more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11

Perpendicular magnetic anisotropy (PMA) Spin and Charge TransportAbstract:The equilibrium magnetization of a ferromagnetic film is inplane, because of the demagnetization field. Due to the demagnetization field, the magnetic energy is smaller when the magnetization direction is inplane than perpendicular to plane. However, at an interface an electron may have an additional magnetic energy due to the spinorbit interaction. This additional energy may be substantial and it makes the total magnetic energy smaller in the perpendiculartoplane direction. As a result, the direction of the equilibrium magnetization becomes perpendiculartoplane. This effect is called the perpendicular magnetic anisotropy (PMA).The spinorbit 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.
Contentclick on the chapter for the shortcut1.Energy or magnetic field. What is the PMA?1a. What is the PMA energy?1.c. Spinorbit interaction and PMA() Model of perpendicular magnetic anisotropy. oversimplified Neel description() Model of perpendicular magnetic anisotropy. full description based on properties of the spinorbit interaction() Comparison of the oversimplified Neel description and full description of PMA2. Origin of PMA2.1 Calculation of PMA in a thin film3. Calculation of Anisotropy field3.1 Case 1: There is no perpendicular external magnetic field3.2 Case 2: There is a perpendicular external magnetic field3. anisotropy field4. Measurement of PMA() Internal magnetic field inside a nanomagnet5. Nonlinear PMA6. Explanation video7. Video:(1) Measurement of coefficient of spin orbit interaction k_{SO} and anisotropy field H_{ani} in a nanomagnet.(2) Measurement of strength of spinorbit interaction(3) measurement of Anisotropy field.() Questions & Comments(7) about dependency of spinorbit interaction on an external magnetic field.
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See detailed calculations of anisotropy field and PMA energy here CalculationHanisPMA.pdfPerpendicular 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 inplane of the filmQ. What is the reason the magnetization of the nanomagnet becomes perpendicular to the plane?A. The spinorbit interaction (fact) The PMA or the spinorbit interaction can be originated either at a nanomagnet interface or in the bulk of the nanomagnet. The interfaceoriginated PMA is stronger and more interesting, because engineering of only the one atomic layer at interface, a huge PMA and unique properties can be achieved. Q. Why is the nanomagnet with PMA important for applications?A. It is because a huge energy is accumulated in only one atomic layer at interface. This allows to fabricated experimentally small nano sized devices with an unique functionality. When the volume of an object is reduced, the object energy reduces proportionally to the volume. In order for the object to function as a device (a memory cell, a sensor cell etc), the magnetic energy between its two stable states should be sufficiently larger than the thermal energy kT. Otherwise, it will be continued random thermal switching between two states and the object cannot function as a device.
Key properties of the spinorbit interaction, which lead to the existence of PMA(key property 1): The magnetic field of the spin orbit interaction H_{SO} is proportional to an externally applied magnetic fieldThe SO by itself cannot break the time inverse symmetry and, therefore, to manifest itself. The H_{SO} exists only when there is an external magnetic field. (key property 2): The spinorbit interaction substantially increases (changes) when the electron orbital is deformedWhen atomic orbital is deformed, the electron becomes closer to the atomic nucleus, where the electrical field is much stronger. As a result, the SO becomes larger.
Devices, functionality of which is based on PMA
(device 1): Hard diskIn order to memorize a huge amount of data onto a small surface of the hard disk, the surface area of one bit is very small ~10 nm x10 nm. Only the SO is able to hold magnetic date in such a tiny area. (uniqueness of PMA): There is no known effect except PMA, which allows to store magnetic data in area of 10 nm x10 nm, protecting data from the unwanted thermal switching (device 2): Magnetic random access memory (MRAM)In MRAM, one bit of data is memorized by two stable magnetization of the nanomagnet. In order to compete with other memory schemes (e.g. DRAM), the volume of nanomagnet should be smaller than ~30 nm x 30 nm x 1 nm (uniqueness of PMA): There is no known effect except PMA, which allows to store magnetic data in volume of 30 nm x 30 nm x 1 nm, protecting data from the unwanted thermal switching (device 3): Molecular recognition sensorIn the molecular recognition sensor, a surface touching event by a molecule or an virus is detected by a measure of a change of spinorbit interaction the at the sensor interface. In order to practical the detection area of the sensor should be comparable with the size of the molecule or virus. The practical size is ~30 nm x30 nm. See more details here. (uniqueness of PMA): There is no known effect except PMA, which allows to store magnetic data in the area of ~30 nm x30 nm protecting data from the unwanted thermal switching. (device 4): Nanosized microwave generatorIn this device the microwave is generated by a stable magnetization precession in a nanomagnet. The A stable singledomain magnetization oscillation can be achieved in a nanomagnet of the volume ~300 nm x 300 nm x 1 nm (usefulness of PMA): Even though the stable magnetization oscillation can be achieved in a nanomagnet without PMA, the microwave generator made of a nanomagnet with PMA is more reliable and stable. (device 5): Nanosized magnetic sensorA nanomagnet as a part of a tunnel magnetic junction (MTJ) can be used as a nano sensor of a magnetic field. The spatial resolution of the sensor is better for a smaller size of the nanomagnet. Calculation of PMA & Magnetic Anisotropycalculation of magnetization direction in a nanomagnet with PMA under an external magnetic fieldSee detailed calculations of anisotropy field and PMA energy here CalculationHanisPMA.pdfHow the magnetization direction is calculated: (step 1) The magnetic energy is calculated for the nanomagnet (step 2) The magnetic energy is minimized with respect to the magnetization direction  step 1 Calculation of Magnetic energy In the case of zero orbital moment, the magnetic energy of an electron is a product of the magnetization M (or the same the total spin of localized d electrons) and the total magnetic field.
