Dr. Vadym Zayets

(2d) In short. Measurement of size of nucleation domain

Measurement method of all parameters of thermo- activated switching (coercive field, retention time, parameter Δ, size of nucleation domain)

main idea of measurement method: magnetization switching time is measured vs. applied magnetic field. The dependence of Log(switching time) vs magnetic field is linear. It makes the fitting very precise and allows to evaluate all switching parameters with a very high precision

It is main measurement method to evaluate all parameters of thermo- activated switching

In comparison with other method, only this method is free of systematic errors and allows to achieve a high- precision, high-repeatability and high- reliability.

(what is measured) The magnetization switching time vs magnetic field, which is applied opposite to the magnetization

(which parameters are evaluated) Coercive field Hc, retention time τret, size of nucleation domain and parameter Δ

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(3) In short. Transformation of Hysteresis loop under different effects (VCMA, SOT, in-plane magnetic field)

(4) In short. Magnetization-reversal time t_switch influenced by different effects (VCMA, SOT, in-plane 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 thermally-reversed? 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 single-domain 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 single-domain 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 direction

about 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 domains

Part 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 spin-up to down and spin-down to spin-up states

(17) FORC method to analyze magnetic properties of multi particles systems

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(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 in-plane magnetic field

..............

Measurement method of coercive field, retention time, parameter delta, size of a nucleation domain
(measurement method): A magnetic field is applied opposite to the magnetization and the time until the magnetization switching is measured. The switching time is exponentially depends on the intensity of the external magnetic field. From this dependence all parameters of the thermally activated switching are measured, which include coercive field, retention time, parameter delta, size of a nucleation domain
(thermally-activated switching): There is an energy barrier between two stable states for magnetization of a nanomagnet. The magnetization switches between the states only by assistance of a thermal fluctuation of an energy higher the barrier height. The probability of the thermal fluctuation is exponentially proportional to its energy. The magnetic field reduces the energy barrier. This is the reason why the switching time exponentially proportional to the magnetic field.
(features of the method): high precision; high repeatability, high reliability
Note. I have developed this measurement method in 2017-2018
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Thermally-activated magnetization switching & storage time & magnetic recording

Existence of a coercive loop as indication of a thermally - activated process -> =

It doesn't matter whether the coercive loop is large or small, square - shaped or some complex shape, the existence of a coercive field is always due to a thermally- activated process

Existence of a hysteresis loop means that there is an energy barrier, which prevents a continuos change of some parameters. A thermal fluctuation is required to overcome the energy barrier.

The statement is applied to a micro- or nano- sized object (e.g. a domain, domain wall), in which the energy barrier comparable with the energy of a thermal fluctuation (~kT)

Only reliable method of measurements of any parameter of thermally activated switching is a measurement of switching time vs. applied field. (reason why:) A probability of a thermal fluctuation is proportional to the measurement time. The probability is larger for a longer time. As a result, any measurement type (except the measurement of switching time) depends on measurement time and therefore often contains a systematic error.

A switching field can be found from the hysteresis loop. However, each measurement gives a slightly different value due to the variation of the measurement time

The statement is applied also to more complex measurements of thermally activated processes like FORC

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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 memory

A 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 below-described measurement method, the data storage time can be measured with a very high precision in both cases.

recording method of magnetic memory

In 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 higher-energy spin-down level exceeds the number of electrons on the lower-energy spin-up 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 descriptions

Magnetization reversal occurs when:

(Quantum description): number of spin-down electrons exceeds the number of spin-down electrons

(Classical description): the electron moves over or tunnel through an energy barrier, which separates the spin-down and spin-up 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 description

It 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 higher-energy level exceeds the number of electrons on a lower-energy level.

Thermal mechanism of the magnetization reversal. Quantum description

In the quantum description the magnetization reversal is described as a quantum transition between the spin-up and spin-down energy levels.

