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

Thermallyactivated magnetization switching. Measurements of coercive field, retention time, Δ Spin and Charge TransportAbstract:
A magnetic material has two stable magnetization directions. An external magnetic field may switch the magnetization between these two states. The magnetic field, at which the magnetization is switched, is called the coercive field Hc. This magnetization switching is a thermally activated process. Therefore, it depends on temperature and is assisted by thermal photons, phonons and magnons. Because of its thermal nature, the magnetization switching is probabilistic. It means that for each scan of a magnetic field, the magnetization switching occurs at a slightlydifferent magnetic field.In most cases, the magnetization switching is initialized by a magnetization reversal in a very small area, which is called the magnetic domain, and the next expansion of the domain over all ferromagnetic material (over all nanomagnet). There are two types of the magnetic domains: the static magnetic domain, which is stable in time, and the nucleation domain, which is unstable and exists only for a very short time of about a few nanoseconds during the magnetization switching. Features of thermally activated switching of the magnetization are described. A new highprecision measurement method of the coercive field, retention time, Δ and the size of a switching nucleation domain is described below.paper on this topic is here: V. Zayets, "Thermally activated magnetization reversal in a FeCoB nanomagnet. Highprecision measurement method of coercive field, delta, retention time and size of nucleation domain" arXiv:1908.08435 (2019)Note: experimental method, which is described below, allows to measure the coercive field with a precision better than 1 Oe, retention time (data storage time) in rage from a few minutes to a few billions years, the size of nucleation domain with a precision of a few nanometers, delta Δ with a precision of a few percents.Note. I have develop this experimental method in 20172018(note) Even though the data shown below is for nanomagnets with the Perpendicular Magnetic anisotropy (PMA), the methods is working very well for a nanomagnet with inplane magnetization.Contentclick on the chapter for the shortcut() Thermal mechanisms of magnetization reversal. Quantum vs. classical descriptions() Quantum Mechanism of Thermal magnetization reversal.() Classic or Néel Mechanism of Thermal magnetization reversal.(2). Measurement method(2a) In short. Measurement of coercive field H_{c}(2b) In short. Measurement of retention time τ_{retention}_{}(2c) In short. Measurement of parameter Δ_{}_{}(2d) In short. Measurement of size of nucleation domain
(3) In short. Transformation of Hysteresis loop under different effects (VCMA, SOT, inplane magnetic field)(4) In short. Magnetizationreversal time _{}t_{switch} influenced by different effects (VCMA, SOT, inplane magnetic field, temperature)(6). Néel model of thermally activated magnetization switching(6.1) Step 1. Calculation of energy barrier E_{barrier } between two stable magnetization states(6.2). Step 2. Arrhenius low & transition state theory (TST)
Measurement of coercive field, size of nucleation domain & retention time in FeCoB nanomagnet
Questions & Answers ( ) Are any other methods except the thermally activated switching used as a recording method of a magnetic memory?about static and nuleation domains ( )() Why does the magnetization reversal occur in a small area (in a magnetic domain) instead of a simultaneous reversal over the whole area of the sample or nanomagnet? Why the size of a magnetic domain cannot be very large?() Can only one spin be thermallyreversed? Why the size of a magnetic domain cannot be very small?( ) Why does the magnetization switching mechanism depend on the nanomagnet size? Which parameter determines the nanomagnet size of each switching mechanism?( )( ) What is the difference between nucleation and static domains?( ) In the case of a static domain, why does the domain wall constantly move with an increase of magnetic field?() Since the nucleation domain is unstable and the energy of singledomain state is smaller than energy of state with nucleation domain, why the magnetization reversal involves the state with nucleation domain, but it is not reversed directly into the singledomain state?( ) nucleation domain and pinning of domain wall() Why do static domains exist in a larger nanomagnet, but magnetization of a smaller nanomagnet is aligned in one direction?( ) about measurement of energy of barrier for domain wall movement( ) Stability of magnetic domain vs size of a nanomagnet. Why does type of switching mechanism dependent on nanomagnet size?( ) reason why magnetization of all regions across a nanomagnet is in the same directionabout coerecive field H_{c}_{}() Why does magnetization not reverse when a magnetic field is applied opposite to its direction?(b) difference of coerecive field H_{c} in a plain film & a nanomagnet(c) film quality vs. coerecive field H_{c}(d) H_{c} of a bulk material without defects (19) Questions & Answers (e) simulation of hysteresis loop. Case of static domainsPart for advanced reader:(7) Switching probability(a) In a constant magnetic field(b) in linearly ramped magnetic field(8) Distribution of switching probabilities(a) time distribution(b) field distribution(8) Measurement of delta Δ(9) What is effective magnetization M_{eff}. Nucleation domain for magnetization switching(10) Measurement of the size of the nucleation domain for magnetization switching(11) Influence of MgO/FeCoB interface on magnetic properties of FeCoB nanowire(12) Relation between delta and retention time(13) Temperature dependence of delta Δ, anisotropy field H_{anis}, effective magnetization M_{eff}(16) Difference of switching fields between spinup to down and spindown to spinup states(17) FORC method to analyze magnetic properties of multi particles systems (18) Known tricks and methods of fake and highlight research(trick 1) Tracing a small change of coercivity field H_{c} using a coercivity loop(trick 2) Measuring properties of magnetization switching without a measurement of retention time(trick 3) Reporting an "efficient" magnetization switching under bias inplane magnetic field..............
Thermallyactivated magnetization switching & storage time & magnetic recording
A thermal fluctuation may switch the magnetization of a nanomagnet between its two opposite stable directions. data storage time or retention time of a magnetic memoryA bit of recorded data of any magnetic memory can be destroyed due to unwanted thermally activated switching of the magnetization. The data storage time can be as short as a minute or can be as long as a billion years. Using the belowdescribed measurement method, the data storage time can be measured with a very high precision in both cases. recording method of magnetic memoryIn order to record a bit of data into a magnetic memory, the magnetization of one memory cell should be reversed. It is done by applying a recording magnetic film, which reverses the magnetization. The mechanism of the switching is a thermally activated magnetization switching. Understanding of features of a thermally activated magnetization switching is important for both the data and data recording of the magnetic memory!!! Are any other methods except the thermally activated switching used as a recording method of a magnetic memory?A. Yes. An irreversible magnetization reversal occurs after an event when the number of the electrons on the higherenergy spindown level exceeds the number of electrons on the lowerenergy spinup level. Additionally, a thermally activation, the magnetization reversal may occur due to a spin injection (see here) or a parametric resonance (See here)
Thermal mechanisms of magnetization reversal. Quantum vs. classical descriptionsMagnetization reversal occurs when:(Quantum description): number of spindown electrons exceeds the number of spindown electrons (Classical description): the electron moves over or tunnel through an energy barrier, which separates the spindown and spinup stable electron states. This means that the electron spins is inclined more than 90 deg with respect to its stable position by a thermally activated random magnetic field. Quantum descriptionIt is only a correct description, which explains perfectly all experimental features.In the quantum description the magnetization reversal is described as a quantum transition between the energy levels. Magnetization reversal occurs at a moment of time when the number of electrons on a higherenergy level exceeds the number of electrons on a lowerenergy level.
(fact about total spin) within a nucleation domain or a singledomain nanomagnet, directions of all spins are glued together by a very strong exchange interaction. The total spin or magnetization (shown as a blue ball) is the sum of spins of all localized electrons within the nucleation domain. At any time during magnetization reversal the magnitude of the total spin remains the same, the direction of the total spin changes. A nucleation domain should be considered as a single quantum object with a single spin. (fact about required small size of nucleation domain) The larger the size of the nucleation domain is and, therefore, the larger the number of the spins is, the smaller the probability of thermallyactivated magnetization reversal becomes. If the probability for a quantum transition from the spinup to the spin down level for one electron equals p, then the probability for the transition of two electrons in the same time is p·p. The the probability for the transition of n electrons in the same time is p^{n}. If the volume of the ferromagnetic domain is V_{domain} , the number of the spin in the domain is M ·V_{domain}. Then, the probability of a reversal of the nucleation domain is p_{domain}=p^{M ·V_domain} (fact about the retention time) The longer the waiting time is, the larger the probability of thermallyactivated magnetization reversal becomes.
Classical description or Néel descriptionThe classical description has some limitations and, explains a part of experimental features.In the classical description the magnetization reversal is described as as a jump over or tunneling of an electron through the energy barrier, which separates two stable spinup and spindown states, which are separated by an energy barrier. The height of the barrier is lowered by an external magnetic field. When the magnetic field equals to the coercive field, the energy barrier bacomes sufficiently low for the electron to tunnel to another stable state.
According to how the energy of perpendicular magnetic anisotropy (PMA) described, there are two classical descriptions: (classical description 1): oversimplified Neel description In this model the simplest dependency of the PMA anergy is assumed. This model does not fit all experimental data, but it is a good approximation in many cases See detailed calculations of anisotropy field and PMA energy by this description here NeelPMA.pdf or hereMagnetic energy: Energy barrier:
(classical description 2): full description based on properties of the spinorbit interaction This model correctly described all features of the perpendicular magnetic anisotropy (PMA). See detailed calculations of anisotropy field and PMA energy by this description here CalculationHanisPMA.pdf or hereMagnetic energy: Energy barrier: (problem 1of classic description): quantum tunneling The classic description is based on tunneling through the energy barrier between 2 stable states. The quantum tunneling may occur only between two different quantum states, which are spatially separated. In contrast, the magnetization reversal means the spin reversal for one single quantum state. Only the degree of the broken time inverse symmetry is changing during the magnetization reversal. (problem 2 of classic description): one electron tunnelin vs. magnetization reversal of whole nucleation domain The tunneling occurs only for a single electron. However, the magnetization reversal occurs simuteneously for a whole nucleation domains, which contains billions of electrons (spins). (problem 3 of classic description): Large difference between anisotropy field and coercive field. The anisotropy field is the magnetic field, at which the spin turns perpendiculary to the easy axis and whch corresponds to the top of the energy barrier. The coercive field is the magnetic field, at which the magnetization is switched. In a Fe nanomagnet, the anisotropy field may reach 10 000 Gauss when the coercive field may be only about 400 Gauss. It means that such a small coercive field reduces only slightly the hight of the energy barrier. Still it triggers the magnetization reversal.