Energy or magnetic field. What is the PMA?What object or effect does the PMA describe?
The PMA describes the magnetic field H_{PMA}=H_{SO} H_{demag}, which is a sum of demagnetization magnetic field H_{demag} and the magnetic field of spin orbit interaction H_{SO}. In the case of a PMA sample, the direction of H_{SO} is along magnetization M, the H_{demag} is always opposite to the M and H_{SO}>H_{demag}. As a result, H_{PMA}>0 and M ( the total spin of localized electrons (magnetic moment)) is aligned along H_{PMA}. and perpendicularly to the film surface. The PMA energy E_{PMA }is the product of H_{PMA} and M (magnetic energy of localized electrons). In some cases it is preferable to use the PMA energy E_{PMA} instead of the magnetic field H_{PMA}. An example is the thermally activated magnetization reversal, when the magnetization reversal occurs when the energy of a thermal fluctuation is larger than E_{PMA}. In some cases it is preferable to use the magnetic field H_{PMA} instead of the PMA energy E_{PMA} . An example is calculation of a torque in external magnetic field H_{ext}, when the magnetization procession M occurs around the field, which is the vector sum of H_{ext} and H_{PMA}. What is the PMA energy?The PMA energy E_{PMA} is defined as a difference between magnetic energies for the cases when when the magnetization is perpendiculartoplane and inplane (it is important) The primary parameter of the PMA effect is the magnetic field. Even the use of the PMA energy is convenient for some applications.
Thickness dependence of H_{PMA} and E_{PMA}H_{PMA} and E_{PMA} are substantially different at the interface and in the bulk of a ferromagnetic material.The difference of H_{PMA} is because of difference of the orbital symmetry. The orbital symmetry at interface is substantially different that in the bulk because of a surface reconstruction at interface and the asymmetry of interaction with atoms from below and above for the interface atoms. Example: Fe film: at interface > H_{SO}>H_{demag}, but in the bulk H_{SO}< H_{demag}. As a result, equilibrium magnetization of a thinner Fe film is perpendicular to  film plane and of a thicker film is in plane (see here or here). FeTbB film: in the bulk H_{SO}< H_{demag}. Equilibrium magnetization of a thicker FeTbB film is perpendicular to  plane Averaging over film thickness The measured parameters of the PMA magnetic field H_{PMA} and PMA energy E_{PMA} where t is the film thickness (fact) In case when is substantially different at interface and bin the bulk, the magnetic properties of a thin film substantially depend on the film thickness (see here or here).Dependence of H_{PMA} and E_{PMA} on magnetization directionBoth the demagnetization magnetic field H_{demag} and magnetic field of spin orbit interaction substantially depend on the magnetization direction. It makes H_{PMA} substantially dependent on the magnetization direction(direction dependence of H_{demag}_{}) The demagnetization magnetic field H_{demag} is always directed perpendicularly to film surface and opposite to the magnetization direction. Its magnitude is proportional to the perpendicularto plane component of the magnetization. (direction dependence of H_{SO}_{}) The magnetic field os spin orbit interaction H_{SO} is directed along to the magnetization direction (along the magnetic field applied to the atomic orbital). Its magnitude depends on the orbital symmetry (or degree of orbital deformation). Usually atomic orbital is deformed along the film normal. As a result, the H_{SO} may be substantially larger when magnetization M is perpendicular to the film surface in comparison to the case when the magnetization is in the plane.
Why the magnetization direction depends on the magnetization direction with respect to the film normal?There is a discontinuity at the film interface. As a result, atomic orbitals are deformed at the interface. Since the effective magnetic field of spinorbit interaction H_{SO}. depends on the orbital deformation (See SO interaction), H_{SO} becomes larger when the magnetization is perpendiculartoplane and H_{SO} becomes smaller when the magnetization is inplane. Correspondingly, the magnetic energy becomes different for the perpendiculartoplane and inplane magnetization directions. Why do we care whether the equilibrium magnetization is inplane or perpendiculartoplane?The energy of the spinorbit 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 spinorbit 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 spinorbit interaction is largest for the film with the PMA ( the direction of the equilibrium magnetization is perpendiculartoplane).
