measure the strength of the holding force.
(short content 2:)  the measured oscillations of Hani and why this oscillation is an important characteristic of a nanomagnet
(short content 3:)   how the deformation of the orbital at the interface creates the perpendicular magnetic anisotropy (PMA)   (short content 3:) experimental measurements of Hint in FeB nanomagnets.   (short content 3:)   the experimental setup and main idea behind the proposed measurement   (short content 3:)   why the spin-orbit interaction depends on the polarity of the magnetic field and how to measure it.
Other parts of this video set is here
click on image to play it

Content of this page represents my personal view and it is reflected my own finding. It may slightly different from the "classical" view on the spin-orbit interaction, which is described in following references