Abstract:

In this method, the spin precession (e.g. Ferromagnetic resonance (FMR) ) is excited by illumination of sample by RF radiation. At resonance conditions, there is a magnetization precession and the magnetization direction changes with the RF frequency. It modulates the magneto- transport feature (e.g. Hall angle or magneto- resistance). In the case, when an additional magneto- transport parameter (for example, current) is modulated by the RF radiation, the frequency beating gives DC voltage, which is proportional to RF power. In this method, the features of the magneto-transport are evaluated from the DC voltage.

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Content

1.

 

() Ferromagnetic

6. Explaination video

 

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(main idea): The sample is illuminated by microwave at frequency of the Ferromagnetic resonance (FMR). The microwave excites the spin precession and additionally the microwave excites the electrical current. Since the Hall voltage proportional to both the spin direction and the current, which are both modulated by microwave, there are frequency beating of these two contribution. As a result, there is a DC component of the Hall voltage which is measured.

 

(merit of the method): It is possible to separate a studied Hall contribution from other contribution and measure a really weak Hall effect. For example, it is possible to measure a very weak ISHE effect in a paramagnetic metal.

 

 

 

 

 

 

 

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(Method 1). Interaction of RF current with magnetization

Measurement of 1st order magneto-transport effects: AHE and ISHE

AHE: Anomalous Hall effect; ISHE: Inverse Spin Hall effect

(what is modulated by RF):spin direction; electrical current

(where it is used) a non-magnetic material in order to detect the spin of spin-polarized conduction electrons,, which are externally created.

RF measurement of Inverse-Spin Hall effect (ISHE) in a nonmagnetic material
Experimental setup Measurement of DC Hall voltage       In an equilibrium, conduction electrons in InAs is not spin-polarized. When an external magnetic field is applied, the spins of conduction electrons are aligned along the magnetic field and the conduction electrons becomes spin- polarized. Under illumination by microwave radiation, the spins starts to precess around the magnetic field, which makes the Hall current. The microwave radiation excites the electrical current jz along z- direction. Additionally, the The microwave radiation excites the spin precession for the conduction electrons. Due to the spin precession the x component Sx of the spin at RF frequency. The Hall current flows along y- direction perpendicularly to Sx and jz. The frequency beating between Sx and jz creates a DC Hall current, which is detected by the DC nano- voltmeter. InAs Ge DC Hall voltage VDC vs applied magnetic field measured in bulk InAs. Temperature1.3o K. Microwave frequency is 9.2 GHz DC Hall voltage VDC (dispersion derivative) vs applied magnetic field measured in bulk Ge. Temperature 2.2o K. Microwave frequency is 9.3 GHz J. N. Chazalviel and I. Solomon, “Experimental evidence of the anomalous hall effect in a nonmagnetic semiconductor,” Phys. Rev. Lett., vol. 29, no. 25, pp. 1676–1679, 1972. J. N. Chazalviel, “Spin-dependent Hall effect in semiconductors,” Phys. Rev. B, vol. 11, no. 10, pp. 3918–3934, 1975.   (experimental setup) A nonmagnetic InAs is illuminated by RF microwave radiation (blue/ green wave).The microwave radiation is produced by the RF generator (left-upper box) and the microwave antenna. The microwave RF radiation produces a Hall voltage, which is measured by the nano- voltmeter. Two metallic contact on the bulk InAs are used to measure DC Hall voltage. The FMR resonance is clearly detected in non-magnetic materials. It can be only the case when an external magnetic field makes the conduction electrons spin- polarized and the microwave radiation creates a spin precession.   (note) Sample are usually measured inside a microwave resonator. The FMR frequency depends on material parameters (e.g. the g- factor) and the external magnetic field. In a common FMR measurements, the microwave frequency is fixed and the magnetic field is scanned until the resonance condition is found.
click on image to enlarge it

 

(How does it work?)Microwave RF radiation excites RF current in the sample. When RF frequency is close to the frequency of ferromagnetic resonance (FMR), the microwave excites the spin precession of both the localized d- electrons and spin-polarized- conduction electrons. The current experience the AHE effect due to the spin of localized electrons and the ISHE due to the spin of conduction electrons. Due to frequency beating between the RF- modulated spin and the RF - modulated current, there id a DC component of Hall voltage, which is measured.