(note): There are 3 possible mechanisms of magnetization reversal: the mechanism of domain wall shifting and expansion of static domains; the mechanism nucleation and mechanism of single domain magnetization reversal (sample size > reversal mechanism) The mechanism of magnetization reversal depends on the sample size. The reversal mechanism for a continuous film and a large sample is the domain wall shifting of static domain. The reversal mechanism for a sample of a moderate size (~ from 50 nm to 20 μm) is by a nucleation domain. The reversal mechanism for a smallest sample (<50 nm) is the single domain reversal.
3 mechanisms of magnetization reversal(magnetization reversal) Mechanism 1: Domain wall movement of static domainssamples: largesize nanomagnet, magnetic film In this case there are static domains of two opposite magnetization directions. When magnetic field is applied, the domain wall slightly moves making the larger the area of domains of magnetization along magnetic field and the smaller the area of domains of the magnetization opposite to the magnetic field. As the magnetic field increases, the area of domain becomes larger and larger and correspondingly the area of opposite domain becomes smaller and smaller until it disappear. At each magnetic field the domain wall is pinned (magnetization reversal) Mechanism 2: Creation of a nucleation domain following the domain wall movement.samples: moderatesize nanomagnet In this case the magnetization of the whole nanomagnet remains opposite to applied external magnetic field until the critical field (the switching field), at which the magnetization of small area (the nucleation domain) coherently reversed and becomes parallel to the external magnetic field. Immediately after that the domain wall moves expanding the nucleation domain over the whole sample. Main feature of this mechanism is that the domain wall is not pinned. At the moment, at which the nucleation domain is created, the domain wall moves without any obstacles until the magnetization of the whole nanomagnet becomes parallel to the applied magnetic field. (magnetization reversal) Mechanism 3: Coherent magnetization rotationsamples: smallestsize nanomagnet In this case the magnetization of the whole nanomagnet remains opposite to applied external magnetic field until some critical field (the switching field), at which the magnetization of the whole nanomagnet rotates to become parallel to the external magnetic field. Through the whole switching process the magnetization at any point of the nanomagnet remains parallel. There is no any magnetic domains.
Why does the magnetization switching mechanism depend on the nanomagnet size? Which parameter determines the nanomagnet size of each switching mechanism?A. It is because of the size dependence of the magnetostatic interactions. (exchange interaction vs. magneticdipole interaction). There are two interactions between spins of different electrons. The shortrange verystrong exchange interaction and the longrange moderate (low)strength magneticdipole interaction. In a ferromagnetic metal, the exchange interaction aligns atoms parallel to each other. The dipole interaction aligns spins opposite each other. Even though it is very strong, the exchange interaction exists only between electrons of neighboring atoms and, therefore, the exchange interaction does not accumulate with an increased number of atoms. For a small number of atoms, the exchange interaction aligns all spins to be parallel. When the number of atoms increases, the exchange interaction remains unchanged, but dipole interaction is accumulated and, therefore, increases. When the number of atoms exceeds some critical number, all spins within a part of the nanomagnet reverse their direction and a magnetic domain is formed. The balance between the magnetostatic interaction and the exchange interaction determines the size of the magnetic domain. (domain volume vs domainwall) The magnetic energy is proportional to the number of spins and, therefore, to the volume of the magnetic domain. In contrast, the energy of the domain wall is proportional to the surface area of the domain. The balance between the magnetic volume energy and the domain wall determines the sizes and distribution of the magnetic domains. When the nanomagnet size is smaller than the minimum domain size, the nanomagnet is in a singledomain state. The strength of the exchange and the dipole interactions, the number of imperfections and defects in both inside of the nanomagnet volume and at the nanomagnet surface, all substantially influence the domain size. Why does the magnetization reversal occur in a small area (in a magnetic domain) instead of a simultaneous reversal over the whole area of the sample or nanomagnet? Why the size of a magnetic domain cannot be very large?A. In order to reverse its spin, an electron should interact with a nonzero spin particle like a photon or a magnon. For example, if the probability to reverse the spin of one electron equals p, then the probability to reverse the spin of two electrons is smaller and equals p·p. The probability to reverse the spin of n electrons is even smaller and equals p^{n}. Therefore, the probability is higher to reverse a smaller amount of the spins. Can only one spin be thermallyreversed? Why the size of a magnetic domain cannot be very small?A. In a ferromagnet, directions of all spins are glued together by very strong exchange interaction. It is impossible to reverse only one spin. The energy of the domain wall would be too large. It requires some amount of spins so that the dipole interaction overcomes the energy of the domain wall. It determines the minimum size of the magnetic domain or, the same, the minimum amount of the spins, which can be simultaneously reversed. The magnetization M of a ferromagnet is defined as a number of the spins per the ferromagnet volume. If the volume of the ferromagnetic domain is V_{domain} , the number of the spin in the domain is M ·V_{domain}. Then, the probability of a reversal of the nucleation domain is calculated as (see previous question) p_{domain}=p^{M ·V_domain}^{} Therefore, the probability of switching is proportional to the volume of the nucleation domain. What is the difference between nucleation and static domains?A. The static domains are stable. In the case of static domains, there is a balance between magnetostatic and exchange energies. At different external magnetic field, the size of domains changes, but always the balance is possible. The balance can exists for a relatively large nanomagnet, which area is sufficient to fit several largearea domains. The nucleation domain is not stable. For the nucleation domain the balance between magnetostatic and exchanges energies cannot be achieved. It is the case of a small nanomagnet, in which magnetostatic is smaller the domain wall energy for any possible domain configuration. As a result, it is energy favorable to remove domain wall to minimize the exchange energy since the magnetostatic energy has a small contribution. Therefore, as soon as a nucleation domain is formed, its domain wall immediately expands over whole nanomagnet. In the case of a static domain, why does the domain wall constantly move with an increase of magnetic field?A. It is because at any applied external magnetic field there is a balance between magnetostatic and exchanges energies. For each value of the external field, there is an optimum domain size for the balance. E.g. let us consider bubble domains of the circle shape and radius R in a thin film. The magnetostatic energy is roughly proportional to the area of domain (~R^{2} · E_{MS}), the energy of domain wall is proportional to length of the domain wall (~R· E_{EX}) and the energy of magnetic interaction with magnetic field is is roughly proportional to the area of domain (~R^{2} · E_{H}· H). When the external magnetic field increases, the radius of the bubbles domains becomes larger. As a results, the energy of domain wall increases and the magnetostatic energy decreases. At any field there is a balance: R^{2} · E_{MS} R· E_{EX}  R^{2} · E_{H}· H =constant (main reason for an expansion of a magnetic domain (a movement of a domain wall):) A domain wall moves when the total energy decreases for a larger domain size. It is always the case for a single domain nanomagnet, because the dipole interaction in a single domain nanomagnet is insufficient to stabilize a two domain state. However, sometimes the domain wall may be stopped and stick to a defect or an imperfection (domainwall pinning). (Expansion of a static domain): At each value of the external magnetic field H, it should be a balance between magnetic energy of interaction of magnetic field with magnetization of each domain, energy of domain walls and dipole interaction. This balance determines the sizes of static domains. Since the nucleation domain is unstable and the energy of singledomain state is smaller than energy of state with nucleation domain, why the magnetization reversal involves the state with nucleation domain, but it is not reversed directly into the singledomain state?A. It is because the energy barrier between singledomain and nucleation domain states is smaller than the energy barrier between two singledomain states of opposite magnetization. The energy barrier is proportional to the area of the nucleation domain (See here) and becomes smaller as the size of the nucleation domain is reduced .(magnetization reversal) Mechanism 1: Domain wall movement of static domainssamples: largesize nanomagnet, magnetic film In this case there are static domains of two opposite magnetization directions. When magnetic field is applied, the domain wall slightly moves making the larger the area of domains of magnetization along magnetic field and the smaller the area of domains of the magnetization opposite to the magnetic field. As the magnetic field increases, the area of domain becomes larger and larger and correspondingly the area of opposite domain becomes smaller and smaller until it disappear. At each magnetic field the domain wall is pinnedExcellent explanation of domain wall movements of static domains as magnetization reversal mechanism from Cao et.al. JMMM (2015). Click here to expand.
citation from Cao et.al. Hysteresis in single and polycrystalline iron thin films: Major and minor loops, first order reversal curves, and Preisach modeling. JMMM (2015)
Part 2.1 pp.362 "The connection between microstructural defects and magnetic properties has long been known, and the common terminology for magnetic materials as “hard” and “soft” stems from this [16]. Increased concentration of dislocations in single crystal ferromagnetic metals results in increased major loop coercivity and decreased initial and reversible susceptibilities [3,17]. The initial magnetization curve (i.e., from a demagnetized state) of a ferromagnet can be seen schematically to be divided into three stages under external field H before saturation (Fig. 1). The initial stage (low applied fields) involves reversible domain wall displacement and bowing (Fig. 1(a) and (b)). The domain wall will return to its initial position if the external field is removed (Fig. 1(a)). The domain wall is pinned by some dislocations or other defects (black spots) in this stage (Fig. 1(b)) and then bends due to the applied field. At higher fields the domain wall breaks away from the defects and the magnetization jumps discontinuously, generating Barkhausen noise [18]. The domain with the easy axis magnetization vector having a component in the same direction as the applied field direction grows (Fig. 1(c) and (d)). Ultimately and ideally, the material will consist of a single domain (Fig. 1(e)) at the end of the second stage. Finally, in the third stage at the highest applied fields, the domains will rotate away from their magneto crystalline easy axis to align with the external field and the magnetization becomes saturated (Fig. 1(f)). Processes (a) to (b) can be seen as reversible, as domain walls have not moved through pinning defects. Similarly, processed (e) to (f) are reversible, as it is just a rotation of the magnetic moment. These reversible processes generate what is known as the defectfree or an hysteretic magnetization [18]. Processes (c) to (d), however, are irreversible and result in hysteresis, as they involve the magnetization moving over an energy barrier (the pinning defect), discontinuously acquiring magnetization energy. It is apparent, then, that the character of the defects (concentration, size, shape, and magnetic nature) will affect the domain wall pinning and thus the processes in the (c) to (d) region. A higher defect concentration should lead to a smaller slope in magnetization in this region (i.e., a higher field is required to advance the magnetization by a given amount). Note that a similar argument can be made with consideration of a major hysteresis loop, rather than an initial magnetization curve as was described here. In principle, then, a given set of defects in a magnetic material should result in characteristic hysteresis behavior when evaluated in a range of field histories, such as with major and minor loops and FORCs. These same defects generate Barkhausen noise due to discontinuous jumps in magnetization as domain walls move past defects, and thus a simulation containing the effects of defects on hysteresis should be able to predict the Barkhausen noise spectrum generated as part of a NDE measurement of a magnetic steel" please check the original paper to check the used citations
Magnetization reversal type: domain wall movement of static domains. Two mechanisms of movement of domain wall.(mechanism 1: reciprocal) :Magnetostatic Domain wall moves in order to decrease the magnetostatic energy. The area of domain, which magnetization is along to external magnetic field, becomes larger. The area of domain, which magnetization is opposite to external magnetic field, becomes smaller. There is no coercive field for this mechanism. The change of the total magnetization is smooth and continuous with increase of the external magnetic field. The change is fully reversible. The mechanism is time independent. (mechanism 2: non reciprocal) : Thermo activation When the domain wall is moving, it can stick to a fabrication defect or a border irregularity. The domain wall overcomes such energy barrier by a thermoactivation mechanism. It means it assisted by a thermo fluctuation. The change of the total magnetization is steplike with some coercivity. The change is irreversible. The mechanism is time dependent. It means that the probability of a thermo fluctuation, which assists the domain wall to overcome the energy barrier, is higher for a longer waiting or measuring time (as for any thermo activated switching)
Two types of magnetic domains: ( type 1): Static domainsThe static domains exist even without any external magnetic field (See fig. 1(a)) or under a bias magnetic field fig. 1(b). The hysteresis loop of a sample having static domains is smooth and does not have steplike features. ( type 2): Nucleation domainsA nucleation domain exists for a very short moment during the event of the magnetization reversal. As soon as a nucleation domain is created, its domain wall moves expanding the domain over the whole sample. The hysteresis loop of a sample having a nucleation domain has steplike features.
