Dr. Vadym Zayets

Ferromagnetic resonance (FMR) is a spectroscopic technique in which a microwave is absorbed by a ferromagnetic material when its frequency matches the Larmor precession of the material's magnetization. The magnetization precession is driven by the oscillating magnetic field of the microwave. However, because the precession of the magnetization is not synchronized with the oscillating magnetic field, the precession angle remains small during FMR. Although the magnetic field exerts a torque on the magnetization, the lack of synchronization causes the torque to periodically reverse polarity, resulting in an overall small net torque. Consequently, FMR does not lead to magnetization reversal or even a significant increase in the precession angle, even under high-power microwave excitation.

In contrast, Parametric Magnetization Reversal achieves full synchronization between the phase and frequency of the oscillating magnetic field and the magnetization precession. This synchronization ensures that the torque induced by the oscillating field is consistently applied in the same direction, leading to a significantly larger precession angle, which can ultimately result in magnetization reversal. The synchronization is typically achieved through a feedback loop, often involving the magnetoresistance of the ferromagnetic material.

 

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Content

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() Mathematics of FMR

() Step 1: From Landau- Lifshitz equations to the equations of FMR dynamics (Rigorous solution for FMR)

() Step 2: Solution of FMR dynamics

()Step 3: Kittel effect

() Step 4: Reduction of Larmor frequency with increase of precession angle

() FMR vs. Optical Quantum Transition. Similarities and Differences.

() Only possible direction of the spin precession

() Reversal of precession direction under reversal of magnetic field

() Kittel effect

 

 

 

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Mathematics of FMR

Step 1: From Landau- Lifshitz equations to the equations of FMR dynamics (Rigorous solution for FMR)

It is my calculation, which I mostly finished in 2025.Jan and later slightly corrected in 2025 Oct.

Main Calculations: right circularly-polarized pump. FMR is pumped by a right circularly-polarized microwave, whose magnetic component rotates in the same direction as the spin precession. It is the case of the most efficient FMR pumping.

See pdf file here: FMR_CirclarRight.pdf

Linear polarization of pump. FMR is pumped by a linearly-polarized microwave. The pumping efficiency in this case is half that achieved with a right-circularly polarized pump. It is because a linearly polarized wave can be decomposed into two equal components of right- and left-circular polarization. Only the right-circularly polarized component effectively excites the magnetization precession, while the left-circularly polarized component has no influence on FMR.

See pdf file here: FMR_Linear.pdf

Left circularly-polarized pump. FMR is pumped by a left circularly-polarized microwave, whose magnetic component rotates in the opposite direction to the spin precession. FMR is not exciting in this case . See pdf file here:

See pdf file here: FMR_CirclarLeft.pdf

 

Kittel effect. The Kittel effect describes the influence of the demagnetization field and the field of the spin-orbit interaction on the Larmor precession of the magnetization.Kittel effect leads to (1) the change of the Larmor frequency; (2) the change of precession trajectory from circular to elliptical; (3) appearance of the 2nd and 3rd harmonics of the precession

It is my calculation, which I mostly finished in Dec. 2025.

See pdf file here: 9.KittelEffect.pdf

: Reduction of Larmor frequency with increase of precession angle

It is my calculation, which I

Quantum- mechanical description of FMR dynamics beyond LL equations. The

It is my calculation, which I mostly finished in Dec. 2025.

See pdf file here:

Quantum- mechanical description of the Kittel effect beyond LL equations.. The

It is my calculation, which I mostly finished in Dec. 2025.

See pdf file here:

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FMR vs. Optical Quantum Transition. Similarities and Differences.

Similar to the optical transition, Ferromagnetic resonance (FMR) describes the dynamics of spin transitions between energy levels: from the lower-energy state, where the spin is aligned parallel to the external magnetic field, to the higher-energy state, where the spin is antiparallel to the external field.

 

Similarities:

(similarity 1) Both optical transitions and ferromagnetic resonance (FMR) describe an electron transition between lower and higher energy levels.

