Dr. Vadym Zayets

(1) Measurement of coefficient of spin- orbit interaction k_SO and anisotropy field H_ani in a nanomagnet.

(2) Measurement of strength of spin-orbit interaction

(3) measurement of Anisotropy field.

() Questions & Comments

(7) about dependency of spin-orbit interaction on an external magnetic field.

.........

Measurement of strength of Perpendicular Magnetic Anisotropy (PMA)
(measurement the strength of PMA) FeB nanomagnet is fabricated on top of Ta nanowire. A pair of Hall probes is aligned to the nanomagnet. When an electrical current through the nanomagnet, there is a Hall voltage, which is linearly proportional to to perpendicular component of the magnetization. The nanomagnet magnetization is shown as the green ball with arrow. The equilibrium magnetization direction is perpendicularly to plane. When an external in-plane magnetic field H|| is applied, the magnetization turns towards H||. The field, at which the magnetization turns fully in-plane, is called the anisotropy field Hani and it is linearly proportional to the PMA energy. The left figure shows measured in-plane component of magnetization M|| vs H||. The dependence is linear. From its slope the Hani is evaluated. To evaluate features of the PMA, an additional external perpendicular magnetic field Hz is applied during a scan of H||. The right figure shows Hani vs Hz . The Hani increases under Hz.
( Hani as a measure of PMA energy) Due to the PMA, the lowest - energy states (easy axis) is when the magnetization is perpendicularly to the plane, and the highest - energy states (hard axis) when the magnetization is in- plane. The in- plane magnetic field reduces the in-plane energy. When the in-plane energy becomes smaller than perpendicularly -to plane energy, the magnetization turns fully in-plane.
(additional measurement: measurement of SOT effect) The line of measured dependence M|| vs H|| (left figure) does not through axis origin. It indicates that additionally to the external in-plane magnetic field ||, there is an intrinsic in-plane magnetic field Hoff. The Hoff is proportional to electrical current. The Hoff originates by the effect of Spin- Orbit Torque (SOT). (See details here).
Note. I have developed this measurement method in 2019-2020
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See detailed calculations of anisotropy field and PMA energy here CalculationHanisPMA.pdf

Perpendicular Magnetic Anisotropy describes the magnetic anisotropy, in which the direction of easy axes is perpendicular to the film surface and the direction of the hard axis is in-plane of the film

Q. What is the reason the magnetization of the nanomagnet becomes perpendicular to the plane?

A. The spin-orbit interaction

(fact) The PMA or the spin-orbit interaction can be originated either at a nanomagnet interface or in the bulk of the nanomagnet. The interface-originated PMA is stronger and more interesting, because engineering of only the one atomic layer at interface, a huge PMA and unique properties can be achieved.

Q. Why is the nanomagnet with PMA important for applications?

A. It is because a huge energy is accumulated in only one atomic layer at interface. This allows to fabricated experimentally small nano- sized devices with an unique functionality.

When the volume of an object is reduced, the object energy reduces proportionally to the volume. In order for the object to function as a device (a memory cell, a sensor cell etc), the magnetic energy between its two stable states should be sufficiently larger than the thermal energy kT. Otherwise, it will be continued random thermal switching between two states and the object cannot function as a device.

Key properties of the spin-orbit interaction, which lead to the existence of PMA

(key property 1): The magnetic field of the spin- orbit interaction H_SO is proportional to an externally- applied magnetic field

The SO by itself cannot break the time inverse symmetry and, therefore, to manifest itself. The H_SO exists only when there is an external magnetic field.

(key property 2): The spin-orbit interaction substantially increases (changes) when the electron orbital is deformed

When atomic orbital is deformed, the electron becomes closer to the atomic nucleus, where the electrical field is much stronger. As a result, the SO becomes larger.

Key properties of the spin-orbit interaction, which lead to the existence of PMA
(key property 1): The magnetic field of the spin- orbit interaction HSO is proportional to an externally- applied magnetic field Hext
In absence of external magnetic field   In presence of external magnetic field no SO magnetic field. HSO=0   strong SO magnetic field. HSO ≠0   The electron experiences the magnetic field HSO of the spin-orbit interaction, when it is orbiting the nucleus. However, the magnetic field HSO is opposite, when the electrons moves in clockwise and counterclockwise directions. As a result, in total the electron experiences no spin- orbit interaction.   In the presence of an external magnetic field Hext, the electron experience the Lorentz force FLor, which is opposite for the clockwise and anti- clockwise paths of the electron orbital. The opposite Lorentz forth makes clockwise and anti- clockwise paths of electron orbital to become different. Since during the counter-clockwise movement, the electron path becomes closer to the nucleus, the electron experiences a larger the electrical field of nucleus and, as result, a larger magnetic field HSO of the spin-orbit interaction. In contrast, during the clockwise movement, the electron path becomes further from the nucleus, the electron experiences a smaller the electrical field of nucleus and, as result, a smaller magnetic field HSO of the spin-orbit interaction. Since the HSO becomes unequal for opposite rotation directions, in total, the electron experiences a non-zero magnetic field HSO of the spin-orbit interaction. Any orbital of a localized electron consists of orbital paths of clockwise and counter-clockwise rotations. The time-reversal symmetry requires that the path of the clockwise rotation to be exactly equal to the path of counter- clockwise rotation.   (proportionality of HSO to the external magnetic field Hext) When the external magnetic field Hext becomes larger, the Lorentz force FLor, becomes larger, which leads to larger difference between clockwise and counter- clockwise orbitals and, as a consequence, a larger magnetic field HSO of the spin-orbit interaction.
The green arrows show the direction of the effective magnetic field of the spin-orbit interaction HSO The red arrows show the direction the Lorentz force. The blue arrow shows direction of the external magnetic field Hext.
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(key property 2): The spin-orbit interaction increases (changes) when the electron orbital is deformed
deformed orbital: HSO is large   not- deformed orbital. HSO is small   When electron orbital is deformed, the part of the electron path becomes very close to the atomic nucleus, where the electrical field of the nucleus is very large. Since the magnetic field of spin-orbit interaction HSO is proportional to the electrical field, the HSO becomes substantially larger. Even though, at same time the HSO reduces from to be small to become negligibly small at the orbital part, which become far from the nucleus, in average the HSO substantially increases for the deformed orbital   When electron orbital is centrosymmetric, the magnetic field of spin-orbit interaction HSO is equally small at all points of the orbital. In average, the HSO is small.
(note: simplified schematic): the figures shown above are simplified schematic representation of the electron orbital aiming to demonstrate the operation principle of the effect.
(note about PMA) In the case of the Perpendicular Magnetic Anisotropy (PMA) is only important how the HSO changes under applied magnetic field, but not the absolute value of. At parts closer to nucleus the influence of the the Lorentz force FLor may be weaker. Therefore, for the PMA a complex consideration on an increase of HSO and on a possible decrease of FLor should be considered to analyze the deformed orbital.
(note about atomic neighbors) In the case of a crystal, a deformation of the orbital and its influence on the HSO depend on neighbor atoms. Proximity of a neighbor can either push out or pull closer the electron orbital to the atomic nucleus. It can influence both the HSO and the orbital deformation due to the Lorentz force FLor.
(note interface and bulk) at an interface the neighborhood of the atom is different from that of the bulk and the orbital reconstruction (deformation) always occurs at the interface. As a result, the HSO and, therefore, the PMA is very different for the interface and the bulk of the material.
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Devices, functionality of which is based on PMA

(device 1): Hard disk

In order to memorize a huge amount of data onto a small surface of the hard disk, the surface area of one bit is very small ~10 nm x10 nm. Only the SO is able to hold magnetic date in such a tiny area.

(uniqueness of PMA): There is no known effect except PMA, which allows to store magnetic data in area of 10 nm x10 nm, protecting data from the unwanted thermal switching

(device 2): Magnetic random access memory (MRAM)

In MRAM, one bit of data is memorized by two stable magnetization of the nanomagnet. In order to compete with other memory schemes (e.g. DRAM), the volume of nanomagnet should be smaller than ~30 nm x 30 nm x 1 nm

(uniqueness of PMA): There is no known effect except PMA, which allows to store magnetic data in volume of 30 nm x 30 nm x 1 nm, protecting data from the unwanted thermal switching

(device 3): Molecular recognition sensor

In the molecular recognition sensor, a surface touching event by a molecule or an virus is detected by a measure of a change of spin-orbit interaction the at the sensor interface. In order to practical the detection area of the sensor should be comparable with the size of the molecule or virus. The practical size is ~30 nm x30 nm. See more details here.