Below the features of thermallyactivated magnetization switching is described for switching mechanisms 2 and 3, which hysteresis loop is of the rectangular shape. These switching mechanism is the feature of a moderatesize and largesize nanomagnet, which is important for the memory applications
Hysteresis loop of a nanomagnet.Figure 1 shows the schematic diagram of a hysteresis loop of a ferromagnetic nanomagnet. It shows the dependence of the nanomagnet magnetization M on the applied magnetic field. The magnetic field is scanned from a negative to positive value and back to negative. In the case of a sufficiently large magnetic field, the magnetization is always aligned along the magnetic field. However, at a smaller field just after a reversal of the external magnetic field, the magnetization does not follow the reversal and remains in the opposite direction to the magnetic field until the magnetic field reaches the threshold field, at which the magnetization is reversed to be again parallel to the external magnetic field. The threshold magnetic field, at which the magnetization is reversed, is called the coercive field H_{c} (See Fig.1). A hysteresis loop for the magnetization switching exists for the following reason. The state, in which the magnetization is opposite to the direction of an external magnetic field, is in an unstable equilibrium. The state, in which the magnetization is parallel to the magnetic field, is more energetically favorable. However, there is an energy barrier between the "up" and "down" magnetization states and the magnetization reversal may occur only when the magnetization overcomes the barrier. The assistance of a thermal fluctuation is required in order to overcome the energy barrier. Because of the critical dependence of the reversal event on the existence of a thermal fluctuation, this type of magnetization reversal is called thermallyactivated magnetization switching. The properties of the thermallyactivated magnetization switching are important for magnetic data recording and magnetic data storage. Why does magnetization not reverse when a magnetic field is applied opposite to its direction?There is an energy barrier between two stable states, when the magnetization is parallel and antiparallel to the external magnetic field. The magnetization can overcome this barrier only with assistance of a thermal fluctuation.
How to reverse magnetization direction? Which parameters influence the magnetization reversal?(method 1) Increase magnetic field. The external magnetic field lowers the height of the energy barrier and makes the probability of a magnetization reversal higher. When the increasing magnetic field reaches H_{c}, the barrier height becomes sufficiently low and the magnetization is reversed. (method 2) wait a longer time. A thermal fluctuation of a higher energy is required to overcome a higher energy barrier. Waiting for a longer time makes the probability of the required higherenergy fluctuation greater. Methods of either applying a stronger magnetic field or waiting a longer time both lead to the magnetization reversal. For example, the reversal probabilities may be equal for the cases when the field is smaller but the waiting time is longer or when the field is larger but the waiting time is shorter.
Why the hysteresis loop of the hysteresis loop of a multi particle system or a magnetic film with complex domain structure is not of the rectangular shape as in the case of a nanomagnet?case 1: (a multi particle system): It consists of many nanomagnets. The magnetization of each nanomagnet is reversed at a slightly different magnetic field. It makes the gradual change of the average magnetization. case 2: (a magnetic film with complex domain structure): Between two states, when the magnetization is fully parallel to the external field, there is a state of complex domain structure. The amount of domain area is changed as the external magnetic field changes. It makes the gradual change of the average magnetization. case 3: (intermediate magnetization direction): Additionally to the two states, when the magnetization is parallel and antiparallel to the external magnetic field, there might be additional intermediate state of local minimums of magnetic energy, when the magnetization is at some angle (between 0 and 180 deg) with respect to the magnetic field.
Statistical nature of the thermally activated magnetization switchingThe thermally activated magnetization switching is statistical process. It is described by statistical parameters and formalism like the average and statistical distribution. Every time you measure, the magnetic field of the magnetization switch (singleshot measurement) Don't do a singleshot measurement for parameters of thermally activated magnetization switching like H_{c}
fact 1: Since the magnetization switching is a thermally activated process, the distribution of switching fields is described by the thermal statistics Average of the switching field is defined as the coercive field. fact 2: The longer the scan of magnetic field is, the shorter the coercive field is. Figure 3 explains this fact. When a magnetic field is applied opposite to the magnetization direction, for a while the magnetization is wiggling around its direction. next, it turns along magnetic field. The average magnetizationreversal time is fixed and it is proportional to the applied magnetic field.
Method of a high precision measurements of parameters of the thermallyactivated switching
It gives a measurement precision of the coercive field at least 1 Oe (often better than 0.1 Oe). Excellent repeatability, reproducibility have been verified for many samples ( more than 200 nanomagnets were measured as 2019.08).This method also measures coercive field H_{c} , retention time τ_{reten}, parameter delta Δ and size of a nucleation domain for magnetization reversal.In this method dependence of the magnetization switching time on the external magnetic field is measured. From this measurement, coercive field Hc, retention time τ_{reten}, parameter delta Δ and size of a nucleation domain for magnetization reversal.The magnetization switching time was measured as follows. A magnetic field H was applied opposite to the magnetization direction and the time interval, after which the magnetization is reversed, was measured. The measurement was repeated 200 times and a statistical analysis was applied to find the average of tswitch. Figure 10 shows the measured tswitch as a function of the magnetic field. On a logarithmic scale, the magnetization switching time is linearly proportional to the magnetic field as it is predicted by the Néel model of thermally activated magnetization switching Can you explain the details how measurements of Fig.10 are done?Detailed steps of a measurement of dependence of magnetization switching time on the magnetic field (Fig.10 ) : (step 1) (reset of the magnetization): Applying a large magnetic field. The magnetization of the nano magnet is fully saturated. There no domain or pinned domain. (step 2) (apply magnetic field in opposite direction) Applying magnetic field opposite to the magnetization. For example, apply H=240 Oe. (step 3) (measure time until reversal) wait until the magnetization is reversed the time interval from a moment, when the magnetic field is applied in the opposite direction to the magnetization, till the moment, when the magnetization is reversed, is the measured switching time. (step 4) (repeating of measurements) repeat the same measurement 70 times. calculate the average switching time. (step 5) (scan the magnetic field) repeat steps 14 for field 241 Oe... and so forth I usually measure in the interval of magnetic field when the switching time changes from 0.5 s to several minutes The Néel model calculates the magnetization switching time t_{switch} as where the magnetization M_{eff } is the magnetization of the nucleation domain (case of multidomain switching) or the total magnetization of the nanomagnet(case of singledomain switching) The retention time τ_{retention }is the magnetization switching time, when the magnetic field is not applied H=0. Linear dependence of log(t_{switch}) perfectly fits to experimental data.From Eq.(a2) all parameters of thermallyactivated switching can be evaluated How many there are free parameters thermallyactivated magnetization switching?Two. the Néel model is relatively simple and only has two free parameters: the energy barrier E_{barrier } and the rate of interaction f_{inter} of the nanomagnet with the magnetizationreversing particles (photon, magnon). Alterternatively, any other pair of free parameters may be used (for example, the coercive field H_{c} and retention time τ_{retention} or M_{eff} may be used as one of the two free parameters). However,there are only two independent parameters of the thermallyactivated magnetization switching. Additionally, there are parameters, which are related to the magnetostatic properties of a nanomagnet. For example, the parameter Δ is proportional to the anisotropy field H_{anis} and the volume of the nucleation domain is proportional to the magnetization M of the nanomagnet. Merits of measurement of parameters of the thermallyactivated switching by method of Fig.10(merit 1)(it is the most direct measurement) It is because the magnetization switching time is the primary parameter of the Neel model of the magnetization switching (merit 2)(it is free of a possible systematic error) Since it is the direct measurement and the measured dependence in Fig.10 is always linear, a possible systematic error is always clear and can be avoided. See here for details of the systematic errors of other used measurement methods. (merit 3)(high measurement precision) The measurement precision of the described method substantially better than the precision of other used measurement methods. The required precision can be reached with a smaller of statistical measurements. In short. Measurement Coercive field H_{c}
The precise value of H_{c} is the field, at which measured line of log (t_{switch }) vs H crosses the xaxis gives the coercive field H_{c}.The definition of coercive field H_{c} :(old,incomplete): The magnetic field, at which magnetization is switched between its two stable magnetization directions is called the coercive field (correct): The coercive field is defined as the magnetic field, at which the average magnetization switching time is equal to 1 second. Highprecision measurement of H_{c} The measured dependence of log(t_{switch}/1 sec) vs H is a line. The coercive field H_{c} corresponds to magnetic field, at which t_{switch}=1 sec or log(t_{switch}/1 sec)=0. A linear fitting experimental data gives the H_{c} with a very high precision. Why a measurement of H_{c} from a hysteresis loop should be avoided and should be used only as a rough estimate of H_{c}(reason 1) Each repeated measurement dives a slightly different value of H_{c} (See Fig.8) (reason 2) A measurement at scanning rate of the magnetic field gives different value H_{c}(See Fig.9) (a bad scientist) (1) When showing a graph with numbers, it is better to show the measurement unit! (2) When showing data of measured H_{c}, it is better to show the measurement time!All my measurements of H_{c} (all data of H_{c} shown on my web site) are done for the measurement time of 1 second.Can we trust old published data of the measured coercive field H_{c }, when the measurement time is not indicated? Since coercive field H_{c} is smaller for a longer measurement time and it is larger for a shorter measurement time?Yes, this statement is fully correct. However, we can can guess the approximate measurement time of these old measurements. I guess for magnetometer measurement or Hall measurements it may be between 1 and 10 seconds. Therefore, these data can be considered as a rough estimate. Can the coercive field be negative?Yes, it is the case of a magnetically soft nanomagnet, in which τ_{retention}<1 second (See telegraphic noise below)
In short. Measurement of the retention time τ_{retention}The precise value of τ_{retention }is measured as a point where the line of log (t_{switch }) vs H crosses the yaxis.The definition of retention time τ_{retention}:The retention time is defined as the average time, after which the magnetization is reversed in the absence of a magnetic field due to a thermal fluctuation. τ_{retention} is referred to the maximum data storage time of a magnetic memory. Using the described method, the τ_{retention} can be measured in the range from one minute to billions of years. The measurement precision is very high (~0.01 %) How it is possible to measure the retention time of one billion years?Experimentally, the magnetization reversal time t_{switch} is measured under external magnetic field. In my experimental setup I can measure t_{switch} in the range from 0.3 s to a few minutes. I adjust the external magnetic field for to be in this range. Since t_{switch} increases exponentially when H decreases, the t_{switch} can easily reach a million or billion years when extrapolated to H=0. It depends of the slope of Fig.10 and H_{c} Highprecision measurement of τ_{retention} The measured dependence of log(t_{switch}/1 sec) vs H is a line. The retention time τ_{retention }corresponds to the magnetization switching time, when the magnetic field is not applied H=0. A linear fitting experimental data and extrapolating the line to H=0 give the τ_{retention} with a very high precision. The Néel model calculates the magnetization switching time t_{switch} as Eq.(a1) can be re written in a linear form as The linear fitting the experimental data of Fig.10, Fig.12, Fig.13 gives the τ_{retention} with a very high precision.
