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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.


 


Content

1. Spin-orbit interaction and PMA

2. Origin of PMA

3. anisotropy field

4. Measurement of PMA

5. Non-linear PMA


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

(feature 1) The spin-orbit interaction and the orbital deformation

The Einstein theory of relativity states that an electron, which moves in a static electrical field, experiences an effective magnetic field. This effective magnetic field is called the effective magnetic field of the spin-orbit interaction HSO (See here for details).   

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

The electrical field of atomic nuclear may induce a substantial HSO, because in an atomic orbital the electron is moving very close to the atomic nuclear at a high speed. However, the value of HSO substantially depends on the orbital symmetry. It is because on its orbital path the electrons experience two opposite directions of  HSO due the different angles between electron movement direction and the electrical field of the nuclear on different parts of the orbital path.  For example, in the case of a spherical orbital the two contributions to HSO are equal. As a result, the electron of the spherical orbital does not experience any spin-orbit interaction.  The case is the different when the orbital is deformed from the spherical shape. In the case when the orbital is elliptical or/and the orbital center is shifted from the nuclear position, the HSO may be substantial and it is proportional to the degree of the orbital deformation.

Additional condition for the HSO to be a non-zero is that an external magnetic field Hext should be applied to the orbital. The HSO is linearly proportional to the Hext and the direction of HSO is along the direction of the Hext. The HSO is a non-zero only when the Hext is applied along the direction of the orbital deformation. Figure 1 describes the spin-orbit interaction for different mutual directions of orbital deformation and Hext. When the orbital is spherical (Fig.1(a)), the HSO equals to zero independently on the direction of the Hext. When the orbital is deformed and the Hext is applied along the orbital deformation, (Fig.1(b)),  the HSO is non- zero and in direction of  the Hext. When the orbital is deformed and the Hext is applied perpendicularly the orbital deformation (Fig.1(c)), the HSO becomes zero again, because along the Hext. the orbital is symmetrical. In the cases of Figs. 1(a,c), the magnetic energy Emag of the electron equals to

where S is the electron spin and mB is the Bohr magneton.

In the cases of Figs. 1(b), the absolute value of Emag  is larger.  The Emag equals to

The electron magnetic energy is larger in the case Fig.1(b) and smaller in cases Fig.1 (a) and Fig.1 (c). Therefore, the deformation of the orbital changes substantially the electron magnetic energy and therefore the magnetic properties of a magnetic film. This property is used in disclosed invention to detect the proximity of the tested molecule near surface of a nanomagnet.

(feature 2) The origin of the PMA

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
Case 1: Magnetization is in-plane. There is no PMA

Figure 2 shows the cross-section of a nanomagnet is shown as an array of its electronic orbitals. The arrow shows the equilibrium magnetization direction of the nanomagnet. Figure 2(a) shows the case when the orbitals at interface are spherical and not deformed. In this case, the equilibrium magnetization of the magnetic film is in-plane. It is because the total magnetic energy of all the bulk and interface electrons are smaller when magnetization direction is in-plane magnetization than perpendicular-to-plane. The electron experience two types of magnetic fields. The first field is the intrinsic magnetic field HM, which is directed along the magnetization. The second field is the demagnetization field Hdemag, which is directed perpendicular to interface and along the directed perpendicular to the interface component of the magnetization. Therefore, the magnetic energy in the case of magnetization along and perpendicular to interface is calculated as

Since, the magnetization for the case of Fig.2(a) is aligned in-plane.

Case 2: PMA. Magnetization is perpendicular-to-plane.

Figure 2(b) shows the case when the orbitals at interface are deformed perpendicularly to the interface. In this case, the equilibrium magnetization of the magnetic film is perpendicularly to the plane. It is because the magnetic energy of the interface electrons becomes smaller in the case of the perpendicular magnetization. The magnetic energy of the interface electrons in the case of magnetization along and perpendicular to interface is calculated as

Even for the case of a small deformation. As a result, and the magnetization may be aligned perpendicular-to -plane. It depends on the film thickness. The orbitals in the bulk of the film are not deformed and their magnetic energy is described by Eqs.(3). The total magnetic energy Emag,total is the sum of the magnetic energies of the bulk and interface electrons and can be calculated as [2]

where t is the film thickness, is Emag,interface is the magnetic energy of the interface electrons and Emag,bulk is the magnetic energy of the bulk electrons per film thickness

Figure 3 shows the magnetic energy of a FeB as function of its thickness for two cases of different Emag,interface. As was explained above, the value of Emag,interface  depends on the orbital deformation. The magnetization of a thinner film is perpendicular-to -plane. The magnetization of a thicker film is in -plane. The thickness, at which the magnetization changes the direction, depends on the value of Emag,interface , therefore on the degree of the orbital deformation at the interface. For example at the thickness of 1 nm, the magnetization direction can be switched between in-plane and perpendicular-to –plane by the modulation of the orbital deformation.

[2]  M.T. Johnson,et.al., Reports Prog. Phys. 59, 1409 (1996)

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.


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.


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);

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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