(uniqueness of PMA): There is no known effect except PMA, which allows to store magnetic data in the area of ~30 nm x30 nm protecting data from the unwanted thermal switching.

(device 4): Nano-sized microwave generator

In this device the microwave is generated by a stable magnetization precession in a nanomagnet. The A stable single-domain magnetization oscillation can be achieved in a nanomagnet of the volume ~300 nm x 300 nm x 1 nm

(usefulness of PMA): Even though the stable magnetization oscillation can be achieved in a nanomagnet without PMA, the microwave generator made of a nanomagnet with PMA is more reliable and stable.

(device 5): Nano-sized magnetic sensor

A nanomagnet as a part of a tunnel magnetic junction (MTJ) can be used as a nano sensor of a magnetic field. The spatial resolution of the sensor is better for a smaller size of the nanomagnet.

Calculation of PMA & Magnetic Anisotropy

calculation of magnetization direction in a nanomagnet with PMA under an external magnetic field

See detailed calculations of anisotropy field and PMA energy here CalculationHanisPMA.pdf

How the magnetization direction is calculated:

(step 1) The magnetic energy is calculated for the nanomagnet

(step 2) The magnetic energy is minimized with respect to the magnetization direction

------------------- step 1 Calculation of Magnetic energy

In the case of zero orbital moment, the magnetic energy of an electron is a product of the magnetization M (or the same the total spin of localized d- electrons) and the total magnetic field.

Energy or magnetic field. What is the PMA?

What object or effect does the PMA describe?

Magnetic anisotropy in a Fe film
thickness dependence
thick film (t>1.5 nm) thin film (t<1.5 nm) Equilibrium magnetization is in-plane. External magnetic field of more than 10 kG is required in order to turn the magnetization to perpendicularly to surface PMA. Equilibrium magnetization is perpendicular- to -plane. External magnetic field of about ~6 kG is required in order to turn the magnetization to the in-plane direction HSO< Hdemag. Energy favorable to minimize the demagnetization magnetic field Hdemag. HSO>Hdemag,. Energy favorable to maximize the magnetic field of spin- orbit interaction HSO bulk contribution is larger than interface contribution interface contribution is larger than bulk contribution
Magnetic anisotropy in a Fe film depends on its thickness.

The PMA describes the magnetic field H_PMA=H_SO- H_demag, which is a sum of demagnetization magnetic field H_demag and the magnetic field of spin orbit interaction H_SO. In the case of a PMA sample, the direction of H_SO is along magnetization M, the H_demag is always opposite to the M and H_SO>H_demag. As a result, H_PMA>0 and M ( the total spin of localized electrons (magnetic moment)) is aligned along H_PMA. and perpendicularly to the film surface.

The PMA energy E_PMA is the product of H_PMA and M (magnetic energy of localized electrons).

In some cases it is preferable to use the PMA energy E_PMA instead of the magnetic field H_PMA. An example is the thermally- activated magnetization reversal, when the magnetization reversal occurs when the energy of a thermal fluctuation is larger than E_PMA.

In some cases it is preferable to use the magnetic field H_PMA instead of the PMA energy E_PMA . An example is calculation of a torque in external magnetic field H_ext, when the magnetization procession M occurs around the field, which is the vector sum of H_ext and H_PMA.

What is the PMA energy?

The PMA energy E_PMA is defined as a difference between magnetic energies for the cases when when the magnetization is perpendicular-to-plane and in-plane

(it is important) The primary parameter of the PMA effect is the magnetic field. Even the use of the PMA energy is convenient for some applications.

Thickness dependence of H_PMA and E_PMA

H_PMA and E_PMA are substantially different at the interface and in the bulk of a ferromagnetic material.

The difference of H_PMA is because of difference of the orbital symmetry. The orbital symmetry at interface is substantially different that in the bulk because of a surface reconstruction at interface and the asymmetry of interaction with atoms from below and above for the interface atoms.

Example:

Fe film: at interface -> H_SO>H_demag, but in the bulk H_SO< H_demag. As a result, equilibrium magnetization of a thinner Fe film is perpendicular- to - film- plane and of a thicker film is in- plane (see here or here).

FeTbB film: in the bulk H_SO< H_demag. Equilibrium magnetization of a thicker FeTbB film is perpendicular- to - plane

Averaging over film thickness

The measured parameters of the PMA magnetic field H_PMA and PMA energy E_PMA

where t is the film thickness

(fact) In case when is substantially different at interface and bin the bulk, the magnetic properties of a thin film substantially depend on the film thickness (see here or here).

Dependence of H_PMA and E_PMA on magnetization direction

Both the demagnetization magnetic field H_demag and magnetic field of spin- orbit interaction substantially depend on the magnetization direction. It makes H_PMA substantially- dependent on the magnetization direction

(direction- dependence of H_demag) The demagnetization magnetic field H_demag is always directed perpendicularly to film surface and opposite to the magnetization direction. Its magnitude is proportional to the perpendicular-to plane component of the magnetization.

(direction- dependence of H_SO) The magnetic field os spin- orbit interaction H_SO is directed along to the magnetization direction (along the magnetic field applied to the atomic orbital). Its magnitude depends on the orbital symmetry (or degree of orbital deformation). Usually atomic orbital is deformed along the film normal. As a result, the H_SO may be substantially larger when magnetization M is perpendicular to the film surface in comparison to the case when the magnetization is in the plane.

Why the magnetization direction depends on the magnetization direction with respect to the film normal?

There is a discontinuity at the film interface. As a result, atomic orbitals are deformed at the interface. Since the effective magnetic field of spin-orbit interaction H_SO. depends on the orbital deformation (See SO interaction), H_SO becomes larger when the magnetization is perpendicular-to-plane and H_SO becomes smaller when the magnetization is in-plane. Correspondingly, the magnetic energy becomes different for the perpendicular-to-plane and in-plane magnetization directions.

Why do we care whether the equilibrium magnetization is in-plane or perpendicular-to-plane?

The energy of the spin-orbit interaction of only one interface layer may be huge and it may substantially exceed the total magnetic energy of all other electrons in a ferromagnetic film. This large energy of the spin-orbit interaction is used to store a data in small volume of magnetic medium (e.g. a nanomagnet of a MRAM cell, a magnetic domain of a hard disk). Since the electron orbital at the top of a ferromagnetic metal are deformed in the direction of the interface, the energy of spin-orbit interaction is largest for the film with the PMA ( the direction of the equilibrium magnetization is perpendicular-to-plane).

Origin of PMA
Spin-orbit interaction due to the orbital deformation
Fig.1. The effective magnetic field of spin-orbit interaction HSO. The depends on the degree of the orbital deformation and the deformation direction with respect of the applied external magnetic field Hext. (a) spherical orbital HSO=0. (b) deformed orbital and an external magnetic field Hext is applied along the deformation. HSO ≠0. (c) deformed orbital and an external magnetic field Hext is applied perpendicularly to the deformation. HSO =0. The electron magnetic energy is larger in the case (b) and smaller in cases (a) and (c)
for more details, see here

Why do we need a high magnetic energy in order to reduce the volume of data storage media?

In a magnetic media the data are stored by means of two opposite equilibrium magnetization directions. The energy of barrier between these two equilibrium states should be substantially (at least 20-50 times) larger than the thermal energy kT. Otherwise, the magnetization can be thermally switched and the date can be lost (See thermally-activated magnetization switching). The barrier energy is linearly proportional to the volume of the storage cell. In order to make the storage cell smaller, the barrier and consequently the magnetic energy should be larger.

How the spin-orbit interaction makes the equilibrium magnetization perpendicular-to-plane?

At the interface the electron orbital is substantially deformed towards the interface. Due to this deformation, the spin-orbit interaction is significantly enhanced. This means that there is a large effective magnetic field of the spin-orbit interaction H_SO, when the magnetization is along the deformation (perpendicular-to-interface). However, there is no such field, when magnetization is in-plane. The magnetic field H_SO may become larger than demagnetization field and the negative magnetic energy become smaller for perpendicular direction than for the in-plane direction. As a result, the perpendicular-to-plane direction of the magnetization becomes energetically favorable.

How to measure the strength of the PMA?

The anisotropy field is used to measure the strength of the PMA. The larger the anisotropy field is, the stronger the PMA is.

What are the reasons why some ferromagnetic films have in-plane magnetization and some films have perpendicular-to-plane magnetizations?

Two factors are factors are important to understand the equilibrium magnetization direction of a ferromagnetic film: (1) the directional dependence of the spin-orbit interaction; (2) the orbital deformation at an interface

Model of perpendicular magnetic anisotropy. oversimplified Neel description

This classical model is based on some oversimplified assumptions. This model ignores the specific features of the spin-orbit interaction, which creates the magnetic anisotropy. However, this model is a very simple and can be used for a rough estimate.