Inshort. Measurement of the size _{}of nucleation domain for magnetization reversal(What one needs to measure the domain size): magnetization of the sample + the slope of the log of the switching probability. Substution of the data into Eq.(a4) below gives the volume of the nucleation domain (a simple idea beyond measurement: ) The log of switching probability is proportional to the number of the spins in the nucleation domain, which is a product of the volume of the nanomagnet and the number of spins per volume (= magnetization of the ferromagnetic material). The external magnetic field changes the magnetic energy of each spin and, correspondingly, the switching probability. Consequently, the change of log of probability under the external magnetic field is linearly proportional to the number of the spins in the nucleation domain and, therefore, the volume of the nucleation domain. The size _{}of nucleation domain is evaluated from the measured slope of the line of log (t_{switch }) vs external magnetic field H and from the magnetization, which, for example,is measured by a magnetometer.When the dimensions of a nanomagnet are sufficiently small, the magnetization reversal occurs in a single domain. It means that the magnetization at all points of the nanomagnet rotates coherently and the magnetization in different parts of the nanomagnet remains parallel during the rotation. When the dimensions of the nanomagnet become larger, the type of the magnetization reversal is changed to the multidomain type In the case of a larger nanomagnet, it is more energetically favorable when at first the magnetization of only a small domain is reversed following by domain wall movement expending the region of the reversed magnetization over the whole nanomagnet. Simple math beyond measurement of the domain sizeIf the probability to reverse the spin of one electron equals p, then the probability to reverse the spin of two electrons equals p·p. The probability to reverse the spin of n electrons equals p^{n}. The magnetization M of a ferromagnet is defined as a number of the spins per the ferromagnet volume. If the volume of the ferromagnetic domain is V_{domain} , the number of the spin in the domain is M ·V_{domain}. Then, the probability of a reversal of the nucleation domain is calculated as p_{domain}=p^{M ·V_domain} or log(p_{domain}) =M·V_{domain} The switching probability of one spin is larger for a longer waiting time and proportional to the energy barrier E_{barrier} (classical model) or the Zeeman energy E_{Zeeman} (quantum model) t_{switch}~e^{E/kT} The magnetic energy of the spin in an external magnetic field H equals to E=H·μ=H·g·μ_{B}·S The log switching time for one spin is log(t_{switch})~ H·g·μ_{B}·S The switching time of the n electrons depends on the external magnetic field as log(t_{switch})~ H·g·μ_{B}·S·n=H·slope, where g is the gfactor and μ_{B} is the Bohr magneton Therefore, the slope is linearly proportional to the number of the spins in the nucleation domain._{} The slope of Fig.13 gives the magnetization of the nucleation domain. From the known (measured) magnetization of the ferromagnetic metal per its volume , the volume of nanomagnet is calculated The log magnetization switching time log (t_{switch }) is linearly proportional to the external magnetic field (See here): where the slope of this dependence is calculated as the magnetization M_{eff } is the magnetization of the nucleation domain (case of multidomain switching) or the total magnetization of the nanomagnet(case of singledomain switching) Magnetization of the ferromagnetic metal of the nanomagnet M_{magnet} per volume V_{magnet} of can be measured by a magnetometer (or checked by literature). Since the magnetization of the nucleation domain is measured, the volume of the nucleation domain is calculated as in conventional units the nanomagnet volume is calculated as the number 51717 is a product of the fundamental constants (see above) in the shown units. (note) The magnetization, which is measured by a magnetometer, is a sum of the total spin of the localized electrons and the total spin of the conduction electrons. In Eq.(a4), only the total spin of the localized electrons should be used. However, the precise measurement of the total spin of the spinpolarized conduction electron is still challenging (See here). The use of the measured magnetization in Eq.(a4) gives a good estimate of V_{domain}, which is slightly smaller than the real value of V_{domain}. Static magnetic domain vs switching magnetic domain
They are two very different objects!(Static magnetic domains): The domains or regions of the different (often opposite) magnetization directions.It is the feature of a nanomagnet or a magnetic film of a larger area. The static magnetic domains usually exists in the absence of the external field. However, sometimes some external magnetic field is required in order to the static magnetic domain. The static magnetic domains are formed to make a balance between the exchange force, which is trying to align all magnetization of all regions to be in one direction, and magnetitic dipole force, which is try to align the magnetization in neighbor regions to be antiparallel. Note: The FeB and FeCoB nanomagnets of size of a few micrometers and smaller do not have any static domains. The magnetization of the whole nanomagnet is in one direction. Note: A nanomagnet free of static magnetic domains have a rectangular shape hysteresis loop (See Fig.1). The hysteresis loop of a nanomagnet with static domains is not rectangular shape. It is either of shape shown in Fig.2 or with some steps. Magnetization switching mechanism of a nanomagnet with static domains: domain wall expansion of static magnetic domains. (switching magnetic domain): The region, which the coherently rotates during of the first step of the magnetization reversal.The switching domain exists in a moderatesize nanomagnet, which are free of static domains. When an external magnetic field is applied opposite to the magnetization direction. At first, the magnetization of a small region (of the switching domain) rotates to be parallel to the external magnetic field. Next, its domain wall of the switching domain expands over the whole nanomagnet. The life time of the switching domain is short ( less than o millisecond). Note: The size of the switching domain in FeB and FeCoB nanomagnets is varied from 30 nm to 90 nm depending on the material and structural defects in the nanomagnet. The size of the switching domain in nanomagnet of reasonablygood quality is 4555 nm. Why the hysteresis loop of a nanomagnet, which are free of static magnetic domains, is of a rectangular shape, but hysteresis loop of a nanomagnet with static magnetic domains is of more complex nonrectangular shape?
Inshort. Measurement of the parameter ΔThe precise value of the Δ is measured from the slope of the line of log (t_{switch }) vs H and the anisotropy field H_{anis} of the nanomagnet, which is evaluated from a magnetostatic measurement.The parameter Δ describes the energy required for the magnetization reversal in comparison with thermal energy. The parameter Δ is defined is the ration the energy barrier to the thermal energy Δ = E_{barrier}/kT The parameter Δ is a parameter, which estimates the ability of a memory cell to withstand a thermal fluctuation and the ability to withstand the temperature rise without loss of the stored data. It is defined as the ratio of energy barrier Ebarrier in the absence of an external magnetic field to the thermal energy kT. In the case of multidomain switching, the magnetic energy is the energy of the nucleation domain. In the case of singledomain switching, the magnetic energy is the energy of the whole nanomagnet. The magnetic energy or the PMA energy can be calculated as (See PMA) where M_{eff } is the magnetization of the nucleation domain (case of multidomain switching) or .the total magnetization of the nanomagnet(case of singledomain switching) and H_{anis } is the anisotropy field. The H_{anis} is the feature of the PMA and the same for the whole nanomagnet and the nucleation domain. From Eq.(a10) the parameter Δ can be calculated as
The retention time can be calculated from Δ as In short. Transformation of Hysteresis loop under different effects
In short. Magnetizationreversal time_{} _{}t_{switch }influenced by different effects
Classical or Néel model of thermally activated magnetization switchingL. Néel, Adv. Phys., 1955Just a fact: The classical Néel model describes the thermally activated magnetization reversal as a thermallyassisted jump or tunneling of an electron through an energy barrier, which separates the spinup and spindown stable energy states.
