(2). Analytical solution of Landau-Lifshitz Equations

(3) Spin precession

(3a) Number of spin-up/spin-down electrons vs. precession angle

(4) Damping of the spin precession

() Magnetization reversal by spin injection. Spin- transfer torque.

() Dependence of Magnetization, Zeeman energy splitting, FMR frequency (precession frequency), internal magnetic field on the precession angle and the rate of the spin pumping

(5) Landau-Lifshitz Equations in a nanomagnet with PMA

(6) Spin damping due to emission of a photon

() Stable magnetization precession. Precession angle.

() Magnetization reversal. Magnetization reversal time.

(10) Precession of orbital moment in a magnetic field. Landau-Lifshitz Equations for orbital moment

Questions & Answers

() Relation between precession damping and exchange. Spin relaxation: individual for each spin (electron) or collective for all electrons (spins) simultaneously?

() spin wave & spin precession

() spin wave as a source of the spin damping

() strength of the exchange interaction

() spin dumping for an individual electron

() spin of one individual electrons vs. the spin as a component of the total spin

() magnetic domain & spin damping

.........

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 click here to enlarge	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 click here to enlarge

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 H_eff is the total magnetic field, which includes the external magnetic field, demagnetization field and effective magnetic field of spin-orbit interaction; λ is the damping coefficient.

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

Analytical solution of LL equation:

Analytical solution of LL equations Zayets arXiv:2104.13008 (2021) Appendix 1 TorqueSimple.pdfDampingTorqueCalculation.pdf

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 and is a general feature of the broken time-inverse symmetry.

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). It a quantum state, in which the electron spin is between its two equilibrium states: (equilibrium state 1) a lower- energy state, in which spin is along the external magnetic field and (equilibrium state 2) a higher- energy state, in which spin is opposite to the external magnetic field. The is a spin-conserving effect. As soon as an electron does not interact with another non-zero particle, theoretically the precision can be going forever (See note below). However, the the oscillations of magnetic moment, which are due to the spin precession, causes an emission of a photon and therefore the damping of the spin precession.

(note) The spin precession equally means the precession of the magnetic moment and the precession (the change) of the magnetic moment is process, which emits a circularly -polarized photon. As a result, the spin precession is always interacts with a non-zero-spin particle (a circularly -polarized photon). However, this interaction can be greatly suppressed in a small-size particle (e.g. spin of nucleus). It results of a very law spin damping (e.g. a low damping of NMR) (Details see below)

(2) the second term describes the spin damping. This is a process of interaction of the electron with a non-zero particle (e.g. a photon, a phonon, another electron, a magnon(spin wave)). The spin damping is a spin- non-conserving process. The electron spin is aligned along one direction and therefore it is changed.

(Part 9b): Spin dynamic: Landau-Lifshitz Eq vs Quantum mechanic
(short content 1:) Landau - Lifshitz (LL)  equitations vs. Quantum Mechanical description of spin dynamic
(short content 2:) Why the 1st precession term of LL eq. describes breaking of the time-inverse symmetry and why the 2nd precession damping term of LL eq. describes the quantum transition between spin-up and spin-down energy levels.
(short content 3:)   why the newly-introduced field-like torque and the reasons why it severely violates Laws of Quantum mechanics
Other parts of this video set is here
click on image to play it

Spin precession

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 Fig.1(a) The frequency is between 1 and 100 GHz. The precession frequency does not depend on precession angle Fig. 1(b) Spin can precess at the same angle for infinite time. An interaction with another particle is required in order to change the precession angle. Fig. 1(c) 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 wavefunction. Such mixture describes electron precession of spin around magnetic field.

click on image to enlarge it

Equation of spin precession

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

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

(origin of the spin precession )

see more details Zayets arXiv:2104.13008 (2021) Appendix 3

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

Number of spin-up/spin-down electrons vs. precession angle
Fig2 Two representations of the identical spin precession (left) as partially filled spin-up and spin-down energy levels; (right) as magnetization precession around magnetic field H:
click on image to enlarge it

(fact): The origin of the spin precession is the Zeeman splitting between energies of the spin-up and spin -down electrons

