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Perpendicular magnetic anisotropy (PMA)

Spin and Charge Transport

Abstract:

The equilibrium magnetization of a ferromagnetic film is in-plane, because of the demagnetization field. Due to the demagnetization field, the magnetic energy is smaller when the magnetization direction is in-plane than perpendicular- to- plane. However, at an interface an electron may have an additional magnetic energy due to the spin-orbit interaction. This additional energy may be substantial and it makes the total magnetic energy smaller in the perpendicular-to-plane direction. As a result, the direction of the equilibrium magnetization becomes perpendicular-to-plane. This effect is called the perpendicular magnetic anisotropy (PMA).

The spin-orbit interaction and consequently the PMA becomes strong only when the electron orbital becomes asymmetric (deformed) in one direction. When the orbital deformation is due to the breaking periodicity at an interface, the effect is called the interfacial PMA. When the orbital deformation is due to the crystal spacial asymmetry, the effect is called the bulk PMA.


measurement method

External magnetic field H|| is applied in-plane and perpendicularly to the magnetization (along a hard axis). As a result, the magnetization turns towards the magnetic field. The magnetic field, at which the magnetization turns completely in-plane, is called the anisotropy field Hanis . The Hanis is a measure of the strength of the perpendicular magnetic anisotropy (PMA). In order to characterize the features of the PMA, an additional field Hz is applied perpendicularly to the film
(what is measured) The anisotropy field Hanis vs applied perpendicular magnetic field Hz
method has been developed in 2019-2020 by Zayets
Click on image to enlarge it

 


Content

1. Spin-orbit interaction and PMA

2. Origin of PMA

3. anisotropy field

4. Measurement of PMA

5. Non-linear PMA


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


Energy or magnetic field. What is the PMA?

What object or effect does the PMA describe?

Magnetic anisotropy in a Fe film

thickness dependence

a thick film (t>1.5 nm)

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

The PMA energy EPMA is the product of HPMA and M (magnetic energy of localized electrons).

In some cases it is preferable to use the PMA energy EPMA instead of the magnetic field HPMA. An example is the thermo- activated magnetization reversal, when the magnetization reversal occurs when the energy of thermo fluctuation is larger than EPMA.

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

What is the PMA energy?

The PMA energy EPMA 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 HPMA and EPMA

HPMA and EPMA are substantially different at the interface and in the bulk of a ferromagnetic material.

The difference of HPMA 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 -> HSO>Hdemag, but in the bulk HSO< Hdemag. 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 HSO< Hdemag. Equilibrium magnetization of a thicker FeTbB film is perpendicular- to - plane

Averaging over film thickness

The measured parameters of the PMA magnetic field HPMA and PMA energy EPMA

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 HPMA and EPMA on magnetization direction

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

(direction- dependence of Hdemag) The demagnetization magnetic field Hdemag 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 HSO) The magnetic field os spin- orbit interaction HSO 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 HSO 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 HSO. depends on the orbital deformation (See SO interaction), HSO becomes larger when the magnetization is perpendicular-to-plane and HSO 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 thermo energy kT. Otherwise, the magnetization can be thermally switched and the date can be lost (See thermo-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 HSO, when the magnetization is along the deformation (perpendicular-to-interface). However, there is no such field, when magnetization is in-plane. The magnetic field HSO 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


Calculation of PMA

(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 HSO .The electric field of atomic nucleus may induce a substantial HSO, 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 HSO substantially depends on the orbital symmetry. It is because the electron experiences different directions of the HSO  on different parts of the orbit that may compensate each other. For example, in the case of a spherical orbital the contributions to the HSO are equal and opposite in sign and the resulting HSO=0. In the case of a deformed orbital, when the orbital is elliptical or/and the orbital center is shifted from the nuclear position, the HSO becomes substantial and proportional to the degree of the orbital deformation. Also, the HSO is linearly proportional to an external magnetic field Hext. Figure 1 demonstrates how the HSO 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 HSO equals to zero independently on the direction of the Hext. When the orbital is deformed and the Hext is applied perpendicularly the orbital deformation (see Fig.1b), the HSO is also zero, because the orbital is symmetrical along the direction of the Hext. For the cases of Figs. 1a and 1b, the magnetic energy of the electron Emag is equal to:

where S is the electron spin and is mB  the Bohr magneton. When the orbital is deformed and the Hext applied along the orbital deformation (Fig.1c), the HSO becomes a non‐zero and proportional to the Hext. In this case the Emag   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 inplane or perpendiculartoplane 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 crosssection 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 HSO. 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, μB is the Bohr magneton, HM is the intrinsic magnetic field induced by the magnetization, HD is the demagnetization field, which is directed perpendicular-to-plane and proportional to the perpendicular component of the HM. DEB is the difference of the magnetic energy for two magnetization directions. For the bulk electron, the DEB is negative.

The interface electrons experience the spin-orbit magnetic field HSO, additionally to HM  and HD, because their orbitals are deformed. The HSO 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 DEi  can be calculated as

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

where NB and Ni are the numbers of the bulk and surface electrons, correspondingly. Since ΔEB 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 EPMA of a FeB thin film as function of its thickness for two cases of different orbital deformation at the interface with the interface magnetization ΔEi 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 ΔEi. 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 HSO 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 HSO 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 HSO is small for the p- and d- electrons in this fully centrosymmetric electrical field of one nuclear. 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 HSO.


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.
click on image to enlarge it

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 HSO. A breaking the spacial symmetry is important, but not the polarity of the breaking. The most effective enhancement of HSO is when the center of electron orbital is shift of the position of the nuclear. (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 HSO 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 HSO.