Assumption of the the Néel model. The Néel model assumes that the switching between two stable magnetization states occurs when the energy of a thermal fluctuation becomes larger the energy barrier E_{barrier } between states. The probability of a thermalfluctuation is described by the Boltzmann distribution Facts, which are ignored by the Néel model(Ignored Fact 1) Spin conservation law The spin conservation law requires the participation of a particle with a nonzero spin (a magnon, photon etc) in the magnetization reversal process. The Néel model assumes that particles participate in the magnetization reversal, but only particle energy is used in the calculation of the Néel model. (Ignored Fact 2) Complex dynamic of magnetization reversal The complex dynamics of the magnetization reversal should be described by the LandauLifshitz (LL) equations, and a model of the thermallyactivated magnetization reversal should be based on the LL equation. This dynamic is fully ignored the Néel model. The dynamic of the magnetization reversal is only important for the resonance switching. The Brown is one possible model, which describes the resonance switching (See here) (Ignored Fact 3) Dynamic of movement of domain wall It assumed that after a nucleation domain for the magnetization reversal is created, it has a sufficient energy to move without any pinning over the whole nanomagnet. Despite of the ignorance of the aforementioned facts, the Néel model gives a reasonable description of the thermally activated magnetization switching, which reasonably good fits of the experimental data in many cases.Calculations steps the Néel model(step 1) Calculation of the energy barrier E_{barrier} (step 2) Calculation of the switching probability (step 3) Calculation of the magnetization switching time ( Néel model. Step 1) . Calculation of energy barrier E_{barrier } between two stable magnetization statesThe angledependent part of the energy E of the uniaxial magnetic anisotropy can be written as where θ is the angle between the magnetization M and the film normal, φ is is the angle between the magnetic field H and the film normal, E_{PMA} is the energy of the perpendicular magnetic anisotropy, which includes the energy due to the demagnetization field. Maximum of energy is at θ _{max} with corresponded energy of Maximum of energy is at θ _{min}=0 and 180 degrees
Therefore, the barrier height is The PMA energy E_{barrier }can be express as where E_{barrier }is the anisotropy field (See here). Substituting Eq. (1.7) into Eq.(1.6) gives Realistic case H_{c }<< H_{anisotropy }
In all my measurements of magnetic nanomagnet it always the coercive field H_{c} is about 24 % of the anisotropy field H_{anisotrp}. Therefore, H_{c }/ H_{anisotrp }~ 0.020.04 at measurement time of 1 s. As was aforementioned, the switching field becomes larger as the switching time becomes shorter. In my experimental setup, the shortest measurement time is 100 ms and the condition H_{c }<< H_{anisotropy } is well satisfied. However, in the case of a shorter measurement time, the condition may be not satisfied. The extension of the reported results for that case is straightforward.
Using this assumption, the Eq.(1.8) is simplified as or where M is the magnetization of a nucleation domain for switching (multidomain switching) or the magnetization of nanomagnet (singledomain switching)) (fact):The energy barrier E_{barrier} between two stable states of nanomagnet is linearly proportional to external magnetic field. The proportionality coefficient is the magnetization M of a nucleation domain for switching (multidomain switching) or the magnetization M of nanomagnet (singledomain switching))Note: When the energy barrier is reduced, the magnetization is switched. In the case of a FeCoB nanomagnet, magnetization switching occurs when the barrier height is reduced only by about 6 % at measurement time of 1 secondWhat is the exact meaning of the condition H_{c }<< H_{anisotropy }?It is means that the measurements of the thermally activated switching is done at a moderate or a small magnetic field. In the case of a high magnetic field H ~ H_{anisotropy }, the magnetization switching is very rapid. The magnetization is switched within time of a nanosecond or shorter. It means that the magnetization turns along the external magnetic field almost instantly after it applied. It is not practical to make a measurement for such high field and a short switching time. The practicallymeasurable switching time of a millisecond or a second or a minute is for a small or moderate magnetic field when H << H_{anisotropy}. The H_{c } is defined as the field when the average switching time is one second. ( Néel model. Step 2) . Arrhenius low & transition state theory (TST)Néel model states that the average magnetization reversal time t_{switch} or relaxation time τ is described by the Arrhenius low: where f_{0} is is the socalled attempt frequency associated with the frequency of the gyromagnetic precession; E_{barrier} is the energy barrier between two states when the magnetization is along and opposite to the external magnetic field. The Arrhenius low has its origins in the 1880s when Arrhenius proposed, from an analysis of experimental data, that the rate coefficient in a chemical reaction should obey the law where ΔV denotes the threshold energy for activation of the chemical reaction, f_{0} is the attempting frequency.
Substituting Eq. (1.8) into Eq.(1.11) gives In the case when H_{c }<< H_{anisotrp }, Eq(1.12) is simplified to the retention time τ_{retention} is the average time of magnetization reversal without any external magnetic field. From Eq.(1.13), the τ_{retention} can be calculated as From Eqs.(1.13 and 1.14) , the magnetization reversal time τ is given as: Eq.(1.15) is the main result of the Néel model for the thermally activated magnetization reversalThe linear dependence of log( t_{switch} ) vs H perfectly fit to all experimental measurements. It clearly proves that the classical Néel model fully describes the nonresonance magnetization reversal. (result of the Néel model):The logarithm of the magnetization switching time log( t_{switch} ) vs H is linearly proportional to external magnetic field H. The proportionality coefficient is the magnetization M of a nucleation domain for switching (multidomain switching) or the magnetization M of nanomagnet (singledomain switching) The part below is designed for an advanced reader interested in complex details
Alternative methods to measure H_{c }and Δ(not recommended)The probability of a systematic error for each of belowdescribed methods is very high. The convergence of some of this methods is not fast. Alternative methods to measure H_{c }and Δ : (method 1): graded magnetic field (method 2): graded pulsed magnetic field Common errors of the alternative measurements methods of H_{c }and Δ(common error 1) unjustifiably large number of free parameters is used to fit experimental data. There are only two free parameters for a thermallyactivated magnetization switching. Two additional parameters the magnetization M and the anisotropy field , which should be obtained from a magnetostatic measurements, can be used in description of the thermallyactivated magnetization switchingThe Néel model is relatively simple and only has two free parameters: the energy barrier E_{barrier} and the rate of interaction finter of the nanomagnet with the particles (See details of the Néel model). Alterternatively, any other pair of free parameters may be used (for example, the coercive field Hc and retention time τ_{retention} or size of nucleation domain may be used as one of the two free parameters). However, maximum two parameters should be always used for the data fitting. Additionally, there are parameters, which are related to the magnetostatic properties of a nanomagnet. For example, the Δ is proportional to the anisotropy field H_{anis }(See Eq.(a7)) and the volume of the nucleation domain is proportional to the magnetization M of the nanomagnet (See Eq.(a3)). Both H_{anis} and M can be measured from an independent magnetostatic experiment without the use of any thermallyactivated switching measurements. The magnetization M of a ferromagnetic metal can be measured by a magnetometer. H_{anis} can be measured by applying an inplane magnetic field and monitoring the inplane component of the magnetization (See here). The fact that there are only two free parameters of the Néel model, can be confirmed from measured dependence of t_{switch} on H (See here). A straight line perfectly fits to all experimental data and the line is described by only two free parameters. (common error 2) Assumption that the attempt frequency f is a universal constant of the Néel model and equals to 1 GHz The attempt frequency f is not a constant and does not equal to 1 GHzIt is hard to trace from which reason this assumption came from, but it is completely incorrect assumption (See details of the Néel model). (common error 3) Incorrect use of statistical measurement and statistical analysis. Example: A common systematic error of measurement method of graded pulsed magnetic field (See here)
Details of the Néel modelProof of the validity of the Arrhenius low (Eq.1.10) for the magnetization reversalit was originally an assumption of the Néel model as an experimentally verified empirical fact(step 1): Magnetization reversal as a result of interaction with external particlesThe Néel model assumes that the magnetization reversal occurs only when the spin of the nanomagnet interacts with a "nonzerospin" particle (a magnon, a photon etc.), which energy is higher than the barrier height E_{barrier} between two stable states of the nanomagnet. The temperature is assumed to be sufficiently high so that the energy distribution of the particles is described by the Boltzmann distribution. Therefore, the number of particles, which are able to reverse the magnetization, is calculated from the the Boltzmann distribution as where n_{0} is the total number of the particles, which are able to reverse the magnetization(e.g. the total number of magnons,phots,etc.). When there are more particles, the probability of reversal becomes higher. The frequency, at which the magnetization can be reversed, is proportional to the number of the particles and is calculated as where f_{inter} is the frequency of interaction of one particle with the spin of the nanomagnet. (step 2): Differential equation for the function describing the probability of magnetization reversalNote: It is easier to obtain the differential equation for probability P_{not} that the magnetization is not reversed, than for the probability P_{rever} that the magnetization is reversed. Of course, P_{rever}+P_{not}=1The probability P_{rever}(t,t+dt) of the magnetization reversal in a small time interval between t and t+dt is calculated as where The probability P_{not}(t,t+dt), that the magnetization is not reversed during the time interval dt, is calculated is If the magnetization is not reversed in the interval [t0,t+dt], that means that it is not reversed in both intervals [t0,t] and [t,t+dt]. Therefore, the probability P_{not}(t0 ,t+dt) is calculated as Eq.(5.6) can be simplified as The function P_{not}(t) is defined as the probability of nonreversal of the magnetization in the time interval from t_{0} to t. From Eq. (5.7), the function P_{not}(t) satisfies the following differential equation: (step 3): Solution of the differential equationIn the case when the external magnetic field H is a constant and timeindependent, the energy barrier E_{barrier} and τ are timeindependent as well and Eq. (5.8) becomes a linear differential equation. The solution of Eq.(5.8) gives the probability P_{not}(t) of the nonreversal of the magnetization in the time interval from t_{0} to t as: (step 4): Calculation of the average magnetization switching time t_{switch}Next, the averaging magnetization switching time tswitch is calculated. If the external magnetic field is switched on at time t_{0}=0 in the direction opposite to the magnetization, the probability dp_{switch} that the magnetization is reversed in the time interval between t and t+dt is the difference between probabilities that it is not reversed until time t and until time t+dt: From Eq. (5.10), the averaging magnetization switching time t_{switch} is calculated as The substitution of Eq.(5.4) into (5.11) gives the Arrhenius law (Eq.(1.10)) as
Use of pulse magnetic field to measure magnetization switching time t_{switch}This measurement should be used in case when a time of measurement of magnetization direction is comparable with t_{switch} (e.g. Hall measurement)Note 1 (measurement of magnetization reversal event) An event of magnetization reversal is monitored by a measurement of a material parameter, which depends on the magnetization oft he nanomagnet (e.g. resistance, tunneling resistance, magnetooptical constants, Hall angle etc.) Note 2 (long time for measurement of Hall voltage) The Hall voltage is small and can be measured by a nanovoltmeter. One measurement by a nanovoltmeter takes 1.7 second
In order to measure t_{switch} shorter than 5 second using the Hall setup, the pulsed magnetic field should be used. In order to measure t_{switch} longer than 5 second the use of the pulsed magnetic field is not necessary.
