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Spin Precession. Precession Damping. LandauLifshitz Equations Spin and Charge TransportThe LandauLifshitz Equation describes the precession of the magnetic moment around a magnetic field and alignment of the spin along the magnetic field (spin damping)Content
click on the chapter for the shortcut(1). LandauLifshitz Equations(1a) (video:) Spin dynamic: LandauLifshitz Eq vs Quantum mechanic(2). Analytical solution of LandauLifshitz Equations(3) Spin precession(3a) Number of spinup/spindown 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) LandauLifshitz 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. LandauLifshitz Equations for orbital momentQuestions & 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
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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)
LandauLifshitz 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 spinorbit interaction; λ is the damping coefficient. The Landau–Lifshitz–Gilbert equation is similar, but it describes differently the damping term:
Analitical solution of LL equation:Analytical solution of LL equations Zayets arXiv:2104.13008 (2021) Appendix 1 TorqueSimple.pdfDampingTorqueCalculation.pdfQ. Is the LandauLifshitz Equation the equation of the classic mechanic or the Quantum mechanic??A. Both. The LandauLifshitz Equation describes the Larmor precession , which is the classical effect. Also, The LandauLifshitz Equation describes oscillations between two wavefunction of the spinor, which is a QuantumMechanical effect and is a general feature of the broken timeinverse symmetry. note: LandauLifshitz Equation describes two very different processes: (1) the first term describes spin precession. It is a basic quantum mechanical properties of the spin (the timeinverse 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 spinconserving effect. As soon as an electron does not interact with another nonzero 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 nonzerospin particle (a circularly polarized photon). However, this interaction can be greatly suppressed in a smallsize particle (e.g. spin nuclear). 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 nonzero particle (e.g. a photon, a phonon, another electron, a magnon(spin wave)). The spin damping is a spin nonconserving process. The electron spin is aligned along one direction and therefore it is changed.
Spin precession
Equation of spin precession (fact) The spin precesses around a magnetic field with the Larmor frequency: (fact) The spin precesses counterclockwise about the direction of the magnetic field.
(origin of the spin precession ) see more details Zayets arXiv:2104.13008 (2021) Appendix 3The spin precession is a quantummechanical effect. It describes an electron state when electron energy is between the spinup and spindown energy. The wave function of electron during the precession is intermixture of the wave function spindown and spinup state. The spindown and spinup 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
(fact): The origin of the spin precession is the Zeeman splitting between energies of the spinup and spin down electrons(calculation) Number of spinup/spindown electrons vs. precession angle More details are here Spinor in magnetic field.pdfWhen 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 spinup and spindown 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 spinup state (θ=0^{0} ) and the spindown state (θ=180^{0}). 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}:
the wavefunction of the system of electrons with the spin S can be described as a sum of wavefunctions for spinup and spindown electrons as expression (a6a) corresponds to spinor: Comparison of Eqs. (a7) and (a5) gives the percentage of filling of the spinup and spindown energy level at precession angle θ as
(fact) There is always a spin precession in any multielectron quantum state in a magnetic field, in which both the spinup and spindown 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 LandauLifshitz 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
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 nonzerospin particle, which participates in the spin damping, could be a photon (spin=1) or magnon or nuclears with nonzero 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 delectrons (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 spelectrons (note) All localized (1) emitting of a circularly polarized photon (See here); (2) dephasing of precession; Why the pressesion damping is different for the conduction and localized electrons? Because of their different prbability of scattirings. 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 spinpolarized electrons of the opposite spin direction to the magnetization direction of the nanomagnet are injected into the nanomagent, they induce a magentization 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 spindown 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 spindown electron, and states, which are occupied by a spinup electron, there is spin precession of the total spin. It is feature of a object having broken timeinverse symmetry. The spin describes the breaking of the time inverse symmetry (See details below) (fact: reduction of magnetization M under spin injection) When the spinpolarized 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 spinsdown (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 spindown localized electrons exceeds the number of spinup 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 spd 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