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 2050 times) larger than the thermal energy kT. Otherwise, the magnetization can be thermally switched and the date can be lost (See thermallyactivated 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 spinorbit interaction makes the equilibrium magnetization perpendiculartoplane?At the interface the electron orbital is substantially deformed towards the interface. Due to this deformation, the spinorbit interaction is significantly enhanced. This means that there is a large effective magnetic field of the spinorbit interaction H_{SO}, when the magnetization is along the deformation (perpendiculartointerface). However, there is no such field, when magnetization is inplane. The magnetic field H_{SO} may become larger than demagnetization field and the negative magnetic energy become smaller for perpendicular direction than for the inplane direction. As a result, the perpendiculartoplane 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 inplane magnetization and some films have perpendiculartoplane magnetizations?Two factors are factors are important to understand the equilibrium magnetization direction of a ferromagnetic film: (1) the directional dependence of the spinorbit interaction; (2) the orbital deformation at an interface
Model of perpendicular magnetic anisotropy. oversimplified Neel descriptionThis classical model is based on some oversimplified assumptions. This model ignores the specific features of the spinorbit interaction, which creates the magnetic anisotropy. However, this model is a very simple and can be used for a rough estimate.See detailed calculations of anisotropy field and PMA energy by this description here NeelPMA.pdfIn this model, the total magnetic energy, which is a sum of the energy of the magnetic anisotropy EPMA and the magnetic dipole energy can be calculated as in which the energy of the magnetic anisotropy is assumed to be proportional to a square of the magnetic component M_{z} along the easy axis. where θ is the angle between the magnetization M and the film normal, φ is the angle between the magnetic field H and the film normal
Model of perpendicular magnetic anisotropy. full description based on properties of the spinorbit interactionSee detailed calculations of anisotropy field and PMA energy by this description here CalculationHanisPMA.pdfThe sum of the magnetic energy due to the magnetic field of the spin, the demagnetization field, the magnetic field of the spinorbit interaction and demagnetization field gives the total magnetic energy as where k_{so} is the coefficient of the spinorbit interaction and k_{demag} is the demagnetization coefficient
Comparison of the oversimplified Neel description and full description of PMA
Calculation of PMASee detailed calculations of anisotropy field and PMA energy here CalculationHanisPMA.pdf(step 1) The spin‐orbit interaction and the orbital deformationThe 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 spinorbit (SO) interaction H_{SO} .The electric field of atomic nucleus may induce a substantial H_{SO}, because the electron moves with a very high speed on its atomic orbital in close proximity to the atomic nucleus. However, the value of the H_{SO }substantially depends on the orbital symmetry. It is because the electron experiences different directions of the H_{SO} _{ }on different parts of the orbit that may compensate each other. For example, in the case of a spherical orbital the contributions to the H_{SO} are equal and opposite in sign and the resulting H_{SO}=0. In the case of a deformed orbital, when the orbital is elliptical or/and the orbital center is shifted from the nucleus position, the H_{SO}_{ }becomes substantial and proportional to the degree of the orbital deformation. Also, the H_{SO} is linearly proportional to an external magnetic field H_{ext}. Figure 1 demonstrates how the H_{SO} changes depending on the direction of the orbital deformation and the direction of the external magnetic field. When the orbital is spherical (Fig.1a), the H_{SO} equals to zero independently on the direction of the H_{ext}. When the orbital is deformed and the H_{ext} is applied perpendicularly the orbital deformation (see Fig.1b), the H_{SO} is also zero, because the orbital is symmetrical along the direction of the H_{ext}. For the cases of Figs. 1a and 1b, the magnetic energy of the electron E_{mag} is equal to: where S is the electron spin and is m_{B} the Bohr magneton. When the orbital is deformed and the H_{ext} applied along the orbital deformation (Fig.1c), the H_{SO} becomes a non‐zero and proportional to the H_{ext}. In this case the E_{mag} equals to: The absolute value of the electron magnetic energy is larger in the case shown in the Fig. 1c and smaller in the cases shown in the Figs. 1a and 1b. Therefore, the orbital deformation substantially changes the electron magnetic energy and therefore magnetic properties of the ferromagnetic film.
(step 2) PMA of a thin filmThe equilibrium magnetization of an isotropic ferromagnetic thin film can be either in‐plane or perpendicular‐to‐plane depending on the deformation of the electron orbitals at the film interface and the thickness of the film. The interaction of analyte molecules with interface electrons of the ferromagnetic film leads to the orbital deformation of the interface electrons and consequently to a change of the PMA. Consequently, the change of magnetization direction of the ferromagnetic film due to the change of the PMA is used as the molecular detection mechanism in the disclosed invention. The physical phenomenon of the PMA and the reason, why the orbital deformation defines the strength of the PMA, are explained as follows. Figure 2 shows a cross‐section of a nanomagnet as an array of its electronic orbitals. The magnetization of a thicker film (Fig. 2a) is inplane, while the magnetization of a thinner film (Fig.2b) is perpendiculartoplane. 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 inplane. For the interface electrons the magnetic energy is smallest when the magnetization is perpendiculartoplane. 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 inplane. 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 perpendiculartoplane for the substantially thin film. The reason why the dependence of the magnetic energy on the magnetization direction is different for the bulk and interface electrons, is their orbital shape. The orbital of the bulk electrons is spherical. They do not experience any H_{SO}. The magnetic energy E_{,b} and E_{⊥,b} of a bulk electron for the case of inplane and perpendiculartoplane magnetization can be calculated as where S is electron spin, μ_{B} is the Bohr magneton, H_{M} is the intrinsic magnetic field induced by the magnetization, H_{D} is the demagnetization field, which is directed perpendiculartoplane and proportional to the perpendicular component of the H_{M}. DE_{B} is the difference of the magnetic energy for two magnetization directions. For the bulk electron, the DE_{B} is negative. The interface electrons experience the spinorbit magnetic field H_{SO}, additionally to H_{M} and H_{D}, because their orbitals are deformed. The H_{SO} is a nonzero only when the magnetization direction is along the deformation and therefore perpendiculartoplane. For an interface electron, the magnetic energies E_{,i} and E_{⊥,i } for the inplane and perpendiculartoplane magnetizations, respectively, and their difference DE_{i} can be calculated as The H_{SO} is proportional to the degree of the orbital deformation. Even when the deformation is small, H_{SO} > H_{D} and the ΔE_{i} is positive. The difference of the magnetic energy for the  and ⊥ magnetization directions is called the PMA energy (E_{PMA}) and can be calculated as where N_{B} and N_{i} are the numbers of the bulk and surface electrons, correspondingly. Since ΔE_{B} is negative, Eq. (1.5) can be simplified as