Switching probabilityThis part is designed for an advanced reader onlyIt is a ration of the magnetization switching events to the sum of switching and nonswitching events for a fixed measurement time. The switching probability can be measured experimentally. The probability that the magnetization is not switched by the time t is described as where τ is the relaxation time Correspondingly, the probability that the magnetization is not switched by the time t is described as In the case when H<< H_{anisotrp}, substituting Eq(1.15) into Eqns.(2.12.2) gives
To see how to obtain Eqs, 2.1,2.2, 2.7, 2.8 click here to expand
It is easier and simpler to calculate the probability of not switching P_{not} by time t rather than the probability of switching P_{switch} by time t. The relation between P_{not} and P_{switch} is straightforward P_{not} + P_{switch}=1 The non switching of the magnetization in time interval interval [0,t+dt] means that the magnetization is not switched in both time intervals [0,t+dt] and [t,t+dt]. The probability of nonswitching in time interval [0,t+dt] equals to the product of probabilities of nonswitching in time intervals [0,t+dt] and [t,t+dt]: Probabilities of switching in a small time interval [t,t+dt] is linearly proportional to dt where 1/τ is the coefficient of the proportionality. From Eq.(2.41) we have Substituting Eq.(2.42) into Eq.(2.40) gives From time t to t+dt the value of the nonswitching probability change on Substituting Eq.(2.40a) into Eq.(2.43) gives Case 1. External magnetic field is constant in this τ is constant. The integration of Eq.(2.44) gives
Case 2. External applied magnetic field changes in time Substituting Eq.(1.8) into Eq.(2.47) gives where
Case 3. External magnetic field linearly ramped in time For example, if time dependence of magnetic field is described as Using integral and substituting Eq.(2.6) into Eq.(2.48) and integrating gives
Case 4. External applied magnetic field linearly ramped in time and it is H(t)<<H_{anisotropy} (realistic case) In the case of small field H(t)<<H_{anisotropy}, the following approximation can be used substituting Eq(2.49a) into (2.48) gives In the case of linear ramping integration of (2.49c) gives where M_{eff} is the effective magnetization or the magnetization of a nucleation domain
Case when External magnetic field linearly ramped In this case time dependence of magnetic field is described as See X. Feng and P. B. Visscher, Journal of Applied Physics 95, 7043 (2004)
Alternative measurement method of H_{c}. Distribution of switching probabilities. Theory and measurementsThis part is designed for an advanced reader onlyDistribution as function of timebelow I calculate as switching probability depends on the measurement time ( duration of magnetic pulse). Probability that magnetization is switched only in time interval between t and t+dt is equal to the product of probability that it is not switched where switch probability exactly at time t will be The probability was normalized so that The average magnetizationreversal time is calculated as Alternative measurement method of H_{c}. Distribution as function of magnetic fieldThis part is designed for an advanced reader only
Below I calculate the dependence of the switching probability on magnetic field. Below only the realistic case of H<< H_{anisotrp} is calculated. The switching probability in this case is described by Eqs. (2.3),(2.4) As was show above, the coercive field (switching field) depends on the measurement time τ (see Fig.11). Let us refer to the coercive field as the switching field at measurement time of 1 second τ=1. Than, from Eq.(1.15) Using Eq.(4.1), Eq.(1.15) is simplified to Substituting Eq.(4.3) into Eqs.(2.1) and (2.2) gives the probability P_{nonswitch}([0,H]), that the magnetization is not switched, when magnetic field increases from o to H, and the probability P_{switch}([0,H)), that the magnetization is switched, as The probability P_{switch}([H,H+dH)) that the magnetization is switched at the magnetic field between H and H+dH is proportional to where A is the proportionality constant, which can from normalization condition and it gives the switching probabilities as The average switching magnetic field is calculated as The mean deviation is calculated as to see how to obtain Eqs. (4.8) and (4.9) , click to expand
In normalization the following integral was used to obtain the distribution average switching magnetic fieldThe average field is defined as Substituting Eq.(4.7) gives We used the value of the integral Eq(4.8c) gives mean deviationThe mean deviation is defines as Substituting Eqs. (4.7) and (4.8) gives or
Measurement of delta ΔThis part is designed for an advanced reader only
The Δ is parameter, which characterized the magnetic stability of nanomagnet. It is defined as a ratio of the energy barrier E_{barrier} between two stable magnetization states to the thermal energy. It characterized how much bigger E_{barrier} should be than the thermal energy to avoid an undesirable magnetization reversal due to a thermal fluctuation.According to Neel model, the energy barrier E_{barrier} between two stable magnetization states in absence of magnetic field equals to E_{PMA}. Therefore
The retention time can be calculated as or There are three possible techniques to measure Δ. Each technique requires additional measurement of H_{anisotropy} (see here the measurement details). Measurement of Δ. Technique 1. Using linearlyramped magnetic fieldlow precision & moderate measurement timeThe Δ is evaluated by fitting the distribution of magnetization switching probability Eq.(2. 49b) or (2.8). Measurement of Δ. Technique 2. Using magnetic pulses of gradually increases amplitude (Fig.12)low precision & long measurement timeThe Δ is evaluated from the width of the distribution of magnetization switching probabilities Eq.(4.2) and Eq.(4.7).(See Fig.12) Measurement of Δ. Technique 3 8recomnded) . From magnetization switching time τ (Fig.11)high precision & moderate measurement timeThe Δ is evaluated by from measurements of the dependence magnetizationreversal time vs magnetic field(See here ) This highprecision measurement of Δ requires 3 steps step 1 : Measuring the effective magnetization M_{eff}_{} On log scale, τ is linearly proportional to the applied magnetic field(Fig.11). The slope of the fitting lines is proportional to M_{eff} and the horizontal offset is proportional to τ_{retention } step 2 : Measuring the anisotropy field H_{anisotropy}_{} Method to measure anisotropy field is described here It is a relatively easy to measure the anisotropy field (See here) . Even though it often requires a relatively large inplane magnetic field. Without magnetic field the magnetization is perpendicularlytoplane. Under magnetic field, the magnetization turns toward magnetic field. The field, at which the magnetization turns completely inplane, is called the anisotropy field. The E_{PMA} is calculated from H_{anisotropy} as (See here) where M_{eff} is the total magnetization in the case of a singledomain magnetization reversal or the effective magnetization M_{eff} in the case of multidomain magnetization reversal step 3 : Calculating Δ_{} The Δ can be simply calculated as: Important: Some researchers are trying to find both H_{anisotropy} and Δ only by fitting the distribution of magnetization switching probability with two independent parameters H_{anisotropy}_{} and Δ. It is incorrect and leads to incorrect result. From the Neel model, the distribution has only one independent parameter, which is the ratio of Δ and H_{anisotropy}_{} or M_{eff}. It is important that H_{anisotropy}_{} should be measured independently. Otherwise, the fitting gives incorrect result. The measurement of H_{anisotropy} is relatively simple (See here)Additional method to measure ΔFrom nonlinear dependence of switching time on magnetic fieldit requires a high magnetic field and measurement of a shorter switching time!Any method related to Neel model is based on only one important parameter, which is the barrier height E_{barrier} between two stable magnetic states: There are three component, which are proportional to magnetic field H in power 0, 1,2. The Δ can evaluated from independent measurements of the 2d and 3d components.
What is effective magnetization M_{eff}. Nucleation domain for magnetization switchingThis part is designed for an advanced reader only
Multi domain magnetization switching When size of the nanomagnet is sufficiently large, the magnetization reversal is not coherent over whole nanomagnet. At first, the magnetization is reversed in a small domain. Next the domain wall moves and expands. As a result, the magnetization of whole nanomagnet becomes along the applied external magnetic film. The M_{eff} is magnetization of first magnetic domain, which triggers the magnetization reversal. The reason, why magnetization switching occurs by this mechanism: A thermal activation energy to reverse magnetization of a small domain is much smaller, than the energy to reverse magnetization of the whole film. How small is the size of the nucleation domain The size of the nucleation domain is determined by trade of between the exchange energy and the barrier energy E_{barrier}_{ }for the magnetization switching. The stronger exchange interaction is, the larger size of the nucleation domain become. The size of the nucleation domain is evaluated from measurements of Fig.11
Singledomain magnetization switching In this case the magnetization of whole nanomagnet is reversed coherently. This switching occurs only in a nanomagnet of very small size (diameter ~1040 nm) M_{eff} equals to the saturation magnetization M_{sat} of material multiplied by the sized of the nanomagnet
M_{eff} gives the magnetization of the initial domain, which is first switching during magnetization (case of multidomain switching). In the case of singledomain reversal, M_{eff} equals to product of the saturation magnetization and the volume of nanomagnet.
Singledomain switching and multidomain switchingThis method can unambiguously measure for a tested device whether magnetization switching is singledomain or multidomain. singledomain switching It is the case when the effective magnetization M_{eff}_{} is equal to the saturation magnetization M multiply to volume of the nanomagnet multidomain switching It is the case when the effective magnetization M_{eff}_{} is smaller than the saturation magnetization M multiply to volume of the nanomagnet Measurement of the size of the nucleation domain for magnetization switching
Estimated measurement precision of the nucleation domain size is better than 1 %.The size on the nucleation domain is measured without using any microscope. Only data of Hall measurements are used!!!How to measure?The size of nucleation domain equals the effective magnetization M_{eff} divided per the saturation magnetization M. The saturation magnetization M is measured by SQUID magnetometer before nano fabrication and The the effective magnetization M_{eff} is measured by this method after micro fabrication. Note: M is the magnetization per unit of volume; M_{eff} is the total magnetization of the nucleation domain.Note: The area of the domain measured. The size of domain is estimated as a square room from domain volume assuming a domain of a square shape.Note:I have developed this method in 20172018.
Size of the nucleation domain in different materialsFeB and FeCoB amorphous nanomagnet + anneal and partial recrystallization As can be seen from the right picture, the size of the nucleation domain varies from 30 nm to 60 nm. However, there are nanomagnets with a longer domain size. As Nov. 2018, I have studies more than 100 nanomagnets from 25 FeB, FeBCo, (FeB/W)n samplesCo singlecrystal nanomagnetThe variation is narrow: from 40 nm to 50 nm. H As Nov. 2018, I have studies only 6 nanomagnets from 2 samples
Influence of MgO/FeCoB interface on magnetic properties of FeCoB nanowire
Experiment to test the influence of the interface on magnetic properties of nanowire. The key feature of this experiment: MgO is removed from a half of nanowire and the magnetic properties of both parts are measured. Since all magnetic properties of both are identical except Similar slope of lines of right figure indicate that the effective magnetization M_{eff} is nearly the same for both parts. It means that the etching was stopped just after MgO and FeCoB was not etched Effect of Removal of MgO: 1) Anisotropic field: decreases
Note: The Hall angle and Hall resistance in FeCoB nanowire greatly depend on the proximity of MgO gate
Relation between delta and retention time
Both the delta and retention time characterize the probability of the magnetization switching in absence of a magnetic field. The relation between them is (See here) or Experimentally the retention time and the Δ are measured by two independent experiments (see here and here) All my experimental data (by Nov. 2018) show that the delta linearly proportional to log of the retention time.
note: Experimental data are better fitted by where 0<a<1
Attempt frequency frequency fAll my experimental data (by Nov. 2018) show that attempt frequency f is nearly the same for all samples made of the same ferromagnetic metal. However, it is different for different metals. For example, there is a two order difference between f in samples made of an amorphous FeB and made of a single crystal Co.