(similarity 2) In both cases, the transition occurs only through the interaction with another particle, such as a photon.

Differences:

 

(difference 1) Field to Induce the Transition:

Optical transitions are induced by the oscillating electric field of an electromagnetic wave.

In contrast, FMR is induced by the oscillating magnetic field of an electromagnetic wave.

(difference 2) Nature of the Transition:

In an optical transition, an actual electron moves from one energy level to another. The electron’s symmetry changes, and the number of electrons occupying a given energy level is altered after the transition.

In contrast, during FMR or parametric magnetization reversal, individual electrons do not physically transition between energy levels. Instead, the lower- and higher-energy components of the entire spin system redistribute. Due to strong exchange interactions, all electron spins remain parallel to each other both during and after the transition. The outcome of the transition is a coherent precession of the entire spin system.

(difference 32) Dependence on Phase of electromagnetic wave:

Spin transitions in FMR and parametric reversal are highly dependent on phase matching between the spin precession and the electromagnetic wave. A 180-degree phase shift can completely reverse the nature of the transition: what was previously spin pumping (excitation from a lower to a higher energy level) becomes spin damping (relaxation from a higher to a lower energy level).

In contrast, optical transitions do not depend on the phase of the electromagnetic wave. The wave always induces a transition from the lower to the upper energy level, independently of phase of the electromagnetic wave. (For further details, see discussions on Rabi oscillations and stimulated optical transitions below.)

Optical transition	Spin transition (FMR)
induced by oscillating electric field	induced by oscillating magnetic field
The oscillating electric field interacts with the electron's charge, thereby modulating both its energy and wavefunction. As a result, an electron initially occupying the ground state E0​ acquires a small wavefunction component corresponding to the excited state E1​. This interaction facilitates the electron's transition from energy level E0​ to E1​.	Since spin is fundamentally associated with time-reversal symmetry (T-symmetry), a spin transition that reverses the spin direction requires an external field that itself breaks T-symmetry and directly interacts with the already broken T-symmetry of the electron. A magnetic field applied perpendicular to the spin meets these conditions, as it exerts a torque that tilts the spins, thereby enabling spin transitions.
The magnetic field of an electromagnetic wave is significantly weaker than its electric field, following the relation H=E/c. As a result, its influence on optical transitions is negligible compared to that of the electric field.	In contrast, an electric field cannot induce spin transitions because it does not directly interact with the electron spin.
click on image to enlarge it	click on image to enlarge it

 

 

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Only possible direction of the spin precession

(Key Fact): Unidirectionality of Spin Precession

Spin precession can occur only in one direction—specifically, clockwise with respect to the external magnetic field. This is a fundamental consequence of the time-space symmetry, which dictates the dynamic how the locally broken time-reversal symmetry (T-symmetry) of an electron, which is described by the electron spin, behaves within a globally broken T-symmetry, as defined by the external magnetic field.

Under no circumstances can the spin precess in the opposite (counterclockwise) direction.

Only possible direction of the spin precession
Only- possible clockwise precession of the spin Non-existing counterclockwise precession of the spin When viewed along the direction of the external magnetic field H—in this case, along the z-axis—the spin precession follows a clockwise trajectory. Under no circumstances can the spin precess in the opposite (counterclockwise) direction. Spin precession can occur only in one direction: clockwise with respect to the external magnetic field.
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(fact): Direction of Spin Precession in the Landau-Lifshitz Equation

In the Landau-Lifshitz equation, the direction of spin precession is uniquely determined by the negative sign of the first term on the left-hand side. This sign fixes the precession direction and is a fundamental aspect of the equation's description of spin dynamics.

 

 

 

 

 

 

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Reversal of precession direction under reversal of magnetic field

The direction of spin precession is unambiguously determined by the direction of the external magnetic field. When the magnetic field is reversed, the spin precession direction reverses accordingly.