where t is the film thickness, β is the constant, which depends on the symmetry of crystal lattice. In the simplest case of a cubic lattice, it can be calculated as where a is the lattice constant. Figure 3 shows the energy of the perpendicular magnetic anisotropy E_{PMA} of a FeB thin film as function of its thickness for two cases of different orbital deformation at the interface with the interface magnetization ΔE_{i} of 1.15 and 0.85 mJ/m2. The magnetization of the thinner film is perpendiculartoplane. The magnetization of the thicker film is inplane. The thickness of the film at which the magnetization changes direction depends on the degree of orbital deformation at the interface and therefore on the value of interface magnetization energy ΔE_{i}. For example, at the thickness of 1 nm, the magnetization can be switched between the inplane and perpendiculartoplane directions by the modulation of the orbital deformation. The orbital deformation at interface can be of two types. The orbital can be elongated or shortened along the interface normal. Both types of the orbital deformation lead to the increase of the H_{SO} and the PMA energy. How and why the electron orbital is deformed?The spinorbit interaction is stronger in the case of a nonsymmetrical electron orbital. The less symmetrical orbital is, the stronger H_{SO} the electron experiences. The spinorbit interaction is weaker when induced by centrosymmetric electrical field. Even though the p and dorbital of a hydrogen atom is nonsymmetrical, the H_{SO} is small for the p and d electrons in this fully centrosymmetric electrical field of one nucleus. The case is different, when many atoms interacts. Such interaction makes their orbits nonsymmetrical. 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 nonsymmetric. As a result, the electron of this orbital experience the strong H_{SO}.
How the polarity of the orbital influence the sensor output signal?It does not matter whether orbital is elongated or squeezed. Both deformations induce the same polarity of the change of H_{SO}. A breaking the spacial symmetry is important, but not the polarity of the breaking. The most effective enhancement of H_{SO} is when the center of electron orbital is shift of the position of the nucleus. (See more details in the spinorbit 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 interfacetype 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 spinorbit 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 inplane and a thick FeTbB is perpendicularto 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 wellmatched to all experimental fact. For example, the existence of H_{SO} well explains the linear dependence of inplane component of magnetization as function of applied inplane magnetic field (See anisotropy field) .
Spinorbit (SO) interaction is the origin of Perpendicular magnetic anisotropy (PMA)
for more details see here Fact 1. Relativistic origin of spinorbit interaction An object, which moves in an electrical field, experience an effective magnetic field. This effective magnetic field is 100% magnetic field and it is indistinguishable from any other magnetic field. This magnetic field is called the SO magnetic field H_{SO}. The origin of the SO magnetic field is the relativistic nature of the electromagnetic field. Fact 2. The electrical field of atomic nucleus induces the SO magnetic field, which originates the perpendicular magnetic anisotropy (PMA) Because of its relativistic nature, the SO effect is weak. Only a very strong electrical field may induce sizable SO magnetic field. Only the electrical field of atomic nucleus is sufficient strong to create sufficiently strong SO magnetic field (130 kGauss), which induces the PMA Fact 3. Only an electron in a nonsymmetrical orbital experiences SO interaction
An electron in spherical orbital (s orbital) does not experience any SO interaction (explanation is here). Only when orbital symmetry is broken, the electron experience SO interactions. There are several possibilities to break the orbital symmetry. Each of them induces H_{SO}. The magnitude H_{SO} is proportional to the degree of the breaking symmetry. Fact 4. The electrical field of nucleus cannot break time inverse symmetry. The SO magnetic field exists only when there is external magnetic field. The H_{SO} exists only when there is some external magnetic field. The H_{SO} is always is zero in absence of the external magnetic field. The H_{SO} is always in the same direction as the external magnetic field. The H_{SO} may be significantly larger than the external magnetic field
Fact 5. The SO is direction dependent. It is the source of the PMA When orbital is deformed only in one direction (for example, only along the z direction), the H_{SO} exists only when the external magnetic field is directed along this direction (along the zdirection), but there is no H_{SO} when the external magnetic field is directed in a different direction (along the x or y direction) When the orbital is deformed perpendicularly to the film interface, the H_{SO} is induced only in this direction. The directional dependence of H_{SO} originates the PMA (See below)
Fact 6. The orbital symmetry in close proximity to nucleus determine the strength of the SO magnetic field
A substantial SO is induced only by a very strong magnetic field, which exists only in very close proximity to nucleus. Only this region makes substantial contribution to the H_{SO} and E_{PMA}. For this reason, the orbital symmetry in this region is critically important for the PMA. As a result: H_{SO} is larger when the center of orbital slightly shifted from the position of the nucleus comparing to a slight deformation of Fact 7. Mainly localized d or f electrons contribute to PMA Spins of the localized d or f electrons mainly contribute to the magnetization of a ferromagnetic metal. Therefore, the deformation and symmetry of these orbitals mainly determines the PMA. The contribution of the conduction electrons to the metal magnetization and the PMA exists, but it is very small. It is because the distribution of the spin directions of the conduction electrons is very different from that of localized d or f electrons (See here) The conduction electrons influence the PMA and the magnetization mainly due to the spd exchange interaction. Q. The orbital of d and f electrons are already not spherical. Do they experience the SO interaction and the magnetic anisotropy? A. It is correct. They do. There is magnetic anisotropy along some crystal orientation. Often it is not large. However, at the interface the orbital deformation might be much larger, which induces much larger H_{SO} and E_{PMA}. See also VCMA effect and SO torque Fact 8. The parity symmetry and spinorbit interaction