 

 

 

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Geometry of measurement

(requirement 1): The directions of Hall probe, current and magnetization vibration should all be perpendicular each other.

Since the magnetization does not change its magnitude but only direction, its parallel M_|| and perpendicular M_⊥ components are:

where

α is magnetization angle.

Therefore, the change of the perpendicular component M_⊥ of the magnetization is only contribute to the measurement.

(only possible geometry:) The Hall bar is perpendicular to current. The magnetization either along the current or the Hall bar or in their common plane

E.g. current j_|| is along x-direction, the Hall probe is along the y-direction and is M_⊥along z-direction. It is possible only when the magnetization is along x- or y- directions

 

 

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Importance of phase matching between magnetization vibrations and current

For this measurement it is important to match the phase of current to the magnetization vibration. For example, if current is along x-axis and change of magnetization is along z- axis and the Hall probe is along y- axis:

where φ is phase difference difference between oscillations of the magnetization and current.

The Hall angle is calculated as

where α is the Hall angle of the material

The zero harmonic is calculated

The zero harmonic has maximum and is positive at φ=0 deg; has maximum and is negative at φ=180 deg; and it is zero at φ=90 and 270 deg;

 

 

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(Method 2). Interaction of spins of localized d- electrons and conduction electrons

Measurement of 2nd order magneto-transport effects: Planar Hall effect, AMR, spin-dependent conductivity

Planar Hall effect & Anisotropic magneto-resistance (AMR) are described here; Spin- dependent conductivity is described here

(what is modulated by RF):spins of localized d- electrons; spins of spin-polarized conduction electrons

(where it is used) ferromagnetic materials. Only in a ferromagnetic material both the total spin of conduction electrons and the total spin of localized electrons are non-zero.

(How does it work?)Microwave RF radiation excites a spin precession. The frequency beating of the RF- oscillating components of spins of localized d- electrons and conduction electrons gives the a DC current, which is measured.

 

Let us consider the case when a DC current flows along the x-direction and the Hall probe is along the y-direction. The effects, which can be detected by the RF technique, depends on the direction of the magnetic field with the direction of the electrical current.

(case 1) Magnetic field is along the z- direction. Magnetic field is perpendicular to the current and the Hall bar

For spin S directed along the z-axis, the spin precession is describes as

where S is spin component in the xy-plane.

According to he AMR/ Planar Hall effect (Kondo type), the spin creates a current (See here), which is described as

where S is the total spin of localized d- electrons and s is the total spin of the spin- polarized conduction electrons.

Planar Hall effect:

In the case when the a Hall probe is in the y- direction, the measured Hall voltage is created by the y- component of the current j_mag, which is calculated as

Substitution of Eq.(3.4) into (3.6) gives

or

The DC- component (the non- oscillating component) of Hall current is calculated as

(note): There is a DC Hall voltage when there is a non-zero time leg between the precession of conduction and localized electrons.

AMR effect:

The effect of the Anisotropic Magneto Resistance (AMR) describes the magnetically- created current j_mag (Eq. 3.5) flowing along the bias current j_x. It can be calculated as

Substitution of Eq.(3.4) into (3.10) gives

or

The DC- component (the non- oscillating component) of AMR current is calculated as

 

(case 2) Magnetic field is along the y-direction. Magnetic field is perpendicular to the current and is along the the Hall bar

For spin S directed along the y-axis, the spin precession is describes as

Planar Hall effect:

There is no DC current of the Planar Hall effect for this geometry

See Eq.(3.6)

 

AMR effect:

From Eqs (3.10)(3.15) The DC- component (the non- oscillating component) of AMR current is calculated as

 

(case 3) Magnetic field is along the x-direction.Magnetic field is along to the current and is perpendicular the the Hall bar

For spin S directed along the x-axis, the spin precession is describes as

 

Planar Hall effect:

There is no DC current of the Planar Hall effect for this geometry

See Eq.(3.6)

 

AMR effect:

