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Spin-Orbit Interaction

Spin and Charge Transport

Relativistic origin of the spin-orbit interaction

An electron moving in an electrical field experiences an effective magnetic field, which acts on the electron magnetic moment (spin). The interaction of the electron magnetic moment with the effective magnetic field is called the spin-orbit interaction.

The weak and the strong spin-orbit interaction can be distinguished from the type electrical field, which induces it:

-the weak spin-orbit interaction, which is induced by an external electrical field or an intrinsic electrical field in a junction or in quantum well.

- the strong spin-orbit interaction, which is induced by atomic nuclear. In the vicinity of the nuclear this field is huge and it can induce a substantial spin-orbit interaction.

Breaking of orbital symmetry and time-inverse symmetry are two critical factors, which determine the magnitude of the spin-orbit interaction



Content

click on the chapter for the shortcut

(1). Explanation in short

(2). Relativistic origin of the Spin-Orbit interaction

(3). Lorentz  transformation

(4). Spin-orbit interaction in macro world

(5). Weak and Strong Spin-orbit interaction

(6).Magnitude of the Spin-orbit interaction.

(7). Spin-orbit interaction in the centrosymmetric electric field of atomic nuclear

(8).Spin-Orbit interaction due to the orbital deformation

(9). Time- inverse symmetry and the spin-orbit interaction

(10).Enhancement of magnetic field due to the spin-orbit interaction

(11). g-factor

(12).Perpendicular-to-plane magnetic anisotropy (PMA)

(13). Magneto-elastic effect

(14).Voltage-induced spin-orbit interaction & VCMA effect

(15). Spin-dependent scatterings

(16).Voltage-induced spin-orbit interaction & VCMA effect

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

(18).Spin-orbit interaction obtained from the Dirac equations

(19). Pauli equation & Spin-Orbit interaction

(20).Merits and demerits of the use of the Hamiltonian approach for calculations of the Spin-Orbit interaction.

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Key properties of the spin-orbit interaction

The spin-orbit interaction is relativistic effect (not quantum-mechanical).

The spin-orbit interaction is described by the effective magnetic field HSO of the spin-orbit interaction. The HSO is a real magnetic field, which is indistinguishable from a common magnetic field (An exception, see here)

Only the electrical field of a nuclear can induced a substantial spin-orbit interaction

In the case of a spherical orbital, the electron experiences opposite-sign of HSO in equal amounts. As a result, the electron does not experience any spin-orbit interaction

In order to experience the spin-orbit interaction in the electrical field of an atomic nuclear, two condition should be satisfied.

(condition 1): The electron orbital should be deformed. It should not be spherical. It is better it should be not centrosymmetric.

(condition 2): The time-inverse symmetry for the orbital should be broken. It means that an external magnetic field should be applied.

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What difference the spin-orbit interaction does make? What does the spin-orbit interaction affect and influence?

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.


Which specific changes the spin-orbit interaction does?

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

(Effect 9)- Changing of the Curie temperature in a thin ferromagnetic due to the interface-enhanced spin-orbit interaction.



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 of the Spin-Orbit interaction affects only the electron spin. There are only two interactions:

1) There can be a spin precession

2) There can be a damping of the spin precession, which aligns the electron spin along the effective magnetic field of the spin-orbit interaction

Important Note 1: The effective magnetic field of the Spin-Orbit interaction cannot induce the Lorentz force or the Hall effect.

Important Note 2: The effective magnetic field of the Spin-Orbit interaction does not interact with the magnetic moment induced by the orbital moment of the electron.

 


Q3.Which electrons may experience the Spin-Orbit interaction?All???

Only following electrons

type (1): Electrons, which occupy an electron state by one.