Q. Why conduction electrons do not support magnons and spin waves?
Note:Usually the spin damping is a long process. It takes many spinprecession 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 nonzerospin particle. The atomic nuclear very weakly interacts with photons and electrons. As a result, precession damping of nuclear spin is very weak and therefore the peak of the nuclear magnetic resonance (NMR) is very sharp. Note: The mechanisms of spin damping are different for localized delectrons and conduction electrons.The reason: The different size. The localized delectrons have a size about the size of atomic orbital ~ 1 nm. The conduction electrons have a size of ~31000 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 torqueSee all calculations details here: DampingTorqueCalculation.pdf TorqueSimple.pdfThe LandauLifshitz (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 θIt 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 θ_{0} 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 LandauLifshitz Equationspublished 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.pdfThe LandauLifshitz (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.
(fact 1about solution of 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 (alignment
LandauLifshitz Equations in a nanomagnet with Perpendicular Magnetic Anisotropy (PMA)Read more about Perpendicular Magnetic Anisotropy (PMA) here(fact) The LandauLifshitz 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. (fact) In the most magnetic materials, the PMA is due to the spinorbit interaction. The feature of the spinorbit interaction is that it induces magnetic field H_{SO}, which direction is perpendicular to the nanomagnet interface. The H_{SO} is the largest when the magnetization is perpendicular to the interface and it smallest (absent) when the magnetization is perpendicular to interface. Dependence of the Larmor frequency ω_{L} and damping frequency ω_{D} on the FMR precession angleIn a nanomagnet with the perpendicular magnetic anisotropy (PMA), there is a magnetic field directed perpendicularly to its interface. The magnetic field due to the spin orbit interaction contributes substantially to this intrinsic magnetic field. The feature of SO interaction is that the SO dependents substantially on the magnetization direction. As a result, when the magnetization direction turns out of the perpendicularlyto interface direction, the intrinsic magnetic field decreases. It affects the FMR resonance. As precession angle increases, both the FMR frequency and damping frequency decrease.
Precession pumping
Spin damping due to emission of a photon
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:
Stable magnetization precession. Precession angle.
(fact) Stable magnetization precession occurs at precession angle at which the precession damping is equal to the precession pumping.
(note) Usually precession damping increases and precession pumping decreases when the precession angle increases. (See here and here) At a smaller precession angle, the precession pumping exceeds the precession damping and the precession angle increase. However, at a larger precession angle, the precession damping becomes equal to the precession pumping and the magnetization precession is stabilized.
Magnetization reversal. Magnetization reversal time.
(fact) Magnetization reversal occurs, when the precession pumping is larger than the precession damping at any precession angle θ.
Condition for the magnetization reversal: precession pumping torque is larger than the precession damping torque torque at any precession angle θ.
The total torque should be always positive: Magnetization reversal timeFrom Eq.(6.3) the time, during which the magnetization is reversed, is calculated as: where the initial precession angle θ= 0 deg and at the final (reversed) precession angle is θ= 180 deg.
(fact) The stronger pumping torque and the weaker damping make the magnetization reversal faster.
See detailed calculations of the magnetization reversal time for each mechanism of pumping and damping torque here DampingTorqueCalculation.pdf
The quantummechanical limitation on possible precession angles:
Transversal symmetry and spin precession
Spin damping mechanisms:  emission of a photon interaction with a photon
Magnetic moment induced by the orbital momentIn 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 timeinverse 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 timeinverse symmetry, there could be a significant spinorbit interaction.
Difference between the spin and the orbital moment
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 counterclockwise directions. Therefore, the orbital distribution becomes different for two electrons, which rotates in the opposite directions. The timeinverse 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 innerorbit electrons and the electrons of an inert gas, experiences. This effect contributes substantially to diamagnetic properties of gases and solids.
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 timeinverse symmetry of the orbit.
Precession of orbital moment in a magnetic field. LL equation for orbital moment
To see how the symmetry of the electron orbital is related to the orbital moment , click here
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.
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?
The orbital moment of electrons in a solid can not precess around a magnetic field!!!