(calculation) Number of spin-up/spin-down electrons vs. precession angle

More details are here Spinor in magnetic field.pdf

When an electron in a magnetic field, the electron energy is different when the spin direction is along and opposite to the magnetic field H. The energy difference is called the Zeeman energy and calculated as:

.where g is the g- factor and μ_B is the Bohr magneton. The wavefunction of the spin-up and spin-down electrons can be expressed as

Wavefunctions of (a2) can be expressed using the spinor representation (See here) as

The spinor S of a quantum state, the spin direction of which is describes by angles θ and φ (See Fig) can be described as (See here)

Eq. (a3.) are spinors for for the spin-up state (θ=0⁰ ) and the spin-down state (θ=180⁰). Its comparison with a general case of Eq.(a4) gives the expression for spinor at an arbitrary angle θ as

The spinor Eq.(a5) describes a spin precession at precession angle q and with the Larmor frequency ω_L:

Number of spin-up/spin-down electrons vs. precession angle
Fig.3 Percentage of spin-up and spin-down electrons as a function of the precession angle
click on image to enlarge it

the wavefunction of the system of electrons with the spin S can be described as a sum of wavefunctions for spin-up and spin-down electrons as

expression (a6a) corresponds to spinor:

Comparison of Eqs. (a7) and (a5) gives the percentage of filling of the spin-up and spin-down energy level at precession angle θ as

(fact) There is always a spin precession in any multi-electron quantum state in a magnetic field, in which both the spin-up and spin-down energy states are partly filled.

(fact) The spin precession is an intrinsic state of the breaking of the time- inverse .

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.

(classical (incorrect) view on the spin precession)

E.g. see wiki on this topic

(quantum- mechanical nature of the spin precession) In the classical case, the magnetic field acts on the magnetic moment of spin and creates torque, which acts on the orbital moment of electron. The torque makes a precession of the orbital moment and therefore a spin precession. Such classical mechanism is also described by the Landau-Lifshitz Equations, but this mechanism is not correct. The spin is fully quantum- mechanical feature and the spin precession is a quantum mechanical process (See details here)

Damping of the spin precession

Fig.4 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-Lifshitz 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.

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

The Equation, which describes the spin damping, (LL equation without spin precession part):

in which the damping coefficient λ depends on the precession angle θ.

(fact) 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, a non-zero-spin particle, which participates in the spin damping, could be a photon (spin=1) or magnon or nuclei with non-zero spin.

Mechanisms of the precession damping:

For both the localized and conduction electrons the spin damping is collective process, into which simultaneously many electrons are involved.

Localized d-electrons

(note) All localized electrons are aligned to each other due to the strong exchange interaction. The spins of these electrons are spatially localized to the size of about one atom. As a result, the spins of neighbor electrons swing with respect to each other (similarly as balls connected by springs). E.g. the average swinging angle between to neighbor spins in Ni is 20 degrees at room temperature. (see here). That substantial swinging of all spins with respect to each other is described by an ocean of spin waves and magnons, which are main contributors to the spin damping for localized electrons.

(1) emitting of a circularly- polarized photon; (See here)

(2) interaction with magnons (spin waves)

Conduction sp-electrons

(note) All localized

(1) emitting of a circularly- polarized photon (See here);

(2) dephasing of precession;

Why the precession damping is different for the conduction and localized electrons?

Because of their different probability of scatterings. The conductions electrons are scattered frequently and the scattering of a localized electron is rather a random event. This why the precession damping mechanism are different for those two types of electrons.

Magnetization reversal by spin injection. Spin- transfer torque.

When spin-polarized electrons of the opposite spin direction to the magnetization direction of the nanomagnet are injected into the nanomagnet, they induce a magnetization precession in nanomagnet. The precession angle is

(fact: initial state & internal magnetic field & energy splitting) In an equilibrium, spins of all localized electrons are aligned along the magnetic easy axis. There is an internal magnetic field H_int in a nanomagnet, which holds the spins along the easy axis. There is an unoccupied higher- energy spin-down state. The difference between energy ΔE_FMR=g μ_B H_int is the Zeeman frequency, which corresponds to the FMR resonance

(fact: spin precession) When simultaneously there are states, which are occupied by a spin-down electron, and states, which are occupied by a spin-up electron, there is spin precession of the total spin. It is feature of a object having broken time-inverse symmetry. The spin describes the breaking of the time- inverse symmetry (See details below)

(fact: reduction of magnetization M under spin injection) When the spin-polarized electrons of spins of direction opposite to that of the total spin (magnetization), the total spin decreases, because the total spin equals to the difference between spins- up and spins-down

(fact: reduction of the internal magnetic field H_int under spin injection) Under a spin injection of spins of opposite direction to that of magnetization (the total spin), the magnetization decreases. Since the internal magnetic field H_int is proportional to the magnetization (See here). The H_int decreases following the decrease of the magnetization M.