See detailed calculations of anisotropy field and PMA energy by this description here NeelPMA.pdf

In this model, the total magnetic energy, which is a sum of the energy of the magnetic anisotropy EPMA and the magnetic dipole energy can be calculated as

in which the energy of the magnetic anisotropy is assumed to be proportional to a square of the magnetic component M_z along the easy axis.

where θ is the angle between the magnetization M and the film normal, φ is the angle between the magnetic field H and the film normal

Model of perpendicular magnetic anisotropy. full description based on properties of the spin-orbit interaction

See detailed calculations of anisotropy field and PMA energy by this description here CalculationHanisPMA.pdf

The sum of the magnetic energy due to the magnetic field of the spin, the demagnetization field, the magnetic field of the spin-orbit interaction and demagnetization field gives the total magnetic energy as

where k_so is the coefficient of the spin-orbit interaction and k_demag is the demagnetization coefficient

Comparison of the oversimplified Neel description and full description of PMA

		Neel model		full model		note
energy						the same when kso=0
Mx vs Hx						linear,the same as experimentally observed
anisotropy field
Hani vs Hz						the same when kso=0. The dependence is used to measure kso
energy barrier						the same when kso=0. used in the classical model of magnetization reversal (See here)

Calculation of PMA

See detailed calculations of anisotropy field and PMA energy here CalculationHanisPMA.pdf

(step 1) The spin‐orbit interaction and the orbital deformation

The Einstein theory of relativity states that an electron moving in a static electric field experiences an effective magnetic field, which is called the effective magnetic field of the spin-orbit (SO) interaction H_SO .The electric field of atomic nucleus may induce a substantial H_SO, because the electron moves with a very high speed on its atomic orbital in close proximity to the atomic nucleus. However, the value of the H_SO substantially depends on the orbital symmetry. It is because the electron experiences different directions of the H_SO  on different parts of the orbit that may compensate each other. For example, in the case of a spherical orbital the contributions to the H_SO are equal and opposite in sign and the resulting H_SO=0. In the case of a deformed orbital, when the orbital is elliptical or/and the orbital center is shifted from the nucleus position, the H_SO becomes substantial and proportional to the degree of the orbital deformation. Also, the H_SO is linearly proportional to an external magnetic field H_ext. Figure 1 demonstrates how the H_SO changes depending on the direction of the orbital deformation and the direction of the external magnetic field. When the orbital is spherical (Fig.1a), the H_SO equals to zero independently on the direction of the H_ext. When the orbital is deformed and the H_ext is applied perpendicularly the orbital deformation (see Fig.1b), the H_SO is also zero, because the orbital is symmetrical along the direction of the H_ext. For the cases of Figs. 1a and 1b, the magnetic energy of the electron E_mag is equal to:

where S is the electron spin and is m_B  the Bohr magneton. When the orbital is deformed and the H_ext applied along the orbital deformation (Fig.1c), the H_SO becomes a non‐zero and proportional to the H_ext. In this case the E_mag   equals to:

The absolute value of the electron magnetic energy is larger in the case shown in the Fig. 1c and smaller in the cases shown in the Figs. 1a and 1b. Therefore, the orbital deformation substantially changes the electron magnetic energy and therefore magnetic properties of the ferromagnetic film.

Origin of PMA
orbital deformation at interface & equilibrium magnetization direction
Fig.2. The dependence of magnetization of a ferromagnetic film on  the orbital deformation of interface atom of a ferromagnetic film. The cross-section of a nanomagnet is shown as an array of its  electronic orbitals. The arrow shows the magnetization direction of the nanomagnet. (a) The interface orbitals are spherical and not deformed. The magnetization is in-plane. (b) The interface orbitals are deformed perpendicularly to the plane. The magnetization is perpendicular -to -plane

(step 2) PMA of a thin film

The equilibrium magnetization of an isotropic ferromagnetic thin film can be either in‐plane or perpendicular‐to‐plane depending on the deformation of the electron orbitals at the film interface and the thickness of the film. The interaction of analyte molecules with interface electrons of the ferromagnetic film leads to the orbital deformation of the interface electrons and consequently to a change of the PMA. Consequently, the change of magnetization direction of the ferromagnetic film due to the change of the PMA is used as the molecular detection mechanism in the disclosed invention.

The physical phenomenon of the PMA and the reason, why the orbital deformation defines the strength of the PMA, are explained as follows. Figure 2 shows a cross‐section of a nanomagnet as an array of its electronic orbitals. The magnetization of a thicker film (Fig. 2a) is in-plane, while the magnetization of a thinner film (Fig.2b) is perpendicular-to-plane. The PMA is the reason, why the magnetization changes its direction depending on the film thickness.  The equilibrium magnetization direction is the direction of the smallest magnetic energy of the whole film. For the bulk electrons the magnetic energy is smallest when the magnetization is in-plane. For the interface electrons the magnetic energy is smallest when the magnetization is perpendicular-to-plane. In the case of a thicker film, the number of bulk electrons is larger and the total magnetic energy of the film is smaller when the magnetization is in-plane. In the case of a thinner film, the number of bulk electrons is smaller while the number of interface electrons remains the same. As a result, the magnetization becomes perpendicular-to-plane for the substantially thin film.

The reason why the dependence of the magnetic energy on the magnetization direction is different for the bulk and interface electrons, is their orbital shape. The orbital of the bulk electrons is spherical. They do not experience any H_SO. The magnetic energy E_||,b and E_⊥,b  of a bulk electron for the case of in-plane and perpendicular-to-plane magnetization can be calculated as

where S is electron spin, μ_B is the Bohr magneton, H_M is the intrinsic magnetic field induced by the magnetization, H_D is the demagnetization field, which is directed perpendicular-to-plane and proportional to the perpendicular component of the H_M. DE_B is the difference of the magnetic energy for two magnetization directions. For the bulk electron, the DE_B is negative.

The interface electrons experience the spin-orbit magnetic field H_SO, additionally to H_M  and H_D, because their orbitals are deformed. The H_SO is a non-zero only when the magnetization direction is along the deformation and therefore perpendicular-to-plane. For an interface electron, the magnetic energies E_||,i and E_⊥,i for the in-plane and perpendicular-to-plane magnetizations, respectively, and their difference DE_i  can be calculated as

The H_SO is proportional to the degree of the orbital deformation. Even when the deformation is small, H_SO > H_D and the ΔE_i is positive. The difference of the magnetic energy for the || and ⊥ magnetization directions is called the PMA energy (E_PMA) and can be calculated as

where N_B and N_i are the numbers of the bulk and surface electrons, correspondingly. Since ΔE_B is negative, Eq. (1.5) can be simplified as

Origin of PMA
equilibrium magnetization direction vs film thickness
Fig.3. The total magnetic energy Emag,total of a FeB film as a function of the film thickness. The negative energy means that magnetization is in-plane. Two cases of a different values of the interface energy (Emag,total=1.15 mJ/m2 for solid line and Emag,total=0.85 mJ/m2 for dash line) are shown. In the case of film thickness of 1 nm, the magnetization direction of the film changes from in-plane direction to  perpendicular-to-plane direction when Emag,total increases from 0.85 mJ/m2 to 1.15 mJ/m2
click on image to enlarge it

where t is the film thickness, β is the constant, which depends on the symmetry of crystal lattice. In the simplest case of a cubic lattice, it can be calculated as

where a is the lattice constant.

Figure 3 shows the energy of the perpendicular magnetic anisotropy E_PMA of a FeB thin film as function of its thickness for two cases of different orbital deformation at the interface with the interface magnetization ΔE_i of 1.15 and 0.85 mJ/m2. The magnetization of the thinner film is perpendicular-to-plane. The magnetization of the thicker film is in-plane. The thickness of the film at which the magnetization changes direction depends on the degree of orbital deformation at the interface and therefore on the value of interface magnetization energy ΔE_i. For example, at the thickness of 1 nm, the magnetization can be switched between the in-plane and perpendicular-to-plane directions by the modulation of the orbital deformation.

The orbital deformation at interface can be of two types. The orbital can be elongated or shortened along the interface normal. Both types of the orbital deformation lead to the increase of the H_SO and the PMA energy.

How and why the electron orbital is deformed?

The spin-orbit interaction is stronger in the case of a non-symmetrical electron orbital. The less symmetrical orbital is, the stronger H_SO the electron experiences.