In the case of conduction electrons, they are defined as spin-unpolarized and spin-unpolarized electrons (See here)

The electrons of states, which are occupied by two electrons of opposite spins, do not experience the Spin-Orbit interaction. It is because the spin of such states is zero and the time-inverse symmetry is not broken for such state. Example (1) of no SO : deep-state electrons; Example (2) of no SO: conduction electrons of an energy substantially smaller than the Fermi energy (spin-inactive electrons, See here)

type (2): An electron, which has non-zero speed component perpendicular to an applied electrical field.

type (3): An electron has a non-zero orbital moment .


Fact (1) : ~95% spin-orbit related effects in a solid are induced by the electrical field of atomic nuclear and occur due to the electron orbiting the atomic nuclear

Fact (2) : The spin-orbit interaction is fully described by the effective magnetic field HSO of the spin-orbit interaction


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 nuclears 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)

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): nuclears 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)


Relativistic origin of the Spin-Orbit interaction

Relativistic origin of the Spin-Orbit Interaction
Fig.3 An object (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.

 

The Lorentz force- tween effect with the effect of the spin-orbit interaction

 

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.

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

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

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 nuclear. Another example, an electron current along an interface. Usually, there is an electrical field perpendicularly to the interface, but still the electron may move along the interface (See below)

 

effective spin-orbit magnetic field HSO

 

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

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 nuclear. There the electrical field increase as 1/r, where r is the distance to the nuclear. The nuclear is almost "point-like" object, therefore the electrical field Estatic is huge in close proximity of the nuclear. It makes a large HSO

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

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

In close vicinity of a nuclear an electron experiences a very strong electrical field of the nuclear. However, this field is very symmetric and the electron experience the opposite signs of the spin-orbit interaction on its path around nuclear. 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.


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

Spin-orbit interaction in macro world
Animated figure 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 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 effective magnetic field.

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 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.

 

 

 

 

 

 

 

 

 

 

 

 


Weak and Strong Spin-orbit interaction

Two types of the Spin-Orbit (SO) interaction

In order to experience the Spin-orbit interaction an electron should move, it should move and its movement direction should not be along the electrical field. According to the electrical field, which induces the Spin-orbit interaction, two types of the Spin-orbit interaction can be distinguished:

(1) weak SO interaction induced by an external electrical field (SO is proportional to an electron speed). SO is small

(2) strong SO interaction induced by a centrosymmetric electrical field of atomic nuclear (SO is proportional to an electron symmetry or an orbital moment of electron). SO can be large

 


Movement of a delocalized (electron in a metal

Simultaneously with movement along the crystal the delocalized electron rotates a couple of times around each nuclear. The rotation and the linear movent cause two different types of the spin-orbit interaction.

 

The delocalized (conduction) electrons move simultaneously in the forward direction along lattice and around each atom (nuclear) of the lattice.

 

The localized electrons (-d,-f) do not move along lattice. They only rotates around nuclears.

 

Note: Even though the symmetry of delocalized electrons is -s and/or -p, they may have a little mixture of-d and -f symmetry.

 

 

 

 

 

 

 

 

 

 

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Magnitude of the Spin-orbit interaction.

Generally, the magnitude of the spin-interaction in a solid is small!!!.

For example, when an electron moves across an external electrical field: (a moderate electrical field + a moderate electron speed = a very small spin-orbit interaction)

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

a very strong electrical field + a moderate electron speed = a strong 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.


Example 2.

An electron rotating around a nuclear.

Electron Speed: linear speed of electron rotating around atom : ~2.1E6 m/s

The applied voltage:Coulomb Electrical Field in H atom at 1st orbital (r=0.053 nm) 5.1E9 V/cm

The high voltage is because the orbit is very close to the nuclear !!!

Result:

The effective magnetic field of the 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.

It is important: An electron may experience such large magnetic field only when it is very near to the nuclear and only when the electron is rotating around the nuclear.

 

 

 

 



Q. Both localized (-d,-f) and delocalized (-s,-p) electrons are rotating around nuclears (atoms), is it sufficient for them to experience a strong 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 in the centrosymmetric electric field of atomic nuclear.