Reversal of precession direction under reversal of magnetic field
(case 1) External magnetic field H is along z-axis. Small precession angle (case 2) H is opposite to z-axis. Small precession angle (case 3) H is opposite to z-axis. Large precession angle (larger than 90 degrees).
In all cases, the precession is clockwise direction with respect to the external magnetic field
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(Note about internal magnetic field):

The external magnetic field includes not only externally applied fields but also the magnetic field generated by the spin system itself. In a ferromagnet, this self-generated field is referred to as the internal magnetic field.

In the absence of any additional external magnetic field beyond the internal magnetic field, an interesting phenomenon occurs: when the precession angle exceeds 90 degrees, the polarity of the internal field reverses. This polarity reversal leads to a reversal of the spin precession direction (see below for further details).

 

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Only possible polarization of pumping microwave to exite the spin precession

 

(fact): Spin precession can be excited by a microwave whose magnetic field component rotates in the same direction as the natural direction of spin precession.

 

Mathematical proof of fact that there is only one possible polarization of microwave exciting the spin precession
Spin precession can be excited by a microwave, whose magnetic component rotates in the same direction as the direction of the spin precession.
(case 1) pumping torque by right circularly-polarized pump (case 2 pumping torque by left circularly-polarized pump (case 3) pumping torque by linear polarization of pump FMR is pumped by a right circularly-polarized microwave, whose magnetic component rotates in the same direction as the spin precession. FMR is pumped by a left circularly-polarized microwave, whose magnetic component rotates in the opposite direction to the spin precession. FMR is not exciting in this case . A linearly polarized wave can be decomposed into two equal components of right- and left-circular polarization. Only the right-circularly polarized component effectively excites the magnetization precession, while the left-circularly polarized component has no influence on FMR. It is the case of the most efficient FMR pumping. FMR is not exciting in this case . The pumping efficiency is half that achieved with a right-circularly polarized pump.       See pdf file here: FMR_CirclarRight.pdf See pdf file here: FMR_CirclarLeft.pdf See pdf file here: FMR_Linear.pdf
where the Larmor frequency ωL, which is the precession frequency, ω is the frequency of microwave pump, θ is the precession angle, φ is the precession phase and ΩMW is the precession pumping strength of the pumping microwave.

 

 

 

 

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Precession Pumping/Damping by Circulary- polarized electroMagnetic Field
magnetic field rotation: clockwise magnetic field rotation: counter clockwise in-phase: in- antiphase: damping precession pumping precession damping When the magnetic field of an electromagnetic wave rotates in the same direction as the spin precession, the precession is amplified, and the precession angle increases. When the magnetic field of an electromagnetic wave rotates in the same direction as the spin precession but is out of phase (by 180°), it induces damping, leading to a decrease in the precession angle. no effect on precession   When the magnetic field of an electromagnetic wave rotates in the opposite direction to the spin precession, the rotating magnetic field has no effect on the precession. The rotating magnetic field influences spin precession only when its rotation direction matches that of the spin precession. The rotating magnetic field has no effect on spin precession only when its rotation direction is opposite to rotation of the spin precession.
The robustness of the one possible rotation direction for spin precession serves as strong evidence for the fundamental nature of the symmetrical origin of spin.
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Precession Pumping/Damping by Linearly- polarized electroMagnetic Field
in-phase: in- antiphase: precession pumping precession damping
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4 possible configuration for precession pumping by Linearly- polarized electroMagnetic Field
configuration 1 configuration 2 configuration 3 configuration 4
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Mathematical description of magnetization precession, damping torque and pumping torque

The temporal evolution of magnetization components during precession around the z-axis is described as follows:

(1.10)

where m_x,m_y,m_z are component of magnetization M. The easy magnetic axis, and thus the precession axis, is aligned along the z-axis. The precession angle, θ, varies with time. In the case of ferromagnetic resonance (FMR), θ oscillates periodically around a relatively small average precession angle. In the case of parametric reversal, the average precession angle increases continuously until full magnetization reversal occurs. The precession frequency, ω_L, also varies with time. Typically, ω_Lis larger for smaller precession angles θ and decreases as θ increases. The precession phase, φ, varies with time. When φ is in phase with the oscillations of the magnetic field of the pumping electromagnetic wave, the precession angle θ increases. When φ is out of phase, it describes the change in the precession angle.