(fact: reversal of direction of the internal magnetic field) When under the spin injection the number of spin-down localized electrons exceeds the number of spin-up electrons, the magnetization reverses its direction. Following the reversal of M, the internal magnetic field H_int is also reversed.

(fact: influence of the spin relaxation) The electrons of the upper- energy level relaxes to the lower- energy level. This process is called the spin relaxation or the precession damping. The larger the split ΔE_FMR between the energy levels of the opposite spin, the faster the spin relaxation is. Since split ΔE_FMR between decreases when the the internal magnetic field H_int is decreases, the spin relaxation or the precession damping are faster at initial moment of the spin injection and it decreases as the spin precession angle increases.

(fact: interaction between conduction and localized electrons) There are continuous scattering between pool of the conduction electrons and the localized electrons. Such a scattering substantially contributes to sp-d exchange interaction (See here)

Dependence of Magnetization, Zeeman energy splitting, FMR frequency (precession frequency), internal magnetic field on the precession angle and the rate of the spin pumping

Magnetization (total spin), FMR energy splitting (Zeeman splitting), internal magnetic field & precession angle vs spin pumping
initial state       all electron on spin-up states, small spin pumping / spin injection   spin pumping / spin injection

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

A. please see here

only-possible states of spin in a magnetic field
Temporally stable states Temporally unstable states spin along field spin-inactive states (filled by two electrons of opposite spin) When spin is aligned along magnetic field, the electron energy is smallest Two electrons of opposite spins, which occupy one quantum state, are spin-inactive, because the time-inverse symmetry is not broken for them. As a result, they do not interact with the magnetic field at all. Since the time-inverse symmetry is not broken, there is no spin direction and the cases of left figure and right figure are fully identical. There is no direction for this state. All spin-up states of a lower energy are occupied. All spin-down states of a higher energy are empty. The time-inverse symmetry is not broken for spin- inactive states. As a result, the spin-active states do not interact with the magnetic field H and the energy level is in a middle between spin-up and spin-down levels. spin opposite to magnetic field spin precession Emission of circularly- polarized photon, at first, creates the spin precession and finally the spin alignment along the magnetic field. The direction of the magnetic moment is alternating with the precession frequency. It results in emission of circularly- polarized photons and the spin damping. The spin damping aligns the spin along the direction of the magnetic field. All spin-up states of a lower energy are empty. All spin-down states of a higher energy are occupied. When a spin-down electron moves to spin-up level, a curricular- polarized photon is emitted and the spin precession starts.. Both the spin-up and spin-down levels are partly filled. When a spin-down electron moves to spin-up level, a curricular- polarized photon is emitted and spin precession angle decreases. This process is called the spin damping
Fig. 10 click on image to enlarge it

Note:Usually 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 it is not spin- conserved process and it requires the interaction of the electron with another non-zero-spin particle.

The atomic nucleus very weakly interacts with photons and electrons. As a result, precession damping of nucleus spin is very weak and therefore the peak of the nucleus magnetic resonance (NMR) is very sharp.

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.

Calculation of the damping torque

See all calculations details here: DampingTorqueCalculation.pdf TorqueSimple.pdf

The Landau-Lifshitz (LL) equations without the precession term can be written as

where M is the magnetization, H is total magnetic field applied to the magnetization, which is the sum of external magnetic field and the effective internal magnetic field (See here); λ is damping coefficient, which depends on the precession angle θ.

General solution of Eq.(2.1):

In the case the spin precession around the magnetic field directed in the direction, the damping torque can be calculated as

The integration of Eq.(2.9) gives the temporal evolution of the precession angle θ as

(Case 1): the damping constant λ and H_z are independent of the precession angle θ

It is the case when a large external magnetic field is applied to a nanomagnet and the the internal magnetic field can be ignored.