The spin-orbit interaction is weaker when induced by centrosymmetric electrical field. Even though the p- and d-orbital of a hydrogen atom is non-symmetrical, the H_SO is small for the p- and d- electrons in this fully centrosymmetric electrical field of one nucleus. The case is different, when many atoms interacts. Such interaction makes their orbits non-symmetrical. The case of an atom at an interface is prominent. At two side of the interface the atom experience different electrostatic force, which makes the atom orbital well non-symmetric. As a result, the electron of this orbital experience the strong H_SO.

Perpendicular-to-plane magnetic anisotropy at a Fe/Pt interface

Magnetization is perpendicular to the interface Magnetization is along the interface

Schematic diagram of Pt/Fe interface. The blue and red spheres show the electron orbitals in Fe and Co, respectively. Blue arrows shows the magnetization (spin of localized d-electrons). The green arrows show the effective magnetic field HSO of the spin-orbit interaction. The orbital of the localized electrons are shown.

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How the polarity of the orbital influence the sensor output signal?

It does not matter whether orbital is elongated or squeezed. Both deformations induce the same polarity of the change of H_SO. A breaking the spacial symmetry is important, but not the polarity of the breaking. The most effective enhancement of H_SO is when the center of electron orbital is shift of the position of the nucleus. (See more details in the spin-orbit interaction)

Does the interface roughness influence the PMA?

Yes, very much. The interface PMA exists only at a very smooth interface. Even a moderate roughness of the interface causes the reduction and disappearance of the PMA.

Does the PMA increases when the film thickness decreases?

Yes, in the case of the interface-type PMA, the PMA increases when the ferromagnetic layer becomes thinner. See Fig.3

Is it possible to get film with a large PMA by decrease the film thickness?

Yes, it is possible. See Fig.3. However, often when the ferromagnetic film or layer becomes very thin, the film roughness sharply increase and even the film may becomes discontinuous. It causes reduction and disappearance of the PMA. However, there sever growth techniques (tricks), which allow to grow a very thin or thick layers with a high PMA.

For example, using a conventional sputtering of amorphous FeB on amorphous SiO2 it is only possible to grow film with perpendicular magnetization in the range of thicknesses between 0.8 nm and 1.5 nm. I have made FeB with strong PMA and perpendicular magnetization as thin as 0.1 nm and as thick as 2.5 nm.

Thickness-dependence of PMA

Thicker metal. Magnetization is in-plane Thinner metal Magnetization is perpendicular to plane

Fig.16. Magnetization of CoFeB film grown on MgO. The magnetization of a thick film (thickness >1.5 nm) is in-plane , but the magnetization of a thinner film is perpendicular-to-plane

Can an electron deep in bulk experience the strong spin-orbit interaction?

Yes. It is the case of a single crystal metal with anisotropy axes along the film normal. Another case is the asymmetrical rearrangement of atoms along growth direction. For example, FeTbB has a strong bulk type PMA. Also, it has a weak the the interface PMA. As a result, The equilibrium magnetization of a thin FeTbB is in-plane and a thick FeTbB is perpendicular-to- plane. The slope of Fig.3 becomes positive instead of negative.

Which interface induce the strongest PMA?

There are many possible interfaces with a strong PMA. The famous interfaces are: (1) Co(111)/Pt(111); (2) Fe(001)/ MgO (001); (3) Fe/Ta; (4) Fe/W

The strength of the PMA depends very much on growth technique. It means how thin film and smooth the interface can be obtained.

The model, which is described below, is well-matched to all experimental fact. For example, the existence of H_SO well explains the linear dependence of in-plane component of magnetization as function of applied in-plane magnetic field (See anisotropy field) .

Spin-orbit (SO) interaction is the origin of Perpendicular magnetic anisotropy (PMA)

Relativistic origin of the Spin-Orbit Interaction
An object (red) is moving in a static electric field. In the coordinate system moving together with the object, the static electric field is relativistically transformed into the effective electric field Eeff and the effective magnetic field Heff. The effective magnetic field is called the effective magnetic field of SO interaction HSO . The HSO is absolutely ordinary magnetic field For example, in case if the particle has a magnetic moment (spin), there will be a spin precession around the HSO.
click on image to enlarge it
more details is here

for more details see here

Fact 1. Relativistic origin of spin-orbit interaction

An object, which moves in an electrical field, experience an effective magnetic field. This effective magnetic field is 100% magnetic field and it is indistinguishable from any other magnetic field.

This magnetic field is called the SO magnetic field H_SO.

The origin of the SO magnetic field is the relativistic nature of the electromagnetic field.

Fact 2. The electrical field of atomic nucleus induces the SO magnetic field, which originates the perpendicular magnetic anisotropy (PMA)

Because of its relativistic nature, the SO effect is weak. Only a very strong electrical field may induce sizable SO magnetic field. Only the electrical field of atomic nucleus is sufficient strong to create sufficiently strong SO magnetic field (1-30 kGauss), which induces the PMA

Fact 3. Only an electron in a non-symmetrical orbital experiences SO interaction

An electron in spherical orbital (s -orbital) does not experience any SO interaction (explanation is here). Only when orbital symmetry is broken, the electron experience SO interactions.

There are several possibilities to break the orbital symmetry. Each of them induces H_SO. The magnitude H_SO is proportional to the degree of the breaking symmetry.

Fact 4. The electrical field of nucleus cannot break time- inverse symmetry. The SO magnetic field exists only when there is external magnetic field.

The H_SO exists only when there is some external magnetic field.

The H_SO is always is zero in absence of the external magnetic field.

The H_SO is always in the same direction as the external magnetic field.

The H_SO may be significantly larger than the external magnetic field

Fact 5. The SO is direction dependent. It is the source of the PMA

When orbital is deformed only in one direction (for example, only along the z- direction), the H_SO exists only when the external magnetic field is directed along this direction (along the z-direction), but there is no H_SO when the external magnetic field is directed in a different direction (along the x- or y- direction)

When the orbital is deformed perpendicularly to the film interface, the H_SO is induced only in this direction. The directional dependence of H_SO originates the PMA (See below)

Fact 6. The orbital symmetry in close proximity to nucleus determine the strength of the SO magnetic field

The increase of the spin-orbit interaction due to deformation of the electron orbital

Shift of orbital center: larger HSO elliptical deformation: smaller HSO

Electron orbital. When the orbital is spherical and its center at the position of the nucleus, the effective magnetic field of the spin-orbit interaction HSO is zero. Only when the orbital is deformed, there is the magnetic field of the spin-orbit interaction.

A substantial SO is induced only by a very strong magnetic field, which exists only in very close proximity to nucleus. Only this region makes substantial contribution to the H_SO and E_PMA. For this reason, the orbital symmetry in this region is critically important for the PMA.

As a result: H_SO is larger when the center of orbital slightly shifted from the position of the nucleus comparing to a slight deformation of

Fact 7. Mainly localized d- or f- electrons contribute to PMA

Spins of the localized d- or f- electrons mainly contribute to the magnetization of a ferromagnetic metal. Therefore, the deformation and symmetry of these orbitals mainly determines the PMA.

The contribution of the conduction electrons to the metal magnetization and the PMA exists, but it is very small. It is because the distribution of the spin directions of the conduction electrons is very different from that of localized d- or f- electrons (See here)

The conduction electrons influence the PMA and the magnetization mainly due to the sp-d exchange interaction.

Q. The orbital of d- and f- electrons are already not spherical. Do they experience the SO interaction and the magnetic anisotropy?

A. It is correct. They do. There is magnetic anisotropy along some crystal orientation. Often it is not large. However, at the interface the orbital deformation might be much larger, which induces much larger H_SO and E_PMA.

See also VCMA effect and SO torque

Fact 8. The parity symmetry and spin-orbit interaction

The SO interaction and the PMA depend only the direction of orbital deformation, but not its polarity.

For example, the orbital deformation due to a shift of the center of electron orbit from position of atomic nucleus in + x direction induces absolutely equal H_SO and E_PMA

as the shift in - x direction.

Fact 8. Neither covalent nor ionic bonding is good for PMA and SO. The optimum bonding should be something between.

In both case of the covalent bonding (E.g. Si, Fe) or the ionic bonding (E.g. NaCl, ZnO), the electron orbital is rather symmetric to induce any H_SO and E_PMA. In order to induce a large magnetic anisotropy, the orbital should be deformed asymmetrically. It is the case when the bonding is neither fully covalent nor fully ionic.