Animated figure 6. Spin-Orbit Interaction in the center-symmetrical electrical field of atom nuclear. It is an imaginary case when electron moving in a circular orbit around nuclear (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

 

 

 

 

 

 

 

 

 

 

 

 

 

Q. How is it possible that an electron, while rotating around a nuclear, 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.

When the orbit of an electron is center-symmetrical, the electron does not experience any spin-orbit interaction !!

In order to experience any spin-orbit interaction, the electron orbit should be asymmetrical !!!!

s-orbital. An electron rotates around a nuclear 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.
The electron in atom is similar to a standing delocalized electron wave, which also does not experience the spin-orbit interaction

 

 

 

When an electron rotates in a spherical orbital (s-orbital), it does not experience any spin-orbital interaction. For the s-orbital the orbital moment is zero. That means that the electron rotates in clockwise and anti clockwise directions an equal number of times. The effective magnetic field the electron experiences due the spin-orbit interaction is zero, because it is fully compensated during rotations in opposite directions.

When an electron rotates in an elliptical orbital (d- or p-orbital), the orbital momentum is not zero and the electron rotates in one direction more than in the opposite direction. In this case the electron experiences the effective magnetic field due to the spin-orbit interaction.

 

The following is important

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.

 

 


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).


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 nuclear. 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 nuclear. The nuclear charge is larger for an element of a larger atomic number. Therefore, the electrical field of the nuclear 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 nuclear 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 nuclear field of anion is larger. Therefore, the spin-orbit interaction induced by the nuclear 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 nuclear by the electrons of the inner shell is not centrosymmetric. It makes the spin-orbit interaction stronger.


It is good to know.

1. As

Note: The orbital moment as well as the electron speed are different at different points of the Brillouin zone.

 

2. The magnetic moment of an electron is a quantum- mechanical sum of magnetic moments induced by the spin and induced by the orbital moment. The effective magnetic field acts only on spin and it does not effect the magnetic moment due to the orbital moment.

 



Spin-Orbit interaction due to the orbital deformation

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

click here or on image to enlarge it

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 nuclear

 

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.

 

 

 

 

 



Enhancement of magnetic field due to the spin-orbit interaction

 

Along an applied external magnetic field, an additional magnetic field is induced due to the spin-orbit interaction. Therefore, an electron experience a larger magnetic field than externally applied due to the spin-orbit interaction

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.

 


Time- inverse symmetry and the spin-orbit interaction

Time-inverse symmetry is not broken

click here to enlarge

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 nuclear 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

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Time-inverse symmetry: broken

Time-inverse symmetry is broken

click here to enlarge

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

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

 

An external magnetic field (or exchange 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 nuclear, it experiences the Lorentz force in the magnetic field. The Lorentz force is in opposite directions for electron moving in the clockwise and anticlockwise directions around the magnetic field. The Lorentz force modifies the orbital of electrons. When an electron moves in the anticlockwise direction, it moves closer to the nuclear and it experiences the larger electrical field from the nuclear and the larger corresponded effective magnetic field of the the spin-orbit interaction. When an electron moves in the clockwise direction, it moves more distant from the nuclear and it experiences the smaller electrical field from the nuclear and the smaller corresponded effective magnetic field of the the 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

The increase of the effective magnetic field due to the spin orbit interaction

click here or on image to enlarge it

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.

The deformed electron orbital is shown.

 

 

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 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.

 

Both the external applied magnetic field and the exchange field can be amplified due to the spin-orbit interaction.

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).

 

 

 

 

 


 

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

Example 1: Fe/Pt, Co/Pt

Perpendicular-to-plane magnetic anisotropy at

Fe/Pt interface

Magnetization is perpendicular to the interface

click here or on image to enlarge it

Magnetization is along the interface

click here or on image to enlarge it

 

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 effective magnetic field due to the exchange interaction. The green arrows show the effective magnetic field of the spin-orbit interaction. The orbital of the localized electrons are shown.

