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Landau-Lifshitz Equations

Spin and Charge Transport

The Landau-Lifshitz Equation describes the precession of the magnetic moment around a magnetic field.


 


 

Electron in an Electrical field

In an electrical field an electron accelerates along the electrical field.

The electric field interacts with electron charge, but not spin.

The interaction is described by the Newton's second laws of motion and the Coulomb's law

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Electron in a Magnetic Field

In a magnetic field, there is a precession of the electron spin (the electron magnetic moment) around the magnetic field.

The magnetic field interacts with electron spin (magnetic moment), but not charge.

The interaction is described by the Landau -Lifshitz equation

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It should be noted that due the relativistic nature of the electromagnetic field when an electron moves

in a static electrical field it experiences an effective magnetic field. (See here)

in a static magnetic field it experiences an effective electrical field. (See here)

 

 

 


Landau-Lifshitz Equations:

where γ is is the electron gyromagnetic ratio, M is magnetization and Heff is the total magnetic field, which includes the external magnetic field, demagnetization field and effective magnetic field of spin-orbit interaction.

The Landau–Lifshitz–Gilbert equation is similar, but it describes differently the damping term:

 

 

Q. Is the Landau-Lifshitz Equation the equation of the classic mechanic or the Quantum mechanic??

A. Both.

The Landau-Lifshitz Equation describes the Larmor precession , which is the classical effect.

Also, The Landau-Lifshitz Equation describes oscillations between two wavefunction of the spinor, which is a Quantum-Mechanical effect.

note: Landau-Lifshitz Equation describes two very different processes:

(1) the first term describes spin precession. It is a basic quantum mechanical properties of the spin (the time-inverse symmetry)

(2)

 


Spin precession

Frequency of spin precession of a free electron around a magnetic field

Precession of spin (red arrow) around a magnetic field (blue arrow)

Quantum view on the spin precession

The frequency is between 1 and 100 GHz. The precession frequency does not depend on precession angle Spin can precess at the same angle for infinite time. An interaction with another particle is required in order to change the precession angle. In case of spin is parallel or antiparallel to magnetic field, the spin has a defined energy. For an energy between energies of these states, the electron wavefunction is a mixture of the spin-up and spin-down wavenctions. Such mixture describes electron precession of spin around magnetic field.
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Spin precession

The spin precesses around a magnetic field with the Larmor frequency:

The spin precesses counter-clockwise about the direction of the magnetic field.

The spin precession is a quantum-mechanical effect. It describes an electron state when electron energy is between the spin-up and spin-down energy. The wave function of electron during the precession is intermixture of the wave function spin-down and spin-up state.

The spin-down and spin-up states are the electron states when the electron spin is either parallel or anti parallel to the magnetic field.

 

note: The spin precession does not minimize the energy of an electron in the magnetic field

 

During spin precession the spin direction changes. Does it violate the spin conservation law?

A. No, the spin precession does not violate the spin conservation law. During the spin precession the electron spin does not change.

The electron spin can be either parallel to the applied magnetic field or antiparallel or between these direction. The electron wavefunction for the case when the spin is between parallel and antiparallel directions is a combination of the wavefunction of parallel and antiparallel directions. Such combination describes a state of the spin precession. The states, when electron spin is parallel, antiparallel or at angle to magnetic field and precess, are absolutely equal and describe eigen state of an electron. Even more, it is correct to say that there is a spin precession even for case spin is parallel and antiparallel to the magnetic field, but the precession radius is zero.

 

 

Fig.3 Precession of electron spin and the spin precession damping in a magnetic field. The red arrow represents electron spin. The grey arrow shows the direction of magnetic field. The data was calculated by solving Landau-Lifshiz equation (see here)

Spin precession and precession damping in a magnetic field. During the precession the spin aligns itself along the direction of the magnetic field. Click on the image to enlarge it.

Damping of the spin precession

The spin damping describes the process of alignment of electron spin along a magnetic field.

During the spin damping process the direction of electron spin is changing.