The SO interaction and the PMA depend only the direction of orbital deformation, but not its polarity. For example, the orbital deformation due to a shift of the center of electron orbit from position of atomic nucleus in + x direction induces absolutely equal H_{SO} and E_{PMA} as the shift in  x direction. Fact 8. Neither covalent nor ionic bonding is good for PMA and SO. The optimum bonding should be something between. In both case of the covalent bonding (E.g. Si, Fe) or the ionic bonding (E.g. NaCl, ZnO), the electron orbital is rather symmetric to induce any H_{SO} and E_{PMA}. In order to induce a large magnetic anisotropy, the orbital should be deformed asymmetrically. It is the case when the bonding is neither fully covalent nor fully ionic. A bonding across an interface (the interfacial PMA) or a bonding along some specific crystal direction in a compound crystal (E.g. CoPt, SmCo, GaAs, InP) (bulktype PMA) make optimum orbital deformation and induce a strong H_{SO} and E_{PMA}
Physical Origin of perpendicular magnetic anisotropy (PMA)
The PMA exists, because the electron orbitals in a ferromagnetic metal are deformed in the direction perpendicular to the film interface. When magnetization is along this direction, the intrinsic magnetic field induced a substantial effective magnetic field of the spinorbit interaction H_{SO} , because orbital deformation in this direction. When the magnetization is inplane, there is no H_{SO}, because the orbital is not deformed in this direction. As a result, the absolute value of the magnetic energy increases, when the magnetization is perpendicular to the film, and decreases when the magnetization is in plane. Magnetization is in plane: Total magnetic field= H_{intristic} Magnetic energy= H_{intristic} · M Magnetization is perpendicular to plane: Total magnetic field= H_{intristic}+H_{SO} Magnetic energy= (H_{intristic} +H_{SO} )· M where M is the magnetization or the spin of localized electrons. Since the absolute value of the negative magnetic energy is larger in the case when the magnetization is perpendiculartoplane. As a result, the easy magnetization direction becomes perpendiculartoplane, because of the SO interaction
note: See also about demagnetization field
Anisotropy field H_{anis}
The anisotropy field H_{anis} is the magnetic field, which is applied inplane and therefore perpendicularly to the easy axis of a nanomagnet and at which initiallyperpendicular magnetization turns completely to the inplane direction.The anisotropy field increases under a negative gate voltage and it decreases under a positive gate voltage (see VCMA effect).The anisotropy field is proportional to the PMA energy. A half of product of H_{anis} and nanomagnet magnetization is a common estimate of the PMA energy.
Important feature of PMA: Linear dependence of inplane component of magnetization on in plane magnetic field. See below the math to prove it. Since the fitting of a linear dependence is simpler, is resisted against the noise and other unwanted disturbing factors (like magnetic domain) and can be done with a high precision, measuring of H_{anisotropy} with a high precision is an important step to almost any magneto transport measurement.
Important facts about the spinorbit interaction and PMA are: fact 1: Due to the spinorbit interaction, there is a strong magnetic along the film normal (the zaxis). This magnetic field is called the magnetic field of the SO interaction. It is absolutely real magnetic field, which is generated relativistically due to electron movement in electrical field of atomic nucleus fact 2: The SO magnetic field is generated due to the orbital deformation along the zaxis. In the case of the uniaxial anisotropy it can be simplified that there is a SO magnetic field only along zaxis, but there is no inplane SO magnetic field. There is no field along the xaxis and yaxis. The magnitude of the SO magnetic field is proportional to degree of deformation of electron orbital in close proximity to atomic nucleus. fact 3: The magnitude of the SO magnetic field is linearly proportional to the total magnetic field along the z direction (perpendicularly to interface).
Calculation of Anisotropy field H_{anis}See detailed calculations of anisotropy field and PMA energy here CalculationHanisPMA.pdf(note) The calculation is for a single electron. In the case, when orbital symmetry of the electrons in bulk and at an interface is different, the effective anisotropy field for the whole nanomagnet should be calculated (See here) (how calculation is done) The angle of magnetization turning is calculated by a minimizing the magnetic energy for electron spin. The magnetic energy is the product of the electron spin and a sum of all magnetic fields, which the electron experiences. Magnetic fields, which the electron experiences: (field 1): External magnetic field, which is applied in both the perpendicular to interface direction H_{z} and in the inplane direction H_{x}. (field 2): Magnetic field of magnetization induced by all aligned electrons in a nanomagnet. This field can be imagined as a magnetic field inside the magnetic dipole induced by the electron spin. The average spin is a nanomagnet called the magnetization M. The magnetic field of magnetization is linearly proportional to the electron spin. In simplified units, the magnetic field induced by the spin equals to M. (field 3) Demagnetization field H_{demag}. The magnetic field, which is generated at interface due to broken chain of aligned spins. The direction of H_{demag} is perpendicular to interface and opposite to M. The H_{demag} can be calculated as where k_{demag} is the demagnetization factor. (field 4) Magnetic field H_{SO} of spin orbit interaction, which is a relativistic magnetic field induced by an electrical field of atomic nucleus. It is directed along the orbital deformation (the z direction in this case) and is proportional to the total magnetic field, which is applied along the x direction. where k_{SO} is a parameter describing the strength of the spin orbit interaction, is the intrinsic field in the nanomagnet and M is the magnetization of the nanomagnet (intrinsic field=field2+field3) It is magnetic field, which all atom in nanomagnet equally experience. In contrast, the spinorbit field H_{SO} (field 4) is individual for each atom (not common), because H_{SO} is linearly proportional to the orbital deformation (See here), which is very different for each atom. ( example: a Fe nanomagnet) At an interface orbital deformation of Fe atoms is large and interfacial Fe atoms experience a large H_{SO}. However a few atomic layers below, the orbital of Fe is not deformed and these experience no H_{SO}.