A bonding across an interface (the interfacial PMA) or a bonding along some specific crystal direction in a compound crystal (E.g. CoPt, SmCo, GaAs, InP) (bulk-type PMA) make optimum orbital deformation and induce a strong H_SO and E_PMA

Physical Origin of perpendicular magnetic anisotropy (PMA)

The PMA exists, because the electron orbitals in a ferromagnetic metal are deformed in the direction perpendicular to the film interface. When magnetization is along this direction, the intrinsic magnetic field induced a substantial effective magnetic field of the spin-orbit interaction H_SO , because orbital deformation in this direction. When the magnetization is in-plane, there is no H_SO, because the orbital is not deformed in this direction. As a result, the absolute value of the magnetic energy increases, when the magnetization is perpendicular to the film, and decreases when the magnetization is in plane.

Magnetization is in plane:

Total magnetic field= H_intristic

Magnetic energy= -H_intristic · M

Magnetization is perpendicular to plane:

Total magnetic field= H_intristic+H_SO

Magnetic energy= -(H_intristic +H_SO )· M

where M is the magnetization or the spin of localized electrons.

Since the absolute value of the negative magnetic energy is larger in the case when the magnetization is perpendicular-to-plane. As a result, the easy magnetization direction becomes perpendicular-to-plane, because of the SO interaction

note: See also about demagnetization field

Anisotropy field H_anis

Measurement of anisotropy field Hanis

Measured in-plane magnetization as a function of applied in-plane magnetic field. The arrow shows the direction and magnitude of the applied in-plane magnetic field. The ball shows the magnetization direction. Without magnetic field the magnetization is perpendicularly-to-plane. Under magnetic field, the magnetization turns toward magnetic field. The field, at which the magnetization turns completely in-plane, is called the anisotropy field. The dots of the right graph shows experimental data. Measurement date: May 2018.

Click on image to enlarge it

The anisotropy field H_anis is the magnetic field, which is applied in-plane and therefore perpendicularly to the easy axis of a nanomagnet and at which initially-perpendicular magnetization turns completely to the in-plane direction.

The anisotropy field increases under a negative gate voltage and it decreases under a positive gate voltage (see VCMA effect).

The anisotropy field is proportional to the PMA energy. A half of product of H_anis and nanomagnet magnetization is a common estimate of the PMA energy.

Important feature of PMA: Linear dependence of in-plane component of magnetization on in plane magnetic field. See below the math to prove it. Since the fitting of a linear dependence is simpler, is resisted against the noise and other unwanted disturbing factors (like magnetic domain) and can be done with a high precision, measuring of H_anisotropy with a high precision is an important step to almost any magneto transport measurement.

Important facts about the spin-orbit interaction and PMA are:

fact 1: Due to the spin-orbit interaction, there is a strong magnetic along the film normal (the z-axis). This magnetic field is called the magnetic field of the SO interaction. It is absolutely real magnetic field, which is generated relativistically due to electron movement in electrical field of atomic nucleus

fact 2: The SO magnetic field is generated due to the orbital deformation along the z-axis. In the case of the uniaxial anisotropy it can be simplified that there is a SO magnetic field only along z-axis, but there is no in-plane SO magnetic field. There is no field along the x-axis and y-axis. The magnitude of the SO magnetic field is proportional to degree of deformation of electron orbital in close proximity to atomic nucleus.

fact 3: The magnitude of the SO magnetic field is linearly proportional to the total magnetic field along the z- direction (perpendicularly to interface).

Calculation of Anisotropy field H_anis

See detailed calculations of anisotropy field and PMA energy here CalculationHanisPMA.pdf

(note) The calculation is for a single electron. In the case, when orbital symmetry of the electrons in bulk and at an interface is different, the effective anisotropy field for the whole nanomagnet should be calculated (See here)

(how calculation is done) The angle of magnetization turning is calculated by a minimizing the magnetic energy for electron spin. The magnetic energy is the product of the electron spin and a sum of all magnetic fields, which the electron experiences.

Magnetic fields, which the electron experiences:

(field 1): External magnetic field, which is applied in both the perpendicular- to- interface direction H_z and in the in-plane direction H_x.

(field 2): Magnetic field of magnetization induced by all aligned electrons in a nanomagnet. This field can be imagined as a magnetic field inside the magnetic dipole induced by the electron spin. The average spin is a nanomagnet called the magnetization M. The magnetic field of magnetization is linearly proportional to the electron spin.  In simplified units, the magnetic field induced by the spin equals to M.

(field 3) Demagnetization field H_demag. The magnetic field, which is generated at interface due to broken chain of aligned spins. The direction of H_demag  is perpendicular- to- interface and opposite to M. The H_demag can be calculated as

where k_demag is the demagnetization factor.

(field 4) Magnetic field H_SO of spin- orbit interaction, which is a relativistic magnetic field induced by an electrical field of atomic nucleus. It is directed along the orbital deformation (the z- direction in this case) and is proportional to the total magnetic field, which is applied along the x- direction.

where k_SO is a parameter describing the strength of the spin- orbit interaction, is the intrinsic field in the nanomagnet and M is the magnetization of the nanomagnet

(intrinsic field=field2+field3) It is magnetic field, which all atom in nanomagnet equally experience. In contrast, the spin-orbit field H_SO (field 4) is individual for each atom (not common), because H_SO is linearly proportional to the orbital deformation (See here), which is very different for each atom. ( example: a Fe nanomagnet) At an interface orbital deformation of Fe atoms is large and interfacial Fe atoms experience a large H_SO. However a few atomic layers below, the orbital of Fe is not deformed and these experience no H_SO.

Measurement of anisotropy field

Measurement of the anisotropy field of a magnetic nanowire. Without an external magnetic, the magnetization M is perpendicular to the film due to PMA. When external in-plane magnetic field Hext is applied, the magnetization turns. The magnetic field , at which the magnetization turns fully in-plane is called the anisotropy field

Click on image to enlarge it

The magnetization direction under an external in-plane magnetic field is calculated from minimizing the magnetic energy. In the case of zero orbital moment, the magnetic energy of an electron is a product of electron spin M and a sum of all magnetic fields, which the electron experiences: H_all=H+M+ H_demag + H_SO. The magnetic energy can be calculated as

---------------
Case 1: There is no perpendicular external magnetic field H_z =0

In this case (H_z=0), the magnetic energy Eq.(2.4) is calculated as

Under an in-plane magnetic field H_x, the magnetization M does not change its magnitude only its direction turns. As a result, M is a constant and independent of H_x. Eq.(2.5) can be written as

The equilibrium magnetization direction corresponds to the minimum magnetic energy. Minimizing the energy with respect to M_x gives

The solution of Eq.(2.7) gives the linear dependence of in-plane component M_x of the magnetization on the in-plane magnetic field H_x as

where  the anisotropy field H_anis is calculated as

Anisotropy field. Calculations vs. Measurement
Anisotropy field calculated from Eq.(2.13) Measurement Measurement Anisotropy field calculated from Eq.(2.13). Hanis,0=4 kG;kSO =0.2. All orbitals of nanomagnet experiences the same strength of spin-orbit interaction and demagnetization field. Magnetically- harder nanomagnet. Ta(5nm):FeCoB(1nm) width x length=400nm x 400nm coercive field Hc=480 Oe Magnetically- softer nanomagnet. Ta(2.5nm):FeB(1.1nm) width x length=400nm x 400nm coercive field Hc=60 Oe . Sample:Volt54B. (See here for sample details)
Calculation predicts an nearly- linear increase of the anisotropy field when the external field H increases. The experimental measurements confirms the linear increase with H at at a larger field, but dependence deviate from linear at a smaller field
Click on image to enlarge it

---------------
Case 2: There is a perpendicular external magnetic field H_z≠ 0

In this case the magnetic energy can be calculated as (See Eq.2.4)

where H⁽⁰⁾_anis is the anisotropy field in the absence of the external magnetic field (H_z =0)

The equilibrium magnetization direction corresponds to the minimum magnetic energy. Minimizing the energy with respect to M_x gives

The solution of Eq.(2.7) is

Even though the dependence of Eq.(2.12) deviates from linear, it is close to a linear dependence and can be expressed similar to Eq.(9) as

Fig.30 Anisotropy field. Calculations vs. Measurement
A weak non-linearity of dependence of Hx vs Mx Effective anisotropy field calculated from Eq.(2.15) In-plane magnetization Mx vs in-plane magnetic field calculated from Eqs. (2.13),(2.15). The dependence is slightly deviate from linear, but this deviation is very weak. Effective anisotropy field vs. in-plane magnetic field Hx as calculated from Eq.(2.15) by the iteration method. The effective anisotropy field is slightly increase with , Hx but the is very weak. An assumption of independence of effective anisotropy field of Hx is a relatively good assumption
Parameters used in calculations Hanis,0=4 kG; kSO =0.2. Hz=2 kG;
Click on image to enlarge it

in which the anisotropy field depends on H_z and can be calculated as

Substitution Eq.(2.13) into Eq. (2.14) gives the effective anisotropy field as

From Eq.(2.15), the anisotropy field depends linearly on the external perpendicular magnetic field H_z.