During the spin damping, the spin is not conserved!! Another particle with the spin should interact with the electron in order to conserve the spin during spin damping.

For example, it could be a photon (spin=1) or magnon or nuclears with non-zero spin.

Note:The spin damping is a long process. It takes many spin-precession periods during the spin damping until the spin is aligned along the magnetic field. The spin damping is the long process because the interaction with another particle with

 

Note: The mechanisms of spin damping are different for localized d-electrons and conduction electrons.

The reason: The different size. The localized d-electrons have a size about the size of atomic orbital ~ 1 nm. The conduction electrons have a size of ~3-1000 nm.

 

In case of conductive electrons, the spin damping is the collective process when the different contributions of many conduction electrons causes the spin damping. Many conduction electrons experience the spin damping together at the same.

In case of localized electrons, the spin damping is the individual process when each localized electron experiences the spin damping individually and independently from other localized electrons.

 

Mechanisms of the spin damping:

Localized d-electrons

(1) emitting of photon; (See here)

(2) interaction with magnons (spin waves)

(3)

Conduction sp-electrons

(1) emitting of photon (See here);

(2) dephasing of precession;

 

(3)

 

 

Q. Why conduction electrons do not support magnons and spin waves?

A. please see here

 

 

Spin-depended force, which electron experiences in a gradient of magnetic field

 

According to the Laws of Mechanics, a force acts on an object in the direction, in which the total energy of the is minimized. The electron energy in a magnetic field is S*H/2.

In a gradient of magnetic field, a force acts on an electron. The direction of this force depends on the electron spin.

When spin is parallel to the magnetic field, the force acts so that electron moves in the direction from a smaller to a larger magnetic field.

When spin is antiparallel to the magnetic field, the force acts so that electron moves in the direction from a larger to smaller magnetic field.

 

Note:

This force causes the repelling or attraction between two permanent magnets, which we may experience in everyday life.

 

 

 

 

 

 


 

 

The quantum-mechanical limitation on possible precession angles:

 

Transversal symmetry and spin precession

 


Spin damping mechanisms:

- emission of a photon

-interaction with a photon

 

 


Spin damping due to emission of a photon

 

 

Emission of electromagnetic waves due to alternating of electrical moment or magnetic moment

 

 

alternating of magnetic moment

When the spin (red arrow) precess around a magnetic field (blue arrow), micro waves (violet rings) are emitted and AC electric field is generated.

The alternating magnetic moment is the source of the radiation

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alternating of electrical moment (Microwave antenna)

Electromagnetic waves are emitted from a microwave antenna, because of alternating electrical dipole.

The alternating electrical moment is the source of the radiation

 

 

 

Note: It could be a pumping of the spin precession due to absorption of a photon. The electron magnetic resonance (EMR) and nuclear magnetic resonance (NMR) the nu are based on this effect.

Q. Only circular -polarized wave has spin. Is in EMR and NMR, circular polarized microwave radiation is used.

A. No. The electromagnetic wave, which are used in the EMR and NMR, is not polarized. The spin absorbed the required polarization. The wave of other polarization remains unabsorbed.

 

 

size dependence:

 

 

 

 

 

 

 

 

 

 



 


 

 

 

 

 

 

 

Magnetic moment induced by the orbital moment

In an atom in a gas, both the spin and electron orbital moment contribute to the atom magnetic moment. In crystal the orbital moment usually is ignored. It is only partially true. There can be a large orbital moment for both the localized and delocalized electrons in a crystal, but interaction of the orbital moment with magnetic field is different in the crystal than in a gas.

1) Orbital moment in a crystal does not precess around a magnetic field

2) There is a difference in energies for orbital moment directed along and in opposite to magnetic field (Zeeman effect).

3) Because of the orbital moment, a magnetic field breaks the time-inverse symmetry for the orbitals. The distribution of the orbitals with the orbital moment along and opposite to the magnetic field are different. Because of the breaking of the time-inverse symmetry, there could be a significant spin-orbit interaction.