There is no DC current for the AMR effect for this geometry

See Eq.(3.10)

 

 

 

 

 

 

 

 

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Measurement of time leg between spin precession of conduction and localized d- electrons

(in an equilibrium) The spin directions of localized d-electrons and spin-polarized conduction electron are the same (See here). In a ferromagnetic metal, the conduction electrons are spin- polarized, because of the spin pumping. The electrons scattering are trying to disalign spins of conduction electrons. However, this disalignment mechanism (which is called the spin relaxation) is balanced by the spin pumping (See here). The spin pumping is the mechanism of alignment of spins of conduction electrons along the spins of the localized electrons due to the sp-d exchange interaction and the electron scatterings). Naturally, the spins of localized and spin-polarized conduction electrons are aligned in one direction.

(spin rotation) The spin of conduction electrons aligns to the spin of localized electron fast, but not infinitely fast. When the spin direction of localized

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Frequency Dependence

 

(influence 1) The close the microwave frequency to the FMR frequency, the more efficient the excitation of precession of spins of localized d-electrons and therefore the angle of spin precession becomes larger.

(influence 2) The close the microwave frequency to the FMR frequency, the large the time leg φ between precession of localized d- electrons and conduction electrons. It enhances 2nd order magneto-transport effects, for example, the Planar Hall effect (See above)

 

 

 

 

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Spin- dependent conductivity

 

 

 

 

 

 

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Excitation of ferromagnetic resonance (FMR)

The ferromagnetic resonance (FMR) is the spin precesion under illumination of a nanomagnet by microwave at frequence close to the Larmor frequency.

(note)

 

 

 

 

 

 

 

 

 

 

 

 

 

methods to excite FMR: (method 1) Quantum-mechanical excitation

By circulary polarized microwave.

In this case the spin is transfered from a photon to the manetization of the nanomagnet.

 

methods to excite FMR: (method 2) Parametric excitation by alternating magnetic field.

 

(polarization of RF field) FMR is excired by a linerally- polarized electromagnetic wave.

(geometry of FMR experiment) Magnetic field of the electromagnetic wave is pependicular to the easy axis of smaple (nanomagnet). It means that the linear polarization of microwave is parallel of the easy axis or the propagation direction of the microwave is

 

 

In this case an alternating magnetic of microwave dis align mgnetization spin with respect to the internal (external) magnetic field

 

methods to excite FMR: (method 3) Parametric excitation by alternating magnetic torque

By electrical current.

In this case an alternating magnetic of microwave dis align mgnetization spin with respect to the internal (external) magnetic field

 

 

 

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Resonance self excitatation of FMR. Torque-free magnetization oscilations and torque-free magnetization reversal.

 

 

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Off-resonance excitation

When an external magnetic field is applied to nanomagnet, how much the magnetization of nanomagnet is changed?

When a very large magnetic field is applied, the magnetization of the nanomagnet is aligned along the field.

However, when the magnetic field is weak or moderate, the magnetization is aligned between directions of the easy axis of the nanomagnet and the magnetic field

Each nanomagnet has a magnetic anisotropy and therefore there are the easy and hard axis for any nanomagnet. In absence of an external magnetic field, the magnetization is aligned along the easy axis. In order to align the magnetization along the hard axis, an external magnetic field field should be applied along the hard axis and therefore perpendicularly to the easy axis. The minimum field, at which the magnetization is aligned along the hard axis, is called the anisotropy field H_ani. (see here and here). The feature of the magnetic anisotropy is that the magnetization component M_x along hard axis is linearly proportional to the component of applied magnetic field H_x along this direction (see here)

Then, the angle α of magnetization with respect to its easy axis is calculated as

The anisotropy field H_ani also depends on the component of magnetic filed H_z applied along easy axis (see here). The dependence can be approximated roughly as linear.

At very large magnetic field H_z, γ approaches 1 and the magnetization is aligned along the external magnetic field. Substitution of Eq.(4.3) into Eq.(4.2) gives the magnetization angle as

 

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Ferromagnetic resonance (FMR)

 

 

 

FMR in nanomagnet with perpendicular magnetic anisotropy. Kittel formula.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Video

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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