When  the applied in-plane magnetic field is much smaller than the anisotropy field

Eq.(2.15) is simplified to 

and therefore the effective anisotropy field becomes independent of H_x

When condition (2.16) is not the case, Eq. (2.15) can be solved by an iteration starting from Eq.(2.17)

Figure 30 shows that the independence of the anisotropy field on in-plane magnetic field and the linear dependence of M_x vs. H_x is a good approximation for the most of a realistic nanomagnets.

Experiment. Measurement of anisotropy field

External magnetic field is applied along current External magnetic field is perpendicularly to current

Measurement of the anisotropy field of a magnetic nanowire. The green balls shows the magnetization direction of FeB nanomagnet, the blue arrow shows direction of external magnetic field. The FeB nanomagnet is fabricated on top of Ta nanowire. The magnetization direction is measured by a pair of Hall probe. The measured Hall voltage is largest when the magnetization is perpendicular to the top surface of nanomagnet

Without an external magnetic, the magnetization M is perpendicular to the film due to PMA. When external in-plane magnetic field H|| is applied, the magnetization turns towards the H||. The magnetic field, at which the magnetization turns fully in-plane is called the anisotropy field. The stronger PMA is and the stronger the PMA resists to the magnetization turning, the lager and the anisotropy filed is. Therefore, the anisotropy field is measure of the strength of the PMA.

Click on image to enlarge it

PMA energy

(method 1 of calculation of PMA energy) imaginable switching off the spin- orbit interaction and demagnetization effect

In case when there is no external magnetic field (H_x=0; H_z=0) , the magnetization is perpendicular to plane (M_x=0; M_z=M) and the magnetic energy can be calculated from Eqs. (2.6,2.9) as

If we switch off the spin- orbit interaction k_SO=0 and demagnetization field k_demag=0 (just imagine it), k_SO=0 becomes zero and the magnetic energy is calculated as

The PMA energy EPMA is defined as an energy, which is induced by the spin-orbit interaction minus demagnetization. Comparison of Eqs.(12) and 813) gives the PMA energy as

(note 1) The anisotropy field H_anis is the parameter, which characterizes the PMA energy.

(note 2) In the case, when orbital symmetry of the electrons in bulk and at an interface is different, the effective PMA energy for the whole nanomagnet should be calculated (See above).

(note 1) The PMA energy can be either positive (equilibrium magnetization is perpendicular to plane) or negative (equilibrium magnetization is in- plane). From Eq.(10) the condition when the PMA energy is positive is

(method 2 of calculation of PMA energy) integration over a path of turning magnetization from a perpendicular-to-plane to in-plane direction

The PMA energy can be also defined as an energy, which is required in order to turn the magnetization fully in-plane. A tiny change dH_x of in-plane magnetic field correspond to a change of magnetic energy dH_x · M_x. The integration over the change of H_x from 0 to H_anis using Eq.(2.8) gives the PMA energy as

Internal magnetic field in a nanomagnet

detailed calculations of the internal magnetic field are in this pdf file. InternalMagneticField.pdf

Effective Internal magnetic field in a nanomagnet

It has both in-plane and perpendicular- to- plane components.; nanomagnet easy axis is perpendicular-to-plane; external magnetic field is applied in the in-plane direction

Magnitude of the internal magnetic field (ration to magnetization M) vs. external in-plane magnetic field H|| Angle αHint of the internal magnetic field with respect to film normal vs. external in-plane magnetic field H|| Magnitude of the internal magnetic field (ration to anisotropy field) vs. external in-plane magnetic field H|| Magnitude of the internal magnetic field at a different ratio of the anisotropy field Hani to the magnetization M of nanomagnet. Black line shows the magnetization angle αM and all other lines shows the angle αHint of the internal magnetic field at a different ratio of the anisotropy field Hani to the magnetization M of nanomagnet. Magnitude of the internal magnetic field at a different ratio of the anisotropy field Hani to the magnetization M of nanomagnet. In all cases, the internal magnetic field is larger than the magnetic field induced by the magnetization Hint>M. The difference is the largest The angle of the internal field is smaller than the magnetization angle. The difference is the largest when the magnetization angle is about 45 degrees with respect to film normal.

In absence of external field HII=0, the internal magnetic field Hint equals to Hint=M+Hani and is directed perpendicularly- to film normal. Under a large in-plane magnetic field HII> Hani, both the magnetization and the internal magnetic field are in-plane and Hint=M.

The data was calculated from Eq.(5.20). See here.

Click on image to enlarge it

There is a magnetic field inside nanomagnet, which is called the internal magnetic field .

(note about magnetic field of spin-orbit interaction ): The localized electrons on the surface and in the bulk of the nanomagnet experience a different magnetic field due to the spin-orbit interaction. The magnetic field H_SO of spin- orbit interaction is individual for each electron and depends on its orbital symmetry (See here). The electron at interface experience a strong H_SO. In the contrast, the electrons in the bulk of the nanomagnet experience a weak H_SO. As a result, the same electrons experience a very magnetic field at interface and in the bulk.

(note fixing spins of all electrons in one direction ): localized electron in a ferromagnetic material experience a substantial exchange interaction (See here).The effective magnetic field of the exchange interaction is very large (~1500 T See here). Since it is so large, the exchange field solidifies all spins in one direction. As a result, the whole nanomagnet can be consider as a single spin object with united total spin.

Two types of the internal magnetic field in a nanomagnet

(type 1) Effective internal magnetic field

Even though the electrons at interface and bulk experience a different internal magnetic field, the total spin of nanomagnet experience an average of all those contributions. This magnetic field affects the total spin of the nanomagnet. This is the internal magnetic field, which by the nanomagnet as one object. For example, this internal magnetic field determines properties of the ferromagnetic resonance (FMR) of the nanomagnet.

(type 2) Global internal magnetic field

This is the conventional magnetic filed, which is inside of the nanomagnet (See here). This magnetic field "fills" all the space inside the nanomagnet and all other spin objects (e.g. spin of a nucleus) equally experience this magnetic field.

Internal magnetic field in a nanomagnet. Measurements.

Measurements: Sample Volt40A nanomagnet L43C (all measurement data see here)

effective internal magnetic field Hint vs. in-plane external magnetic field H|| effective internal magnetic field Hint vs. perpendicular-to-plane external magnetic field Black line shows total magnitude of effective internal magnetic field Hint. Green line shows its perpendicular-to-film component. Red shows its in-plane component.   At H||=0, the nanomagnet magnetization is perpendicular- to-plane. When in-plane external magnetic field is greater than anisotropy field H||>Hint, the nanomagnet magnetization is in-plane. Nanomagnet magnetization is always perpendicular- to- plane

Click on image to enlarge it

(fact 1 about field when magnetization is in-plane) When magnetization is in-plane, both the effective internal magnetic field and the global magnetic field equals to M (magnetic field induced by the magnetization) E.g. in the case of FeB nanomagnet (See here) this field is about 14 kGauss. This fact is applied for both types of the nanomagnets with in-plane and perpendicular-to-plane equilibrium magnetization.

(fact 2 about global internal field when magnetization is perpendicular- to-plane ) When magnetization is perpendicular-to-plane the global internal magnetic field is zero. This fact is applied for both types of the nanomagnets with in-plane and perpendicular-to-plane equilibrium magnetization.

(fact 3 about effective internal field when magnetization is perpendicular- to-plane ) (case when the equilibrium magnetization the nanomagnet is perpendicular-to-plane ): When magnetization is perpendicular-to-plane, the effective internal magnetic field equals to M+H_ani (the sum of magnetic field induced by the magnetization and the anisotropy field). E.g. in the case of FeB nanomagnet (See here), M ~14 kGauss and H_SO ~6 kGauss, therefore the effective internal magnetic field is about 20 kGauss=2 T.

(fact 4 about effective internal field when magnetization is perpendicular- to-plane ) (case when the equilibrium magnetization the nanomagnet is in-plane ): When a sufficient external magnetic field is applied so that the nanomagnet magnetization turns to perpendicular-to-plane direction, the effective internal magnetic field is between zero (case of no PMA) to M (case of some PMA). E.g. in the case of FeB nanomagnet, M ~14 kGauss, therefore the effective internal magnetic field is between 0 kGauss (e.g. a thick FeB film) and 14 kGauss. (e.g. a thin FeB film)

(fact 5 about an increase of effective internal field under applied external magnetic field) when the magnetization is perpendicular- to- plane and an external magnetic film is applied in the same direction, the effective internal field increases with increase proportionally to the increase of the external magnetic field due to the increase of the magnetic field of the spin- orbit interaction (See here). The external perpendicular magnetic field does not affect the global internal field.

