My Research and Inventions

click here to see all content or button bellow for specific topic

 

Perpendicular magnetic anisotropy (PMA)

Spin and Charge Transport

Abstract:

The equilibrium magnetization of ferromagnetic metal is in-plane, because of the demagnetization film. However, the magnetization of some ferromagnetic metal is perpendicular-to-plane. It is because these metals have substantial of the PMA effect. The PMA is originated by the spin-orbit interaction.

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.


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 fields) .


Content

1. Spin-orbit interaction and PMA

2. Origin of PMA

3. anisotropy field

4. Measurement of PMA

5. Non-linear PMA


 

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 truly appreciate your comments, feedbacks and questions

I will try to answer your questions as soon as possible

 

Comment Box is loading comments...