The magnetization of a single-material ferromagnetic film is in the film plane. In the case when the film consists of a thin layers of different metals, the magnetization could be out of plane. The example of such 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 in the vicinity of the contact. 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.

In the bulk:

There is an exchange interaction in the bulk of Fe, but there is no exchange interaction in the bulk of Pt. The blue arrows show the effective magnetic field of the exchange interaction is shown

The effective magnetic field of the exchange interaction breaks the time-inverse symmetry. It induces the effective magnetic field of the the exchange interaction (red arrows) along the direction of the exchange field. The effective magnetic field of the spin-orbit interaction is small in the bulk of the Fe, because its near-spherical orbital. In the bulk of Pt the effective field of the spin-orbit interaction is zero, because the time-inverse symmetry is not broken in there.

In the vicinity of the interface:

- the exchange field in Fe becomes smaller, because there are less Fe atoms in the surrounding.

- the exchange field in Pt is non-zero, because of the magnetic proximity effect or the interlayer exchange coupling (See here)

-the effective magnetic field of the spin-orbit interaction is non-zero in Pt. It can be larger than this field in Fe, because the electrical charge of the Pt nuclear (+78) is larger than charge of Fe nuclear (+26) and the distance between atoms nearly the same.

 

Dependence on the magnetization direction

- exchange field does not depend on the magnetization direction.

- the effective magnetic field of the spin-orbit interaction significantly depends on the magnetization direction. Because of the deformation of the orbitals, it is large, when the magnetization is perpendicular to the plane, and it is small, when the magnetization is in the film plane (For the case shown in Fig. 15)

 

Example 2: CoFeB/MgO, Fe/MgO

Perpendicular-to-plane magnetic anisotropy at

CoFeB/MgO interface

Thicker metal.

Magnetization is in-plane

Thinner metal

Magnetization is perpendicular to plane

 

Fig.16. A thick film (thickness >1.5 nm) have in-plane remanent magnetization, but the magnetization of a thinner film is perpendicular-to-plane

(See Ikeda et al. Nature Material 2010)

 

 

It was found experimentally (See here Ikeda et al. Nature Material 2010) that the magnetization of a Fe or FeCoB thin film on MgO depends on the thickness of the CoFeB layer.

 

It should be noticed that the magnetizations of a thin Fe(bcc)(001) on Cu(bcc)(001) is also is perpendicular to plane. Similar to CoFeB/MgO, this film is tensile-strained with only a little strain-relaxation.

 

It might be possible that this kind of the perpendicular-to-film magnetic anisotropy is a feature of a tensile-strained thin CoFe (001) films.

 

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 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 nuclears 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)

Wikipedia page is 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): nuclears 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

Without electrical field

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 or exchange 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 nuclear 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 nuclear. Therefore, at the left side of the nuclear 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 nuclear 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 nuclear decays as 1/r^2 and at left side it increases sharply, but at the right the decrease is small.

It is important: In absence of an external magnetic field, there is no 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

click here or on image to enlarge it

Fig.

 

 

 

 

 

 

 

 

 

 

 

 


Spin-dependent scatterings

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

The total spin of the electron gas is not conserved after each spin-dependent scattering.

An electron scattering becomes spin-dependent when the spin-orbital interaction connected the spin of the electron gas to a spin particle outside of the equilibrium of the electron gas

 

The Spin Hall effect is the effect describing accumulation of the spins at a surface of a metallic wire, when an electrical current flows through the wire due to the Spin-Orbit interaction.

The effects which are originated by spin-dependent scatterings

The effective electrical field, which is induced by the defect, is shown by the green arrows. The electrical field induces the spin-orbit interaction. The direction of the effective magnetic field of the spin-orbit interaction is different for electrons scattered into the left and into the right. This makes different the probabilities of scattering into the left and into the right.