Calculation result:

detailed calculations of the internal magnetic field are in this pdf file. InternalMagneticField.pdf

Effective internal magnetic field:

This field, which keeps the magnetization of the nanomagnet along its easy axis, induce FMR, etc.

perpendicular- to- plane component:

in-plane component:

total magnitude:

Global internal magnetic field:

Other object inside the nanomagnet (e.g. spin of nucleus, spin of a conduction electron) experience this field

perpendicular to plane component:

in-plane component:

where M is magnetization (~14 kGauss for FeB); H_ani is the anisotropy field (~2-10 kGauss for FeB); and H_ext are in-plane and perpendicular-to-plane of applied external magnetic field.

Measurements of the anisotropy field H_anisotropy

3 following experimental method are most often used to measure H_anisotropy

Using a Vibrating-sample magnetometer (VCM) or a SQUID magnetometer.

The magnetometer measures the magnetization. From measurement of the in-plane magnetization as a function of in-plane magnetic field, the linear fitting by Eq.(2.10) gives H_anisotropy.

merit: High-reliability direct measurements

weak-points: Due to sensitivity limitations, only a relatively large samples can be measured.

Using the Anomalous Hall effect

The Hall angle is linearly proportional to perpendicular component of the magnetization (See here). The measurement of the Hall angle as a function of in-plane magnetic field gives dependence M_x/M. The linear fitting by Eq.(2.10) gives H_anisotropy.

merits: (1) Nano-sized object can be measured (2) It can be combined with other magneto-transport measurements

weak-points: (1) Its measurement precision is rather sensitive to existence of magnetic domains. (2) It takes a relative a long time for a measurement.

Using the Tunnel magneto-Resistance (TMR)

The method uses a magnetic tunnel junction (MTJ), in which the magnetization of the “reference” layer is in-plane and the magnetization of the “free” layer is perpendicular-to-plane. When a magnetic field is applied in the in-plane direction, the magnetization of the “free” layer turns toward the magnetic field. From the measurement of the tunnel resistance, the angle between “free” and “reference” layers is calculated. It gives in-plane component of magnetization of "free" layer vs the in-plane magnetic field. From this data, the linear fitting by Eq.(2.10) gives H_anisotropy.

merits: (1) Nano-sized object can be measured (2) It can be combined with other magneto-transport measurements (3) It is fast measurements

weak-points: (1) Comparing with previous two methods, it is more indirect measurement. It is easy to get a systematical error with this method. (2) there is an undesirable influence of the dipole magnetic field from the reference electrode (3) The MTJ configuration is limited to a specific ferromagnetic metal, which has to provide a sufficient magneto resistance.

Anisotropy influenced by different effects

Switching time influenced by different effects

temperature Voltage- controlled magnetic anisotropy, VCMA effect spin-orbit torque, SOT effect Hanis rapidly decreases with increases of temperature The anisotropy field linearly decreases under a gate voltage. The slope is negative. Typical range of the slope is 40-100 Oe/V. The inset shows the in-plane component of magnetization under magnetic field applied in-plane. The anisotropy field is field at which magnetization turns fully in-plane. (Details of VCMA effects are here) Due to SOT effect, the Hanis increases at a negative current and decreases at a positive current. Additionally, the electrical current heats the sample. The heating causes the decreases of Hanis. Since SOT effect is linearly proportional to current, but heating ~I2, at a small negative current the Hani increases, but at a higher current effect of the heating dominates.(Details of SOT effect are here) Sample: Volt58A (L58B)   Sample R64A Volt58B Ta(5):FeCoB(1):MgO

Click on image to enlarge it.

Demagnetization field

Interface-type PMA

Bulk-type PMA

Non-linear effect for spin-orbit interaction and PMA

Doing the Science has never been easy

In contrast, the production of fakes in a larger numbers is much easier

Click on image to enlarge it

Under a strong magnetic field, the dependence the spin-orbit interaction and consequently PMA from perpendicular magnetic field deviates from a linear dependence. There are several reasons for that. The first reason is that the magnetic field may deform the atomic orbital, which enhances the SO and consequently PMA. As a result, the strength G_SO of the spin-orbit interaction (See Eq. 2.1) becomes dependent on the magnetic field. As was explained here, the breaking the symmetry only along the z-direction (perpendicularly to the interface) affects the PMA, the G_SO may be changed only by H_z .Than, Eq. 2.1 can be re- written as

where H_SO is the effective magnetic field of SO interactions; G_SO is the proportionality constant; H_intristic,z is the z-component of the total magnetic field; H_nl is the magnetic field, at which the strength of SO interaction increases in two times. It means the proportionality coefficient between H_SO and H_intristic,z increases in two times. H_nl is usually substantially larger than magnetization M. Usually it is about a few Teslas.

Which parameters are influenced by the non-linear SO interaction?

The dependence of the in-plane component of magnetization vs in-plane magnetic field deviates from linear at the magnetic field close to anisotropy field H_anis (the "non-linear tail")

It changes the value of anisotropy field H_anis

the equilibrium magnetization direction can be calculated from

The effective anisotropy field H_anis can be calculated as

where

is the anisotropy field in case without any non-linear SO component

(Part 0): Introduction. Content.

click on image to watch YouTube video

(Video)Measurement of coefficient of spin- orbit interaction and anisotropy field in a nanomagnet.

This video set explains the details of a high precision measurement method of the anisotropy field H_ani, coefficient of spin-orbit interaction k_SO , internal magnetic field in a nanomagnet H_int and a magnetic field H_off created by a spin-accumulation in a nanomagnet.

I will show you how to do measurement, how to process measured data and how to evaluate each parameter. You can use either my raw measurement data, which you can download, or your own data.

I will describe how to use the measurement data to improve the efficiency of the magnetization reversal either by a electrical current or by a gate voltage for a memory application and for a sensor application

I will explain the details of the magnetization reversal mechanism using the classical mechanics based on LL Eq. and using Quantum mechanics. I will explain the quantum mechanical meaning of the spin torque and the spin precession. I will explain the reasons why the newly-introduced field-like torque severely violates Laws of Quantum mechanics. I will explain about the interpretation problems and possible systematic errors of the popular measurement techniques, which are called the 2nd harmonic measurement and the zero-harmonic measurement. I will give some technological recommendations on how to improve the fabrication of a nanomagnet, which are based on my personal experience.