The torque describes the change in the precession angle. When the torque increases the precession angle, it is referred to as the pumping torque: . When the torque decreases the precession angle, it is referred to as the damping torque: . The damping torque drives the magnetization back toward its initial equilibrium direction, while the pumping torque drives it toward magnetization reversal.

The mathematical description of magnetization precession is given by a solution of the Landau- Lifshitz equation with oscillating magnetic field H_MW:

(1.11)

where m is magnetization, H is unchanged bias magnetic field, H_MW is magnetic component of the microwave ( magnetic field oscillating with frequency ω) and γ is the electron gyromagnetic ratio.

The solution of Landau- Lifshitz equation (Eq.1.11) in form of Eq.(1.10) gives the torque (change the precession angle θ) and the change of precession phase φ as

where

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Reversal of precession direction during magnetization reversal

 

 

 

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Precession Pumping vs. Precession Damping

 

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Kittel effect

The Kittel effect describes ferromagnetic resonance (FMR) under conditions where the bias magnetic field varies in response to the motion of the magnetization during one precession cycle. A common case occurs in ferromagnetic thin films where the equilibrium magnetization lies in the film plane. In this configuration, both the demagnetizing field and the magnetic field of spin–orbit interaction are modulated during each precession period. These fields reach their maximum values when the magnetization is tilted out of the plane and vanish when it lies entirely in-plane. The Kittel effect modifies the Larmor frequency and gives rise to higher-order harmonics of the FMR signal, particularly the second and third harmonics.

 

 

Kittel effect. Calculations by solving the Landau-Lifshitz equations. See pdf file here: 9.KittelEffect.pdf

It is my calculation, which I mostly finished in Dec. 2025.

 

Reason for the Kittel effect:

The Kittel effect occurs because, during spin precession, the total magnetic field acting on the spin is modulated by the precession angle. Two magnetic fields are angle dependent: the demagnetization field H_demag and the magnetic field of spin–orbit interaction H_so​. Both fields reach their maximum values when the magnetization is tilted toward the film normal and vanish when the magnetization lies entirely in the film plane.

How it affects spin precession:

During one precession period, the precession angle is not constant. It typically decreases when the magnetization is tilted toward the film normal and increases when the magnetization lies entirely in the film plane.

it leads to:

(Kittel effect consequence 1) : change of the Larmor frequency

Because of the Kittel effect, the dependence of the Larmor frequency on the external magnetic field becomes nonlinear.

(Kittel effect consequence 2) : change of precession trajectory from circular to elliptical

Precession angle decreases when the magnetization is tilted toward the film normal and increases when the magnetization lies entirely in the film plane.

(Kittel effect consequence 3) : appearance of the 2nd and 3rd harmonics of the precession

The complex precession trajectory causes the oscillations of system parameters to deviate from a purely harmonic behavior. This deviation leads to more complex nonlinear dynamics and results in the generation of second- and third-order harmonics of the precession frequency, in addition to the fundamental Larmor frequency.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elliptical precession trajectory due to Kittel effect
The Kittel effect occurs because, during spin precession, the total magnetic field acting on the spin is modulated by the precession angle. Two magnetic fields are angle dependent: the demagnetization field Hdemag and the magnetic field of spin–orbit interaction Hso​. Both fields reach their maximum values when the magnetization is tilted toward the film normal and vanish when the magnetization lies entirely in the film plane.
Precession geometry. The equilibrium magnetization is in-plane (the z- direction). The external magnetic field Hz​. is applied along the easy axis (the z- direction). The spin creates three types of the magnetic fields: (field 1: HM​.): the magnetic field along spin direction (field 2: Hdemag): Demagnetization field directed along film normal (the y-axis); (field 3: Hso) Magnetic field of spin orbit interaction directed along film normal (the y-axis) and opposite to Hdemag. All magnetic fields are modulated during presession Hdemag and Hso become largest when the magnetization is directed toward the film surface and vanish when magnetization is fully in-plane.
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