The magnetization direction under an external inplane magnetic field is calculated from minimizing the magnetic energy. In the case of zero orbital moment, the magnetic energy of an electron is a product of electron spin M and a sum of all magnetic fields, which the electron experiences: H_{all}=H+M+ H_{demag} + H_{SO}. The magnetic energy can be calculated as  In this case (H_{z}=0), the magnetic energy Eq.(2.4) is calculated as Under an inplane magnetic field H_{x}, the magnetization M does not change its magnitude only its direction turns. As a result, M is a constant and independent of H_{x}. Eq.(2.5) can be written as The equilibrium magnetization direction corresponds to the minimum magnetic energy. Minimizing the energy with respect to M_{x} gives The solution of Eq.(2.7) gives the linear dependence of inplane component M_{x} of the magnetization on the inplane magnetic field H_{x }as where the anisotropy field H_{anis} is calculated as
 In this case the magnetic energy can be calculated as (See Eq.2.4) where H^{(0)}_{anis} is the anisotropy field in the absence of the external magnetic field (H_{z} =0) The equilibrium magnetization direction corresponds to the minimum magnetic energy. Minimizing the energy with respect to M_{x} gives The solution of Eq.(2.7) is Even though the dependence of Eq.(2.12) deviates from linear, it is close to a linear dependence and can be expressed similar to Eq.(9) as
in which the anisotropy field depends on H_{z} and can be calculated as Substitution Eq.(2.13) into Eq. (2.14) gives the effective anisotropy field as
From Eq.(2.15), the anisotropy field depends linearly on the external perpendicular magnetic field H_{z}. When the applied inplane magnetic field is much smaller than the anisotropy field
and therefore the effective anisotropy field becomes independent of H_{x} When condition (2.16) is not the case, Eq. (2.15) can be solved by an iteration starting from Eq.(2.17) Figure 30 shows that the independence of the anisotropy field on inplane magnetic field and the linear dependence of M_{x} vs. H_{x} is a good approximation for the most of a realistic nanomagnets.
PMA energy(method 1 of calculation of PMA energy) imaginable switching off the spin orbit interaction and demagnetization effect
In case when there is no external magnetic field (H_{x}=0; H_{z}=0) , the magnetization is perpendicular to plane (M_{x}=0; M_{z}=M) and the magnetic energy can be calculated from Eqs. (2.6,2.9) as If we switch off the spin orbit interaction k_{SO}=0 and demagnetization field k_{demag}=0 (just imagine it), k_{SO}=0 becomes zero and the magnetic energy is calculated as The PMA energy EPMA is defined as an energy, which is induced by the spinorbit interaction minus demagnetization. Comparison of Eqs.(12) and 813) gives the PMA energy as
(note 1) The anisotropy field H_{anis} is the parameter, which characterizes the PMA energy. (note 1) The PMA energy can be either positive (equilibrium magnetization is perpendicular to plane) or negative (equilibrium magnetization is in plane). From Eq.(10) the condition when the PMA energy is positive is (method 2 of calculation of PMA energy) integration over a path of turning magnetization from a perpendiculartoplane to inplane direction The PMA energy can be also defined as an energy, which is required in order to turn the magnetization fully inplane. A tiny change dH_{x} of inplane magnetic field correspond to a change of magnetic energy dH_{x} · M_{x}. The integration over the change of H_{x} from 0 to H_{anis} using Eq.(2.8) gives the PMA energy as Internal magnetic field in a nanomagnetdetailed calculations of the internal magnetic field are in this pdf file. InternalMagneticField.pdf
There is a magnetic field inside nanomagnet, which is called the internal magnetic field . (note about magnetic field of spinorbit interaction ): The localized electrons on the surface and in the bulk of the nanomagnet experience a different magnetic field due to the spinorbit interaction. The magnetic field H_{SO} of spin orbit interaction is individual for each electron and depends on its orbital symmetry (See here). The electron at interface experience a strong H_{SO}. In the contrast, the electrons in the bulk of the nanomagnet experience a weak H_{SO}. As a result, the same electrons experience a very magnetic field at interface and in the bulk. (note fixing spins of all electrons in one direction ): localized electron in a ferromagnetic material experience a substantial exchange interaction (See here).The effective magnetic field of the exchange interaction is very large (~1500 T See here). Since it is so large, the exchange field solidifies all spins in one direction. As a result, the whole nanomagnet can be consider as a single spin object with united total spin. Two types of the internal magnetic field in a nanomagnet(type 1) Effective internal magnetic field Even though the electrons at interface and bulk experience a different internal magnetic field, the total spin of nanomagnet experience an average of all those contributions. This magnetic field affects the total spin of the nanomagnet. This is the internal magnetic field, which by the nanomagnet as one object. For example, this internal magnetic field determines properties of the ferromagnetic resonance (FMR) of the nanomagnet.