Temperature dependence of delta Δ, anisotropy field H_{anis}, effective magnetization M_{eff}
When temperature rises, all the Hall angle , coercive field, anisotropy field, effective magnetization, saturation magnetization, retention time and nucleation domain size decrease Q. By definition the Δ= E_{PMA} /kT. Does it mean that the temperature dependence of Δ following the law 1/T?A. No. The decrease of the Δ with a temperature rise is more sharp. It is because E_{PMA} substantially decreases with a rising of temperature. Additionally, the size of a nucleation domain for magnetization reversal may change with temperature. That also affects the temperature dependance of Δ.
Difference of switching fields between spinup to down τ_{uptodown} and spindown _{ }to spinup states τ_{downtoup}
In principle, the magnetization switching times for switching from spinup to down state τ_{uptodown} and from spindown τ_{downtoup} to up state should be the same Effects, which make τ_{uptodown} and τ_{downtoup} different:applying an inplane magnetic field Effects, which keep τ_{uptodown} and τ_{downtoup} equal: change of temperature
Note: Any possible systematical error in measurements of relative position of switching times was eliminated by a precise calibration by a Hall measurement for a nonmagnetic metal.A problem of magnetic random access memory (MRAM). Measurements and solutions
A serious problem of the MRAM is a wide variation of magnetic properties from a cell to cell The major variation is due to the variation of the domain nucleation size, which may be influence by technologydependent factors.
4. Néel  Brown modelW.F.Brown (1963), W. T. Coffey, Y. P. Kalmykov (2012)In Néel  Brown model, it is assumed that the magnetization switching occurs not because an energy of a thermal fluctuation exceed E_{barrier}, but because a more complex magnetization dynamic described by the LandauLifshitz equations.
resonance magnetization switching It is the case when the Néel  Brown model should be used instead of the Néel model!When frequency of magnetic field, or electrical current or electrical field is close to the frequency of the ferromagnetic resonance (FMR), the magnetization switching may occur at the energy much smaller than the barrier energy E_{barrier}. Example 1: Microwave assistant magnetic recording (MAMR) to hard disk. When data density in hard disk becomes very high, the required magnetic field to record one data because unacceptably large. In order to solve this problem the MAMR is used. In the case of MAMR, a weak microwave radiation, which frequency is close to the FMR of harddisk media, excites magnetization precession. After that, the magnetic field of recording head reverses the magnetization and records a data bit. The required recording magnetic field is substantially small than in the case without the microwave radiation. Example 2: Data recording of magnetic random access memory (MRAM) using the VCMA effect The VCMA effect is weak effect. At present, it is hard to hard to reverse magnetization by the VCMA effect using DC gate voltage. However, when a pulse of interval close to the reverse of the FMR frequency, the magnetization may be reversed even by small amplitude pulse (Shiota 2012). In this case, the pulse energy is substantially smaller than E_{barrier}.
Details of the Néel  Brown modelIn the Néel  Brown model, the random magnetic field is assumed to act on the magnetization. The magnetization switching conditions are derived from a solution LandauLifshitz equations for the magnetization affected by the random magnetic field. What is the physical meaning of the random magnetic field of the Néel  Brown model?In the model the random magnetic field is a pure mathematical tool. However, the physical meaning of this fields is associated with the interaction of the magnetization with magnons and the electron scattering (spd interaction) between localized d states and states of spinunpolarized conduction electrons.
Which the Néel model or Néel  Brown model is correct?As July 2018, all my experimental measurements fit to the Néel model extremely well (See for example here). Even though I have used the magnetic and electrical pulses at a frequency much smaller than the FMR frequency of my studied samples. The the Néel model is simple and intuitive. The mathematical description of this model is relatively simple. It is based on two simple facts: (1) there is the energy barrier E_{barrier} between two stable magnetization states and (2) the energy of a thermal fluctuation should be larger than E_{barrier}. In contrast, the Néel  Brown model is more complex and less intuitive. The Néel  Brown model should be used only in the case when the Néel model clearly fails to describe the experiment. FORC measurement method to multi particles systems & static domains
This measurement method is used to study features of magnetization reversal Details of The FORC method can be found in the following references A.P. Roberts, et.al. Firstorder reversal curve diagrams: a new tool for characterizing the magnetic properties of natural samples, J. Geophys. Res. (2000) Cao et.al. Hysteresis in single and polycrystalline iron thin films: Major and minor loops, first order reversal curves, and Preisach modeling. JMMM (2015) Mayergoyz, I. D. Hysteresis models from the mathematical and control theory points of view. JAP (1985). Stancu, et.al. Micromagnetic and Preisach analysis of the First Order Reversal Curves (FORC) diagram. JAP (2003).Pike et.al. An investigation of magnetic reversal in submicronscale Co dots using first order reversal curve diagrams, JAP (1999)
(Numerical Analysis) In this method a complex hysteresis loop of a multiparticle system is analyzed as a sum of a simple hysteresis loops of each individual particles (mostly square shape loops). This analysis is called Analysis of Preisach Diagrams. (FORC Measurement) : Magnetic properties are evaluated from a set of scan of magnetic field with a different partial reversal of magnetization. the following is cited from A.P. Roberts, et. al J. Geophys. Res. (2000)"A FORC diagram is calculated from a class of partial hysteresis curves known as firstorder reversal curves or FORCs . As shown in Figure la, measurement of a FORC begins by saturating a sample with a large positive applied field. The field is decreased to a reversal field H_{a}, and the FORC is defined as the magnetization curve that results when the applied field is increased from H_{a} back to saturation. This measurement procedure is repeated for different values of H_{a} to obtain a suite of FORCs (Figure l b). The magnetization at the applied field H_{b} on the FORC with reversal point H_{a} is denoted by M(H_{a}, H_{i}), where H_{i}, > H_{a} (Figure la). Data from consecutive measurement points on consecutive reversal curves (Figure l c; see below) are used to determine the FORC distribution, which is defined as the mixed second derivative: "
Useful tool or a fancy complex method without benefit?Obviously some researcher are using this method and therefore it might be useful. However, as 2004.04 I cannot see any real benefit of this method in my measurements. Does the FORC measurement have a possible systematical error?If it is not addressed, the FORC measurement has a substantial systematic error. Similarly to the measurement of coercive field from a coercive loop, the FORC measurement substantially depends on the measurement time, the variation of the measurement time from scan to scan creates a systematic error.
Magnetization reversal through a vortex stateIt a rare case of the magnetization reversal for nanomagnet of very specific sizes and thickness. Fernandez et. al. Magnetic domain structure and magnetization reversal in submicronscale Co dots. JMMM (1998)
Measurement of coercive field, size of nucleation domain & retention time in FeCoB nanomagnet
Sum of symstimatic measurement in many samples: download origin file: AllSampleHc.opj
Known tricks and methods of fake and highlight research
(trick 1 ) Tracing a small change of coercivity field H_{c} using a coercivity loopE.g. tracing the change H_{c} as function of a gate voltage or bias current or some another parameter of a weak influence on H_{c} (Where is the trick?) Measurement precision of H_{c} from coercive loop is very poor. Also, the width of the coercive is slightly different for each measurement. the distribution of measured can be large. (See here method 3 measurements) . The distribution is wide specially for smaller sample. The width of the H_{c} distribution can be as wide as 50 Oe. Tricks how to generate a dependence on a gate voltage or bias current: When I have started my research on VCMA and SOT effects, I have been very surprised about a large difference in reported data on the H_{c} dependence on gate voltage and bias current. Surprisingly, the opposite polarity of the dependence of H_{c} on gate voltage has been reported for nearly identical samples. Only after I have developed a high precision method of H_{c} measurement, I have understood the trick. For example, a nanomagnet has H_{c} =400 Oe with distribution width of 60 Oe. It means the measured from a hysteresis loop is in the range between 370 Oe and 430 Oe. E.g. under gate voltage of  2 V, the H_{c} increases to 410 Oe. It means that the range of possible H_{c} becomes from 380 Oe to 440 Oe. Measuring hysteresis loop several times, it possible to pick a loop with H_{c}=380 at V_{gate}=2 V and a loop with H_{c}=420 at V_{gate}=0 V and therefore to show the incorrect opposite dependence of H_{c} on V_{gate}. This is how it is possible to make from the same experimental data two opposite dependencies when a low precision measurement method is used. (trick 2 ) Measuring properties of magnetization switching without a measurement of retention timeMany "efficient" magnetization switching by the VCMA or SOT effects has been reported by showing a change of the coercive loop and without a measurement of the change of the retention time (Where is the trick?) The property of thermally activated magnetization switching is that the magnetization can be switched between its two opposite stable directions even without any external magnetic field or a gate voltage or current. The average time, within which the magnetization is switched, is called the retention time. The retention time of FeCoB nanomagnets, which I have been studied, varies from a few seconds to billions of years. For example, the retention time of nanomagnets of sample Volt54A is about an hour. The width of the coercive loop significantly depends on the measurement time. For the case of a fast scan of magnetic field (within 1 second), the loop is wide (It case be assigned as a nonswitching case). For the case of a slow scan of magnetic field (within 30 min ), there is almost no width of loop (it can be H_{c}<0.) (It case be assigned as the switching case). However, nothing basically changed. My point is that for thermally activated switching, the only important parameter if the change of the retention time. E.g. if the retention time is changed from a billion year to 1 millisecond for example, under an gate voltage, it truly means the magnetization switching by the gate voltage. In contrast, a tiny change of the hysteresis loop should not be considered as the magnetization switching.