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

Fact 2. The electrical field of atomic nuclear 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 nuclear 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

Key properties of SO interaction:

Enhancement of magnetic field due to SO interaction

The external magnetic field Hext induces the effective magnetic field of the spin-orbit interaction HSO, which is is in the same direction as the external magnetic field.. Therefore, the total magnetic field, which the electron experiences, becomes larger. The deformed electron orbital is shown. click here or on image to enlarge it

More details are here

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 HSO. The magnitude HSO is proportional to the degree of the breaking symmetry.

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

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

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

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

The HSO 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 HSO exists only when the external magnetic field is directed along this direction (along the z-direction), but there is no HSO 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 HSO is induced only in this direction. The directional dependence of HSO originates the PMA (See below)

 

Fact 6. The orbital symmetry in close proximity to nuclear 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 nuclear, 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 nuclear. Only this region makes substantial contribution to the HSO and EPMA. For this reason, the orbital symmetry in this region is critically important for the PMA.

As a result: HSO is larger when the center of orbital slightly shifted from the position of the nuclear 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 HSO and EPMA.

See also VCMA effect and SO torque

Fact 8. The parity symmetry and spin-orbit interaction

Parity symmetry of SO interaction

The HSO does not dependent on wether the electron orbital is shifted along Hext or in opposite direction. This feature is called the parity symmetry of SO interaction.

click on image to enlarge

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 nuclear in + x direction induces absolutely equal HSO and EPMA

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 HSO and EPMA. 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 HSO and EPMA

 

 

 

 

 


Physical Origin of perpendicular magnetic anisotropy (PMA)

Origin of PMA:

Directional dependence of HSO

The electron orbital is deformed only in z-direction (for example,toward film interface). The external magnetic field Hext induces the effective magnetic field of the spin-orbit interaction HSO, which is is in the same direction as the external magnetic field. When Hext is directed along z-axis, HSO is large, because the orbital is deformed in this direction. When Hext is directed along y- or x-axis, HSO is zero because the orbital is symmetrical in this direction. .

click on image to enlarge it

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 HSO , because orbital deformation in this direction. When the magnetization is in-plane, there is no HSO, 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= Hintristic

Magnetic energy= -Hintristic · M

Magnetization is perpendicular to plane:

Total magnetic field= Hintristic+HSO

Magnetic energy= -(Hintristic +HSO )· 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 Hanis

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

 

Anisotropy field and the dependence in-plane magnetization vs applied in-plane magnetic field reveals key features of the spin-orbit interaction and the PMA (For example, see VCMA effect or SOT effect)

 

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 Hanisotropy with a high precision is an important step to almost any magneto transport measurement.

 

 

As was explained here,

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 nuclear

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 nuclear.

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

 

From the fact 3, the SO magnetic field HSO,z can be described as

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

where GSO is the proportionality constant; H is the external magnetic field and M is the magnetization.

From Eq.(2.1) the magnetic energy can be calculated as:

where

Substituting Eqs. (2.1ab) into (2.2), the magnetic energy is calculated as

In the case when external magnetic field is applied only in-plane (along the x-direction)

The magnetic energy is calculated as

The Eq.(2.5) can be rewritten as

where

The equilibrium magnetization direction can be found from condition of minimum magnetic energy. The energy minimum can be found from Eq. (2.6) as

The solution of Eq.(2.8) gives in plane magnetization Mx as

where the anisotropy field is defined as

The in-plane magnetization Mx is linearly proportional to applied in-plane magnetic field and it is calculated as

Eq.(2.10) is well-matched to the experimental measurements. (See Fig.3)

Using Eq.(2.9a), the magnetic energy can be calculated as

Without external magnetic field the magnetic energy is

The PMA energy is defined as an energy difference between cases when magnetization is in-plane and perpendicularly to the plane. From Eq.(2.3b) it can be calculated

The ratio of the PMA energy to thermo energy is called delta.


Measurements of the anisotropy field Hanisotropy

3 following experimental method are most often used to measure Hanisotropy

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 Hanisotropy.

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 Mx/M. The linear fitting by Eq.(2.10) gives Hanisotropy.

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 Hanisotropy.

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

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 GSO 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 GSO may be changed only by Hz .Than, Eq. 2.1 can be re- written as

where HSO is the effective magnetic field of SO interactions; GSO is the proportionality constant; Hintristic,z is the z-component of the total magnetic field; Hnl is the magnetic field, at which the strength of SO interaction increases in two times. It means the proportionality coefficient between HSO and Hintristic,z increases in two times. Hnl 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 Hanis (the "non-linear tail")

It changes the value of anisotropy field Hanis

 

the equilibrium magnetization direction can be calculated from

The effective anisotropy field Hanis can be calculated as

where

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

click here to expand and see how to obtain Eqs.(5.12a) and (5.14)

Non-linear spin-orbit interaction

The total intrinsic magnetic field is the sum of the extrinsic magnetic field H and the magnetization field M. Therefore, Eq. (5.1) becomes

The magnetic energy can be calculated as

where component proportional to Mz is

and component proportional to Mx is

The total energy is the sum of Eqs. (5.3) and (5.4):

In the case when the magnetic field is applied in-plane (Hx =0), Eq.(5.7) is simplified to

or

where

The minimum of the energy corresponds to the equilibrium magnetization direction, which can be found as

Therefore, the equilibrium magnetization direction can be calculated from

or

where

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

In the case field H is smaller and is not close to Hanis, the following condition is satisfied

From (5.13), (5.12a) and (2.10) the effective anisotropy field can be calculated as

 

 

 

 

 

 


 

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);

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

I am strongly against a fake and "highlight" research

 

 

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