(type 2) Global internal magnetic field This is the conventional magnetic filed, which is inside of the nanomagnet (See here). This magnetic field "fills" all the space inside the nanomagnet and all other spin objects (e.g. spin of a nucleus) equally experience this magnetic field.
(fact 1 about field when magnetization is inplane) When magnetization is inplane, both the effective internal magnetic field and the global magnetic field equals to M (magnetic field induced by the magnetization) E.g. in the case of FeB nanomagnet (See here) this field is about 14 kGauss. This fact is applied for both types of the nanomagnets with inplane and perpendiculartoplane equilibrium magnetization. (fact 2 about global internal field when magnetization is perpendicular toplane ) When magnetization is perpendiculartoplane the global internal magnetic field is zero. This fact is applied for both types of the nanomagnets with inplane and perpendiculartoplane equilibrium magnetization. (fact 3 about effective internal field when magnetization is perpendicular toplane ) (case when the equilibrium magnetization the nanomagnet is perpendiculartoplane ): When magnetization is perpendiculartoplane, the effective internal magnetic field equals to M+H_{ani }(the sum of magnetic field induced by the magnetization and the anisotropy field). E.g. in the case of FeB nanomagnet (See here), M ~14 kGauss and H_{SO} ~6 kGauss, therefore the effective internal magnetic field is about 20 kGauss=2 T. (fact 4 about effective internal field when magnetization is perpendicular toplane ) (case when the equilibrium magnetization the nanomagnet is inplane ): When a sufficient external magnetic field is applied so that the nanomagnet magnetization turns to perpendiculartoplane direction, the effective internal magnetic field is between zero (case of no PMA) to M (case of some PMA). E.g. in the case of FeB nanomagnet, M ~14 kGauss, therefore the effective internal magnetic field is between 0 kGauss (e.g. a thick FeB film) and 14 kGauss. (e.g. a thin FeB film) (fact 5 about an increase of effective internal field under applied external magnetic field) when the magnetization is perpendicular to plane and an external magnetic film is applied in the same direction, the effective internal field increases with increase proportionally to the increase of the external magnetic field due to the increase of the magnetic field of the spin orbit interaction (See here). The external perpendicular magnetic field does not affect the global internal field.
Calculation result:detailed calculations of the internal magnetic field are in this pdf file. InternalMagneticField.pdf
Effective internal magnetic field: This field, which keeps the magnetization of the nanomagnet along its easy axis, induce FMR, etc.perpendicular to plane component: inplane component:
Global internal magnetic field: Other object inside the nanomagnet (e.g. spin of nucleus, spin of a conduction electron) experience this fieldperpendicular to plane component: inplane component: where M is magnetization (~14 kGauss for FeB); H_{ani} is the anisotropy field (~210 kGauss for FeB); and H_{ext} are inplane and perpendiculartoplane of applied external magnetic field.
Measurements of the anisotropy field H_{anisotropy}3 following experimental method are most often used to measure H_{anisotropy} Using a Vibratingsample magnetometer (VCM) or a SQUID magnetometer.The magnetometer measures the magnetization. From measurement of the inplane magnetization as a function of inplane magnetic field, the linear fitting by Eq.(2.10) gives H_{anisotropy}. merit: Highreliability direct measurements weakpoints: Due to sensitivity limitations, only a relatively large samples can be measured. Using the Anomalous Hall effectThe Hall angle is linearly proportional to perpendicular component of the magnetization (See here). The measurement of the Hall angle as a function of inplane magnetic field gives dependence M_{x}/M. The linear fitting by Eq.(2.10) gives H_{anisotropy}. merits: (1) Nanosized object can be measured (2) It can be combined with other magnetotransport measurements weakpoints: (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 magnetoResistance (TMR)The method uses a magnetic tunnel junction (MTJ), in which the magnetization of the “reference” layer is inplane and the magnetization of the “free” layer is perpendiculartoplane. When a magnetic field is applied in the inplane 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 inplane component of magnetization of "free" layer vs the inplane magnetic field. From this data, the linear fitting by Eq.(2.10) gives H_{anisotropy}. merits: (1) Nanosized object can be measured (2) It can be combined with other magnetotransport measurements (3) It is fast measurements weakpoints: (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
Demagnetization field
Interfacetype PMA
Bulktype PMA
Nonlinear effect for spinorbit interaction and PMA
Under a strong magnetic field, the dependence the spinorbit interaction and consequently PMA from perpendicular magnetic field deviates from a linear dependence. There are several reasons for that. The first reason is that the magnetic field may deform the atomic orbital, which enhances the SO and consequently PMA. As a result, the strength G_{SO} of the spinorbit interaction (See Eq. 2.1) becomes dependent on the magnetic field. As was explained here, the breaking the symmetry only along the zdirection (perpendicularly to the interface) affects the PMA, the G_{SO} may be changed only by H_{z }.Than, Eq. 2.1 can be re written as where H_{SO} is the effective magnetic field of SO interactions; G_{SO} is the proportionality constant; H_{intristic,z} is the zcomponent of the total magnetic field; H_{nl} is the magnetic field, at which the strength of SO interaction increases in two times. It means the proportionality coefficient between H_{SO } and H_{intristic,z} increases in two times. H_{nl} is usually substantially larger than magnetization M. Usually it is about a few Teslas. Which parameters are influenced by the nonlinear SO interaction? The dependence of the inplane component of magnetization vs inplane magnetic field deviates from linear at the magnetic field close to anisotropy field H_{anis} (the "nonlinear tail") It changes the value of anisotropy field H_{anis}
the equilibrium magnetization direction can be calculated from The effective anisotropy field H_{anis } can be calculated as where is the anisotropy field in case without any nonlinear SO component click here to expand and see how to obtain Eqs.(5.12a) and (5.14)