(trick 3) Reporting an "efficient" magnetization switching under bias inplane magnetic fieldMany "champion" data have been reported for magnetization switching of a nanomagnet, when a bias magnetic field is applied inplane (along the hard axis of nanomagnet) (Where is the trick?) As it is shown here, the magnetization under bias of inplane magnetic field is very unstable and the switching can be triggered by a smallest external perturbation. However, this method is useless for a realistic practical applications, because such switching is very unstable, repeatability is poor and the switching conditions are greatly different from a nanomagnet to a nanomagnet even for those fabricated on the same wafer. (trick 4 ) ReportingE.g. (Where is the trick?) Measurement
Questions & AnswersHow does the pinning of the nucleation domain influence on the parameters of the thermally activated switching.(nucleation domain and pinning of domain wall) In general, there is no pinning of the nucleation domain during the magnetization reversal. Most of nano magnets, I have been studied, have a perfect squareshaped hysteresis loop. If there is a pinning, the hysteresis loop has large steps. (A very small number of nanomagnets I have with such strong pinning) It requires some special efforts in order to make a domain, which is pinned at some place in a nano magnet or a nano wire. I have studied rather simple nano magnets of round, elliptic and square shapes. There was no any speciallymade domainnucleation spot. In this case, the domain unpinning mechanism or the speed of the motion of the domain wall has no influence on the parameters of the thermal activated magnetization switching. It is because in this case the magnetic energy, which requires for the magnetization reversal in the region of nucleation domain (energy for the creation of the nucleation domain) is much higher than the energy of the pinning. Therefore, after the nucleation domain is created, the magnetic energy is already high. Since the high energy of domainwall motion, after creation of the nucleation domain the domain wall is moving fast and generally it cannot be pinned (except if there is a reallystrong domainpinning site)
Is the coercive loop is the same of a nanomagnet and a continuos film, from which the nanomagnet is made?(difference of H_{c} between film & nanomagnet) The coercive loops are very different. The coercive field H_{c} of continuous film is substantially smaller than H_{c} of a nanomagnet, which is made from this film. It is because of very different thermal energy, which required for switching in each case. In the case of a continuos film, the magnetization reversal is due to expansion of magnetic domains and the moving of the domain wall. The domain expansion is not a thermally activated mechanism and itself does not have a hysteresis loop. However, there may be obstacles (defects, surface imperfections, etc) in the film for continuos movement of the domain wall. In this case thermal activation is required in order to overcome the obstacle and therefore the hysteresis loop appears. The thermal activation energy to overcome the domain pinning is usually small and it causes only small H_{c}. In contrast, in case of a nanomagnet a substantially larger thermal activation energy is required in order to create a nucleation domain. This is the reason why the H_{c} is larger for a nanomagnet.
Since the coercive loop is very different of continuous film and a nanomagnet, should I measure of H_{c} for a continuous film before nanofabrication or it is useless?(film quality vs H_{c}) A measurement of coercive field of continuous film is very useful to estimate the quality of the grown film. The smaller H_{c} is, the better quality is. The smaller H_{c} means that there is a smaller number of defects and the average size of defects is smaller. Therefore, it is easier for the domain wall to move. (note) The above answer is applied to a magnetic film, which have static domain in equilibrium and are not in a single domain state. It mean that the hysteresis loop of the film should not be of a rectangular shape.(note) The film quality should be compared for the film of a similar material.You have mentioned that a ferromagnetic metal without defects doesn't have a coercive loop. How then a coercivity field is associated with a specific material? How are all material divided to the soft magnetic materials (a small coercive field) and the hard magnetic materials ( a large coercive field)( H_{c} of a bulk material without defects) It is correct. A specific coercive field is always associated with bulk ferromagnetic material. For example, H_{c}~30 Oe for bulk Fe and H_{c}~200 Oe for bulk Co. Usually the bulk material has an intrinsic fix density of defects, which determines value of . For example, both the bulk Fe and bulk Co are polycrystalline materials. Domain boundary are good pinning sites for domain wall motion, which need a thermal energy to unpin. An "ideal" ferromagnetic without defects, imperfection or surface roughness doesn't have a coercive loop (H_{c} =0) There are many fabrication techniques, which used a dye or a dopant in a ferromagnetic material to pin or unpin a domain wall, in order to make the ferromagnetic material harder or softer (note) magnetic parameters, such exchange interaction, magnetic anisotropy, size of static domains, strongly influence the H_{c}(note) The above answer is applied to a magnetic film, which have static domain in equilibrium and are not in a single domain state. It mean that the hysteresis loop of the film should not be of a rectangular shape.
Can you simulate numerically the hysteresis loop shown in Fig.33 by blue line?( simulation of hysteresis loop. Case of static domains) There are many rather complex "fancy" methods to simulate the hysteresis loop for the switching by static domain. I use a rather simple, but very effective method. It can be used for a larger sample (not continuous film), in which there are static domains. As the first step, I calculate the loop without the thermally activated contribution (red line of Fig.34). For this purpose I calculate the size of the static domains as a function of an applied external magnetic field. It can be done by minimizing magnetic energy ( energy of magnetostatic interaction between domains minus energy of domain walls. It can be by Comsol or similar software. At the second step, I measure the parameters of the the thermally activated switching (This method is major topic of this page. See fig. 10) and include them into the loop. As a result, the red line of Fig 34 transforms into the blue line.
Why does type of switching mechanism dependent on nanomagnet size? Why magnetic domains are unstable in small nanomagnet?(about stability of magnetic domain vs size of a nanomagnet) A larger nanomagnets have static domain. The magnetization direction of neighbor domains is usually in opposite directions. It minimizes the energy of the magneto static interaction between them. When the size of nanomagnet becomes smaller than the size of a magnetic domain, all spins of localized electrons are aligned in one perpendicular direction and the state of the nanomagnet becomes the singledomain state. The reason why all spins of localized electrons are perfectly aligned in a nanomagnet in one direction can be understood as follows. For existence of a static magnetic domain, the positive exchange energy of a domain wall should be balanced by a negative magneto static energy between dipoles of opposite magnetizations. When shape of nanomagnet is a circle with radius R and domain wall passes through its center, the magneto static energy is proportional to domain area (~R^{2}) and domain wall energy is proportional to its length (~R). When the size of nanomagnet decreases (decrease of R), the magneto static energy decreases faster, at some size it becomes unable to balance the domain wall energy and nanomagnet state becomes a single domain state.
Why all magnetization across a nanomagnet is directed in one direction? Is it possible that some small region has a different magnetization direction?
(reason why magnetization of all regions across a nanomagnet is in the same direction) In case some region of nanomagnet has the magnetization, which direction is different from the rest of nanomagnet, there is a domain wall of a positive energy between this region and the rest of nanomagnet. The state without this region has a smaller total magnetic energy and is therefore more energetically favorable. For most of nanomagnets, which I have studied, the magnetization is aligned perfectly perpendicularly to surface across the whole nanomagnet. However, there are exceptions. The magnetization direction might be very different at last atomic layer at an interface. The exchange interaction at the interface may be very different that that in the bulk. It can be changed even to antiferromagnetic one. The Dzyaloshinskii–Moriya interaction is very common at an interface. The difference of the exchange interaction can balance the existing of a domain wall and the magnetization direction may be a slightly different at the vicinity of an interface.There are many experimental proofs that the magnetization is perfectly aligned in one direction across the whole. Below there are two which I often use: (experimental proof 1: linear dependence of M_{} vs. H_{}): In this measurement (details here and here) an external magnetic field H_{} is applied in plane (along the magnetic hard axis). The magnetization turns towards the inplane field. The dependence of the inplane component of the magnetization M_{} vs H is a perfect line, which is only possible when the nanomagnet is in a single domain state. (proof of perfect alignment): In the case if some region has magnetization different from the perpendicular, the inplane magnetic field H_{} expands this region in order to minimize the magnetic energy and the dependence is substantially different from linear.
(experimental proof 2: existence of a nucleation domain in process of magnetization reversal ): The experimental measurements of thermally activated magnetization switching in the same nanomagnets confirms that the magnetization switching mechanism of the studied nanomagnets is the nucleation domain type. A nucleation domain is an unstable magnetic domain, which exists for a very short time (~a few milliseconds) during magnetization switching. The very existence of the nucleation domain confirms that the sample is mono domain because such nucleation domain can exist only in a single domain nanomagnet in absence of a static domain. Additionally the size of nucleation domain is measured to be between 40 nm and 90 nm for our FeB and FeCoB nanomagnets (details are above). Such measurement is impossible in case when there are static domains. (proof of perfect alignment): the nucleation domain can exists and therefore be measured in a single domain nanomagnet. Otherwise, the pre existed region of different magnetization just expands instead of creation of a nucleation domain.
(experimental proof 3: The stable static domain is at least a few times larger than the unstable nucleation domain. We have exactly similar result (even the measured signal is noisier) for a very small nanomagnets (d<50 nm), which size is even smaller than size of nucleation domain. The existence of a much larger static domain in such a small nanomagnet is absolutely impossible. (proof of perfect alignment): there is a minimum size of static domain. When the size of nanomagnet is smaller that this size, there is no static domains
Why do static domains exist in a larger nanomagnet, but magnetization of a smaller nanomagnet is aligned in one direction?(magnetization of a singledomain nanomagnet) When the size of nanomagnet becomes smaller than the size of a magnetic domain, all spins of localized electrons are aligned in one perpendicular direction and the state of the nanomagnet becomes the singledomain state. The reason why all spins of localized electrons are perfectly aligned in a nanomagnet in one direction can be understood as follows. For existence of a static magnetic domain, the positive exchange energy of a domain wall should be balanced by a negative magneto static energy between dipoles of opposite magnetizations. When shape of nanomagnet is a circle with radius R and domain wall passes through its center, the magneto static energy is proportional to domain area (~R^{2}) and domain wall energy is proportional to its length (~R). When the size of nanomagnet decreases (decrease of R), the magneto static energy decreases faster, at some size it becomes unable to balance the domain wall energy and nanomagnet state becomes a single domain state.
Can your measurement method be used in case when type of magnetization switching is the movement of domain wall?(about measurement of energy of barrier for domain wall movement) Yes it can be used, but it is very effective. In fact, the major parameter, which is measured by above  described method, is the energy of the barrier between two stable states of the nanomagnets. The movement of domain wall needs overcomes some energy barriers due to defects and imperfections. The domain overcomes these energy barriers by a thermally activated mechanism. This reason why the switching by rearrangement of the static domain still has a coercive loop. If the above  described method is applied for a big nanomagnet with static domain, it measures the average energy of barriers for domain wall movement. Usually, this energy is small and difficult to measure.
I am strongly against a fake and "highlight" research

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