(video): Measurement of coefficient of spin- orbit interaction and anisotropy field in a nanomagnet.
(Part 1a): Spin-Orbit (SO) interaction. Perpendicular Magnetic Anisotropy (PMA)   (Part 1c): Internal magnetic field in a nanomagne   (Part 2): Measurement method of Spin-orbit interaction (SO)  & Perpendicular Magnetic Anisotropy (PMA)   (Part 3): Analysis of measured data & Evaluation of anisotropy field Hani and Hoff       (short content 1:)  the properties of spin-orbit interaction   (short content 1:)  the difference between Global, Local & Average Magnetic Fields.   (short content 1:)  how it is possible to measure  the strength of SO and strength of PMA.    (short content 1:) How download analyzing program & measurement data from my Website (short content 2:)   linear relation between the external magnetic field and Hso and about the symmetry and physics arguments, which are behind this linear relation   (short content 2:)  how to measure experimentally the internal magnetic field Hint   (short content 2:)    what holds  the magnetization along the easy magnetic axis and how to measure the strength of the holding force.   (short content 2:) math, which is used for an evaluation of anisotropy field Hani from the raw measurement data. I will explain the problems and the systematic errors, which exist for this procedure. (short content 3:)   how the deformation of the orbital at the interface creates the perpendicular magnetic anisotropy (PMA)   (short content 3:) experimental measurements of Hint in FeB nanomagnets.   (short content 3:)   the experimental setup and main idea behind the proposed measurement   (short content 3:)  how to evaluate, the magnetic field, which is created by the spin accumulation, or Hoff in short
(Part 4a):  1st analysis program & Calculation of Hani and Hoff   (Part 4b): Oscillation of kso due orbital deformation. 2nd calculation program: Calculation of kso,   analyzing Hani & Hoff   (Part 5a): Analyze of anisotropy field Hani. Measurement of coefficient of spin-orbit interaction kso     (Part 5b): Analyze of anisotropy field Hani & kso       (short content 1:) how to evaluate the Hani and Hoff field from raw measurement data.   (short content 1:)  dependence of kSO on the direction of the applied external magnetic field  with respect to the film surface   (short content 1:) analyze the measurement of the anisotropy field   (short content 1:) the relations between Hani and kSO (short content 2:)   how to do automatic fitting using my 1st program   (short content 2:)  peak was experimentally observed, which corresponds to the direction of the orbital deformation of top atoms at interface   (short content 2:)  the measured oscillations of Hani and why this oscillation is an important characteristic of a nanomagnet   (short content 2:) the reasons why the slope of this dependence is positive for a single-layer nanomagnet, but negative of a multilayer-nanomagnet and why this slope is an important characteristic of the fabrication technology of a nanomagnet (short content 3:)   all the specifics of the program operation, data format and how to check the flow of the automatic fitting procedure.   (short content 3:)  the reason why  this observed peak is related to the features of the orbital deformation at the interface.   short content 3:)   why the spin-orbit interaction depends on the polarity of the magnetic field and how to measure it.   (short content 3:)   influence of the demagnetization field on Hani and why does  it depend on the interface quality
(Part 6): Magnetic field Hoff created by spin accumulation. Origin and properties   (Part 7): Dependency of Hani & Hoff on electrical current (SOT effect) & gate voltage (VCMA effect)   (Part 8a): Magnetization reversal by electrical current & Gate voltage. Parametric reversal   (Part 8b): Magnetization reversal due to modulation of Hani or Hoff by electrical current or Gate voltage.       (short content 1:) the origin and properties the magnetic field created by spin accumulation, Hoff   (short content 1:)  the reason why the spin accumulation makes Hani and Hoff dependent on the electrical current.   (short content 1:)  the possibilities of magnetization reversal  utilizing the measured effects of current modulation of Hani & Hoff.   (short content 1:)  required conditions for the parametric magnetization reversal using the current modulation of Hani or using the current modulation of Hoff required conditions for the parametric magnetization reversal using the current modulation of Hani or using the current modulation of Hoff (short content 2:) dependence of Hoff on the strength of the spin-orbit interaction. The reason why Hoff is larger in a multilayer nanomagnet and smaller in a single-layer nanomagnet   (short content 2:)   the experimental data of the current dependency of Hani and Hoff .   (short content 2:)  quantum- mechanical description of the spin- transfer torque   (short content 2:)  ( inefficient parametric resonance 1): FMR measurement; ( inefficient parametric resonance 2):  microwave-assisted hard-disk recording (short content 3:)  dependency of direction and magnitude of Hoff on the external magnetic field.   (short content 3:)    the experimental data of the gate- voltage dependency  of Hani (VCMA effect)   (short content 3:)  the reason  why the parametric mechanism of the magnetization reversal is much more efficient than the spin-transfer-torque mechanism   (short content 3:)  the parametric magnetization reversal using a gate voltage (VCMA reversal).
(Part 9a):Torque, Landau-Lifshitz Eq., 2nd harmonic: Misinterpretation of an indirect measurement   (Part 9b): Spin dynamic: Landau-Lifshitz Eq vs Quantum mechanic   (Part 10a):  Fabrication Technology: Evaluation.   (Part 10b):  Fabrication Technology:  Optimization.       (short content 1:) difficulties for a correct interpretation  2nd-harmonic measurement.   (short content 1:) Landau - Lifshitz (LL)  equatitions vs. Quantum Mechanical description of spin dynamic   (short content 1:)   how the fabrication technology can be evaluated from the 3 dependencies: Hani, amplitude of oscillation of Hani and internal magnetic field as a function of kSO   (short content 1:)  7 technological recommendations on how to fabricate a high-quality interface and, therefore, on how to improve the performance of a nanomagnet. (short content 2:) reasons why two torques, the damp-like torque & field-like torque, were introduced for the explanation of the measurement of the 2nd harmonic.   (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 2:) about the critical thickness, at which the direction of  the equilibrium magnetization of the nanomagnet changes from perpendicular-to-plane to in-plane. The reasons why the critical thickness exists and why the critical thickness is strongly influenced by the interface quality.   (short content 2:)   possible technological optimizations to improve the efficiency of the Spin-transfer magnetization reversal. (short content 3:)  difficulties for a correct interpretation  zero-harmonic measurement.   (short content 3:)   why the newly-introduced field-like torque and the reasons why it severely violates Laws of Quantum mechanics   (short content 3:) how magnetic field of spin-orbit interaction HSO and demagnetization field influence the critical thickness   (short content 3:)   possible technological  optimizations to improve the  parametric reversal
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Content of this page represents my personal view and it is reflected my own finding. It may slightly different from the "classical" view on PMA, which is described in following references

M. T.Johnson et. al. Reports on Progress in Physics(1996) ; P.Bruno PRB (1989);

Erosion in Science

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Questions & Comments

(about dependency of the spin-orbit interaction of external magnetic field)

(from Xian-Peng Zhang) Q. I have a question about the spin-orbit magnetic field. Why it is proportional to the total magnetic field? (see below for details) May you provide me some references?

Your question is why the spin-orbit interaction depends on an external magnetic field and in which reference it was described for the first time.

This fact is known for a very long time and therefore it is more about the history of Science, in which I am not expert. I am sorry if I refer something incorrectly.

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 (SO fact 1) The fact about the relativistic origin of the magnetic field and the existence of the spin-orbit interaction (the existence of the magnetic field of the spin-orbit interaction). This fact is one of basic facts, on which the Special Theory of Relativity was build. Therefore, I think Einstein and Lorentz knew about this fact. Myself, I have read all about this fact (all details and many nice explanations) in Landau textbook "The Classical Theory of Fields"

 (SO fact 2) The fact that the strength of spin- orbit interaction (the SO magnetic field) is proportional to the strength of an externally applied magnetic field. This fact are direct consequence of the another important fact about SO:

(SO fact 3) The spin-orbit interaction cannot break the time- inverse symmetry. As a result, in order to manifest itself, the SO needs an external breaking of the time- inverse symmetry (E.g. by an orbital moment or by an external magnetic field  or an electrical current etc.).

Fact 2 follows very directly from the fact 3. The strength of SO is zero , when the time inverse- symmetry is not broken (case when the orbital moment is zero, there is no external magnetic field etc.) . The SO magnetic field becomes a non-zero only when the time- inverse symmetry is broken. The strength of  the SO ( the strength of the SO magnetic field)  becomes larger when the degree of the breaking of the time inverse symmetry becomes larger (e.g. the orbital moment becomes larger or the external magnetic field becomes larger etc.) .

Once again

zero magnetic field -> time-inverse symmetry is not broken -> spin-orbit is zero

external magnetic field becomes larger -> degree of breaking of time-inverse symmetry becomes larger -> spin-orbit becomes larger.

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(fact 3 explanation)  Why the spin-orbit interaction cannot break the time- inverse symmetry. This feature is the feature of the time-space symmetry and can be obtained from he Special Theory of Relativity (e.g. see Landau textbook " The Classical Theory of Fields  " ). A simplified understanding of the fact 3  can be as follows.  The spin-orbit effect is a relativistic effect and therefore can only occur when the object moves or the field, with which the object interacts, moves. The relativistic effect requires a movement.  The close movement speed is to the speed of the light, the stronger any relativistic effect is.  Any movement means a breaking of the time inverse symmetry. This is why the SO occurs only when the time-inverse symmetry is broken.

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Even though the direct relation between the spin- orbit interaction and the breaking of the time- inverse was understood  a long time ago  as well as the proportionality of the SO strength to the degree of TIS breaking (e.g., to the strength of the orbital moment or  to strength of the external magnetic field), all details and specifics have not been fully understood.

For example, the SO strength  depends on the orbital symmetry. Usually  the orbital is more symmetric in the bulk of a material and the SO is weaker, but at an interface the orbital becomes less symmetrical and the SO becomes larger.

You can read some details  here

This effect is used for molecular-recognition sensor, See here

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About experimental measurements.

The dependence of PMA and therefore SO on an external magnetic field is very strong. The experimental measurement of this dependence is rather simple  and the measured data well fit to the theory. The anisotropy field is a measurable parameter, which characterizes the strength of PMA and SO.

Near for all nanomagnet, I have studied, I measure the dependence of anisotropy field on a perpendicular magnetic field. Such data contains a lot of important information.

Introductory presentation on this measurement method I gave at MMM 2020 conference. You can watch it here

Video

I am strongly against a fake and "highlight" research