Nonlinear spinorbit 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 M_{z} is and component proportional to M_{x} is The total energy is the sum of Eqs. (5.3) and (5.4): In the case when the magnetic field is applied inplane (H_{x} =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 nonlinear SO component. In the case field H is smaller and is not close to H_{anis}, the following condition is satisfied From (5.13), (5.12a) and (2.10) the effective anisotropy field can be calculated as
(Video)Measurement of coefficient of spin orbit interaction and anisotropy field in a nanomagnet.This video set explains the details of a high precision measurement method of the anisotropy field H_{ani}, coefficient of spinorbit interaction k_{SO} , internal magnetic field in a nanomagnet H_{int} and a magnetic field H_{off} created by a spinaccumulation in a nanomagnet. I will show you how to do measurement, how to process measured data and how to evaluate each parameter. You can use either my raw measurement data, which you can download, or your own data. I will describe how to use the measurement data to improve the efficiency of the magnetization reversal either by a electrical current or by a gate voltage for a memory application and for a sensor application I will explain the details of the magnetization reversal mechanism using the classical mechanics based on LL Eq. and using Quantum mechanics. I will explain the quantum mechanical meaning of the spin torque and the spin precession. I will explain the reasons why the newlyintroduced fieldlike torque severely violates Laws of Quantum mechanics. I will explain about the interpretation problems and possible systematic errors of the popular measurement techniques, which are called the 2nd harmonic measurement and the zeroharmonic measurement. I will give some technological recommendations on how to improve the fabrication of a nanomagnet, which are based on my personal experience.
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);
Questions & Comments
(about dependency of the spinorbit interaction of external magnetic field) (from XianPeng Zhang) Q. I have a question about the spinorbit magnetic field. Why it is proportional to the total magnetic field? (see below for details) May you provide me some references?
Your question is why the spinorbit interaction depends on an external magnetic field and in which reference it was described for the first time.
This fact is known for a very long time and therefore it is more about the history of Science, in which I am not expert. I am sorry if I refer something incorrectly.  (SO fact 1) The fact about the relativistic origin of the magnetic field and the existence of the spinorbit interaction (the existence of the magnetic field of the spinorbit interaction). This fact is one of basic facts, on which the Special Theory of Relativity was build. Therefore, I think Einstein and Lorentz knew about this fact. Myself, I have read all about this fact (all details and many nice explanations) in Landau textbook "The Classical Theory of Fields"
(SO fact 2) The fact that the strength of spin orbit interaction (the SO magnetic field) is proportional to the strength of an externally applied magnetic field. This fact are direct consequence of the another important fact about SO: (SO fact 3) The spinorbit interaction cannot break the time inverse symmetry. As a result, in order to manifest itself, the SO needs an external breaking of the time inverse symmetry (E.g. by an orbital moment or by an external magnetic field or an electrical current etc.). Fact 2 follows very directly from the fact 3. The strength of SO is zero , when the time inverse symmetry is not broken (case when the orbital moment is zero, there is no external magnetic field etc.) . The SO magnetic field becomes a nonzero only when the time inverse symmetry is broken. The strength of the SO ( the strength of the SO magnetic field) becomes larger when the degree of the breaking of the time inverse symmetry becomes larger (e.g. the orbital moment becomes larger or the external magnetic field becomes larger etc.) .
Once again zero magnetic field > timeinverse symmetry is not broken > spinorbit is zero external magnetic field becomes larger > degree of breaking of timeinverse symmetry becomes larger > spinorbit becomes larger.  (fact 3 explanation) Why the spinorbit interaction cannot break the time inverse symmetry. This feature is the feature of the timespace symmetry and can be obtained from he Special Theory of Relativity (e.g. see Landau textbook " The Classical Theory of Fields " ). A simplified understanding of the fact 3 can be as follows. The spinorbit effect is a relativistic effect and therefore can only occur when the object moves or the field, with which the object interacts, moves. The relativistic effect requires a movement. The close movement speed is to the speed of the light, the stronger any relativistic effect is. Any movement means a breaking of the time inverse symmetry. This is why the SO occurs only when the timeinverse symmetry is broken.  Even though the direct relation between the spin orbit interaction and the breaking of the time inverse was understood a long time ago as well as the proportionality of the SO strength to the degree of TIS breaking (e.g., to the strength of the orbital moment or to strength of the external magnetic field), all details and specifics have not been fully understood. For example, the SO strength depends on the orbital symmetry. Usually the orbital is more symmetric in the bulk of a material and the SO is weaker, but at an interface the orbital becomes less symmetrical and the SO becomes larger. You can read some details here
This effect is used for molecularrecognition sensor, See here
 About experimental measurements. The dependence of PMA and therefore SO on an external magnetic field is very strong. The experimental measurement of this dependence is rather simple and the measured data well fit to the theory. The anisotropy field is a measurable parameter, which characterizes the strength of PMA and SO. Near for all nanomagnet, I have studied, I measure the dependence of anisotropy field on a perpendicular magnetic field. Such data contains a lot of important information. Introductory presentation on this measurement method I gave at MMM 2020 conference. You can watch it here
Video
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