Questions & AnswersRelation between precession damping and exchange. Spin relaxation: individual or collective? ( from Sky) Q. I have some confusion about the pressesion damping for the localized electrons. There are two statements in this subject: (1)"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)." and (2) "In case of localized electrons, the spin damping is the individual process when each electron experiences the spin damping individually and independently from other localized electrons." In my opinion, the statement (1) means that the pressesion damping of localized electrons is strongly connected with each other, which seems contradict with statement (2). The spin precession and the precession damping are a collective effect, when the directions of all spins are parallel all the time. It is because the exchange interaction between spins is very strong. Exceptions are the spin waves and domain walls. The exchange interaction between localized neighbor electrons is very strong, but is not infinitely strong. As a result, a slight misalignment between neighboring spins are possible (spin waves). Also, when some strength is accumulated over many spins (over millions or billions of spins) the parallel alignment between neighboring spins can be broken (a domain wall). The spin damping is a collective process of the total spin. There is no individual spin damping. The spindown to spinup quantum transition of one electron means only change of one component of the total spin and is not related to individual spin of one localized electron. Since the exchange interaction is not infinitely strong, a slight misalignment between two localized neighbor electrons is possible. Due to such a tiny misalignment, a spin wave exists in a ferromagnetic material. A spin wave is a mixture of an electromagnetic wave and spin precession. The magnetic component of an electromagnetic wave is slightly different at a position of each localized electron. As a result, the spin precession is slightly different between neighboring localized electrons. As you said, the spins of neighboring electrons slightly swing with respect to each other. Even though the spin misalignment between neighboring electrons is very small and the spins of neighboring electrons are still nearly parallel, the misalignment is accumulated with a distance and can be substantial for the electrons separated by a long distance. (spin wave as a source of the spin damping) The spin wave is a particle with a nonzero spin. It interacts with the total spin of the nanomagnet causing an electron transition from the higher energy spindown energy level to the lower energy spinup energy level. This process is called the spin damping and this quantum transition is fully equivalent to the classical precession damping. It is important that the spin wave interacts with the total spin of the whole nanomagnet, but not with individual spin of localized electrons. The interaction is the most efficient when the size of the nanomagnet or size of a magnetic domain matches the wavelength of the spin wave. (strength of the exchange interaction) The strength of the effective exchange magnetic field is possible to estimate from Curie temperature (see my web page on exchange interaction). The magnetic field of the exchange interaction is rather high. It is about 1900 Tesla for Co and 900 Tesla for Ni. For example, a large superconducting magnet produces a magnetic field of about 2040 Tesla. Because of the high strength of the exchange interaction, it is nearly impossible that the spin of one individual localized electron is reversed with respect to the spin direction of all neighboring electrons. Only many electrons can reverse their spins simultaneously and coherently ( a magnetic domain) (spin dumping for an individual electron) All individual localized electrons are so strongly glued to each other by the exchange interaction, they behave as one quantum object. Spin of a localized electron is aligned strongly to be parallel to the spins of its neighboring localized electrons. The total spin behaves as one quantuum object. It precesses as one object or tilts its direction as one object and interacts with spin waves (magnons) and circularly polarized photons as one object. (spin of one individual electrons vs. the spin as a component of the total spin) Even when there is a quantum transition of an electron from the spindown to spinup energy level (spin damping), it does not mean that one localized electron becomes spinup in the surroundings of neighboring spin down electrons. The spins of all neighboring electrons remain parallel (nearly) all the time. The meaning of the transition of one electron from the spindown to spinup energy level means that one component of the total spin is changed and, as a result, the precession angle of the total spin becomes larger. All the time the spin of all localized electrons are glued to each other. All spins precess coherently and are always parallel to each other. (magnetic domain & spin damping) The strong exchange interaction can be broken at a boundary between magnetic domains. Some effects can accumulate for a larger number of localized electrons. When the number of localized electrons reaches some critical number, a domain wall is formed. The behavior of two neighbor domains may be rather independent. E.g., the magnetic dipole interaction makes magnetization of neighbor domains to be antiparallel. Similarly, the spin precession of the neighbor domains can be at slightly different frequency and the precession angle.

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