 

Difference between the spin and the orbital moment

Breaking of the time-inverse symmetry in magnetic field due to an orbital moment

Due to the Lorentz electrical field the orbital distribution become different for electrons with opposite orbital moments.

Red arrow shows direction of the Lorentz force (electrical field). Blue arrow shows the direction of the magnetic field

Time-inverse symmetry is not broken

For electron rotation in any direction, there is equal probability for its rotation in the opposite direction

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Time-inverse symmetry is broken

In magnetic field the electron experience the Lorentz electrical field. For the electron, which rotates in the anticlockwise direction, this field is toward the nuclear and the orbit radius became shorter. For the electron, which rotates in the clockwise direction, this field is outward the nuclear and the orbit radius became longer.

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The spin and the orbital moment interact differently with a magnetic field

If the spin interacts only directly with the magnetic field, the orbital moment additionally interacts with relativistic electrical field (Lorentz electric field) induced by the magnetic field.

 

This field is different for the electrons, which rotate in clockwise and anticlockwise directions. Therefore, the orbital distribution becomes different for two electrons, which rotates in the opposite directions. The time-inverse symmetry is broken !!!

This breaking of the time inverse symmetry may cause a significant enhancement of the magnetic field due to the spin orbit interaction.

Note: For simplicity of understanding, the electron orbit is shown as a 2D circle. The 2D circle can represent a 3D spherical orbit deformed in one direction. This effect exists for any realistic orbit.

Note: Even though the figure shows the classical view of the electron orbit, the quantum mechanical treatment gives exactly the same result.

All electrons, including the inner-orbit electrons and the electrons of an inert gas, experiences.

This effect contributes substantially to diamagnetic properties of gases and solids.

 

 

 

 

 

 

 

Torque induced by the Lorentz force in a magnetic field for an electron with non-zero orbital moment

The electron orbit experiences a torque in a magnetic field applied perpendicularly to the orbital moment. This torque forces the orbital moment to turn to be parallel to the magnetic field. The origin of this torque is the Lorentz force.

Red arrow shows direction of the Lorentz force (electrical field). Blue arrow shows the direction of the magnetic field
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In a crystal the electron orbital can not be rotated, even though the electron may experience some orbital torque (See below). Only the electron spin can precess around a magnetic field

This effect also can break the time-inverse symmetry of the orbit.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Precession of orbital moment in a magnetic field

 

To see how the symmetry of the electron orbital is related to the orbital moment , click here

 

Direct relation between the shape of the electron orbit and the orbital moment

The precession of the orbital moment literally means the precession of electron orbit.

Green arrow shows the orbital moment, electron orbit is

Elliptical orbital

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p-like orbital

 

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Direction and value of orbital moment is directly related to the shape electron orbital. The orbital is asymmetrical in the direction of the orbital moment.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Precession of orbital moment in a magnetic field

The precession of the orbital moment literally means the precession of electron orbit.

Green arrow shows direction of the orbital moment. Blue arrow shows the direction of the magnetic field

Elliptical orbital

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p-like orbital

 

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The precession of the orbital moment literally means the precession of electron orbit as well.

 

The electron

 

 

 

 

 

 

 

 

 


 

Why there can not be a precession of the orbital moment in a crystal?

 

Why there can not be a precession of the orbital moment in a crystal?

The precession of the orbital moment literally means the precession of electron orbit.

Green arrow shows direction of the orbital moment. Blue arrow shows direction of the magnetic field

Electron orbitals in a solid

in crystal the electron orbitals interact with each other and this interaction defines the crystal structure. The electron orbitals can not move or rotates.

The electron orbitals are shown in pink, nuclears are shown in black.

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Imaginary case when an orbital moment of electrons in a crystal precess in a magnetic field

The electron orbits in crystal can not rotate. Otherwise, the bonding between neighbor atoms would alter and the crystal would collapse.

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The orbital moment of electrons in a solid can not precess around a magnetic field!!!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 

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