Anomalous Hall effect

The charge is accumulated, when a spin-polarized drift current flows

click here or on picture to enlarge it

Spin Hall effect

The spin is accumulated, when a spin-unpolarized drift current flows

click here or on picture to enlarge it or another version

Inverse Spin Hall effect

The charge (and spin) is accumulated, when a spin diffusion current flows.

click here or on picture to enlarge it

Under an applied voltage the drift current flows in the metal wire. When the metal is ferromagnetic, the drift current is spin-polarized. Therefore, there are more electrons with spin directed up. It causes more electrons be scattered into the left than into into the right. This is the reason for the charge accumulation at the right side of the wire. Under an applied voltage the drift current flows in the non-magnetic metal wire. The drift current is spin-unpolarized and the electrons have spin in any direction with an equal probability. Since the probability to be scattered to the left is higher for electrons with spin directed up and the probability to be scattered to the right is higher for There is a region of spin accumulation at backside of the wire. The diffusive spin current flows from the region of a higher spin accumulation to the region of a lower spin accumulation. This means that spin polarized electrons (spin directed up (TIA assembly)) flow from back to front of the wire. In the opposite direction the spin-unpolarized electrons (spin directed in all directions (TIS assembly)) flow. The scattering probability of spin-up electrons into the right is higher. This is the reason for the charge accumulation at the left side of the wire.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Detailed explanation of this Figure is here

 



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. However, 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.

e.g. See Landau, Lifshitz. The Classical Theory of Fields

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)

misinterpretations (3): Since the SO interaction can "interact" only with the electron spin, which is a quantum mechanical object, but not with the orbital moment, the SO interaction has 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.

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. 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 nuclear.

 

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. The spin-orbit interaction is the effect of the electromagnetic field, but not of the quantum field of the electrons. The relativistic description of the electromagnetic field is enough to describe the SO interaction. It does not matter whether the quantum field of the electrons is described relativistically (Dirac equations) or non-relativistically (Schrödinger equation with inclusion of the spin). It is matter of convenience whether to use the Dirac equations or Schrödinger equation or Pauli equation for a description of the SO interaction. Note: evaluation of the spin-orbit interaction from the Dirac equations is a very beautiful and educational method.


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.

a perturbation method using 1/c2 as a small parameter.

 

 


Pauli equation & Spin-Orbit interaction

The Pauli equation is the wave equation describing interaction of electron spin with the electromagnetic field. Wiki page is here

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 applied magnetic field.

In the Pauli equation the gauge invariant A is used, which is the invariant for the Lorentz transformation, therefore the Pauli equation (Eq.(3.1))should automatically include the spin-orbit interaction without usage of HSO?

The gauge invariant A is the invariant for the Lorentz transformation, but the used wave function. Therefore, in contrast to the Dirac equation the Pauli equation is not an invariant for the Lorentz transformation. The Pauli equation is the equation valid in only one static coordinate system. When an electron moves in this static coordinate system, it experience the effective magnetic field of the spin-orbit interaction, which should be included into Pauli equation

The Pauli equation is the extended Schrödinger equation, where electron spin properties are included

The Pauli equation is between simpler, but approximate Schrödinger equation and full, but more complex Dirac equation.

The Pauli equation can be obtained from the Dirac equation.

The Pauli equation (classical form) 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

The Pauli equation, which include the Spin-orbit interaction, is


Merits and demerits of the use of the Hamiltonian approach for calculations of the Spin-Orbit interaction

This method has limitations and restrictions, which are discussed below

 

merit (1): The Spin-Orbit interaction can be easily included into a more general Hamiltonian as an additional term

merit (2): It allows to use a calculation based on a minimization of Hamiltonian. It does not require any assumption about orbital symmetry.

 

demerits:

demerit (1): It often approximates that the spin-orbit interaction is small. It is a very rough approximation and it is often incorrect.

 

demerit (2): a small calculation error may lead to a substantial error in the final result. It does not use the important features of the orbital symmetry for the calculation of SO.

 

 


 

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

M. T.Johnson et. al. Reports on Progress in Physics(1996) ; P.Bruno PRB (1989);

 

 

 

 

 

 

 

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