The damping torque can be calculated as

where H_θ=90 is the damping torque at precession angle θ=90 deg

The temporal evolution of the precession angle can be calculated as as

where θ₀ is the precession angle at the time moment t=0

Is it correct to calculate precession damping by solving LL equation without the precession term?

It is correct when the precession damping (or pumping) is independent of the precession frequency and of the precession phase. It is often the case.

As a prove, the damping torque, which is found from analytical solutions of LL equations with and without the precession term are identical for many cases (See below)

However, there are cases when the precession damping does depend on the precession frequency and of the precession phase. The parametric damping and pumping are such cases, in which the precession term in the LL equation should not be ignored. (See here and here and here)

Analytical solution of Landau-Lifshitz Equations

published in Zayets, JMMM (2014)

Detailed description of the solution steps read this pdf file Analytical solution of LL equations Zayets arXiv:2104.13008 (2021) Appendix 1 TorqueSimple.pdfDampingTorqueCalculation.pdf

(Approximation & simplification): precession frequency ω and damping constants λ are constants and independent of the precession angle θ (an incorrect assumption)

(Purpose 1) : To demonstrate that a simple analytical solution exists for a simplest case of Landau-Lifshitz Eq.

(Purpose 2) : To demonstrate that the simplest case when the precession frequency and damping constants are independent of precession angle, gives incorrect results, which do not match to the correct quantum description of the spin precession and which do not fit to the experimental observations.

The Landau-Lifshitz (LL) equations can be written as:

where m is an unit vector directed along the magnetization. |m|=1

The solution of LL equations (1.1) :

Temporal evolution of magnetization is described as

where θ is magnetization angle with respect to direction of the magnetic field H and it is calculated as:

and is the Larmor frequency, is the damping rate.

Solution of the simplest LL equations
(note): This the solution of LL equations under the incorrect assumption that precession frequency ω and damping constants λ are constants and independent of precession angle θ
Temporal evolution of inclination angle θ Temporal evolution of inclination angle θ (log scale) rate of change of the inclination angle θ vs θ Fig.24 (a) The angle θ between magnetization and magnetic field H. At time moment t=0, the magnetization is nearly anti parallel to H (θ=179 deg). After time 10 ωDt, the magnetization is nearly parallel to H (θ=0.66 deg) Fig. 24(b) The angle θ between magnetization and magnetic field H in logarithmic scale. After a slow change until 4 ωDt (θ>130 deg), the decreases nearly exponentially in time. Fig. 24 (c) The rate of angle change as a function of angle θ between magnetization and magnetic field H. The change is slow at beginning when M and H are nearly anti parallel. The change become fast when H and M perpendicular and again becomes slow when H and M are nearly parallel     (contradiction with a correct quantum description): In fact, the damping is largest for θ =0 deg and smallest for θ =90 deg. It is opposite what this the simplest LL Eq. predicts.
(important!) Since the simplest form of LL Eq. does not describe correctly the realistic and experimentally-observed spin precession, this simple analytical solution can be used to trace some general tendencies. Even though it should be done carefully, because in some cases it gives an incorrect tendency (e.g. Fig.24c)
click on image to enlarge it

(fact about a solution of the simplest LL equations): The solution of LL equations is divided into two independent solutions. The first solution describes the spin precession. The second solution describes the spin damping

Landau-Lifshitz Equations in a nanomagnet with Perpendicular Magnetic Anisotropy (PMA)

Read more about Perpendicular Magnetic Anisotropy (PMA) here

(fact) The Landau-Lifshitz equations in form of Eq.(1.1) is a very case of a ferromagnet of a spherical shape without any magnetic anisotropy. The most magnetic materials have a magnetic anisotropy. It means there is an axis, which is called the easy axis. When the magnetization is along the easy, the magnetic energy is smallest. There is an intrinsic magnetic field in a nanomagnet, which holds the magnetization along the easy axis. The strength of the intrinsic magnetic field depends on the magnetization angle. The intrinsic magnetic field is largest when the magnetization is along the easy axis and vanishes or becomes smaller, whent the magnetization direction is along the hard axis.