No external magnetic field   External magnetic field opposite to magnetization. Magnetization reversal.   In absence of an external magnetic field, there is a larger separation between spin-up and spin-down energy levels. The transitions from the spin-up to the spin-down level are rare. The excited spin-down electron quickly relaxed back to the the lower-energy spin-up level. There are a very small number of spin-down electrons.   The separation between spin-up and spin-down energy levels is small, because the total magnetic field Hint-Hext becomes smaller. As a result, the transitions from the spin-up to the spin-down level are frequent and there are a substantial number of spin-down electrons. When the number of spin-down electrons exceeds the number of spin-down electrons the magnetization is reversed.

Magnetization reversal occurs when the number of spin-down electrons exceeds the number of spin-down electrons. As a result, the internal magnetic field reverses its direction and the spin-down state becomes a stable state.

Blue ball shows the magnetization (the total spin) of the nanomagnet (or nucleation domain). Hint is the internal magnetic field, which is aligned either along or opposite to the magnetic easy axis (See details here). . Hext is the external magnetic field. Green balls are exchanged coupled spins of localized electrons. The angle of the magnetization precession depends on a relative number of spin-down and spin-up electrons (See here)

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(fact about total spin) within a nucleation domain or a single-domain 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 thermally-activated magnetization reversal becomes.

If the probability for a quantum transition from the spin-up 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 pⁿ. 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 thermally-activated magnetization reversal becomes.

Classical description or Néel description

The 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 spin-up and spin-down 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.

Thermal mechanism of the magnetization reversal. Classic description

In the classic description the magnetization reversal is described as a jump over or tunneling of an electron through the energy barrier, which separates two stable spin-up and spin-down states..

The green ball shows the magnetization direction, the arrows shows the strength and direction of the external magnetic field H .

Magnetic energy as a function of the magnetization angle with respect to the magnetization direction. The energy is smallest when magnetization angle is along the easy axis (angle=0 and 180 deg). The energy minimums correspond to two stable magtization state. The energy maximums correspond magnetization direction perpendicular to the easy axis (along the hard axis).

The animation parameter is the strength of external magnetic field with respect to anisotropy field H/Hoff . When the external magnetic field is along magnetization the energy dip decreases. When the external magnetic field is opposite to magnetization the energy dip increases.

(reversal mechanism) When the barrier height becomes sufficiently small, the magnetization interacts with an external particle. Due to this thermal process, the magnetization gains the energy and momentum to jump over the energy barrier and reverses own spin direction.

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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 here

Magnetic energy:

Energy barrier:

(classical description 2): full description based on properties of the spin-orbit 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 here

Magnetic 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.

mechanisms of magnetization reversal

There are 3 distinguished mechanisms of the magnetization reversal: (mechanism 1) Domain movement of static domain. This mechanism is a feature of a large-size nanomagnet or a magnetic film, in which a static domain exists. (mechanism 2) Creation of a nucleation domain. This mechanism is a feature of a moderate-size nanomagnet, in which static domains do not exist and the equilibrium magnetization is homogeneous (magnetization is the same direction through all nanomagnet). When magnetic field is applied opposite to the magnetization, at first magnetization of a small area (nucleation domain)is reversed. Next, the area of the reversed magnetization expand over whole nanomagnet. (mechanism 3) Single-domain magnetization reversal. This mechanism is a feature of a smallest-size nanomagnet, which size is smaller the the size of the nucleation domain. As a result, the magnetization is reversed coherently over the whole nanomagnet.

mechanism 1a. Domain wall movement of static domains. Static domain exist at H=0. (Largest nanomagnets and films) mechanism 1b. Domain wall movement of static domains. There is no Static domain at H=0. (Large-size nanomagnet) mechanism 2. Creation of a nucleation domain following the domain wall movement. (Moderate-size nanomagnet) mechanism 3. Coherent magnetization rotation (Smallest-size nanomagnet) Fig. 1a. Top view of domain structure of the magnetic film. The darker and whiter areas show the static domain of 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. Fig. 1b. There is no static domain at H=0. Static domains are nucleated at H1. Next, static domain expand their volume as H increases from H1 to H2. Above H2 the magnetization is parallel to H. Fig. 1c When a magnetic field is applied opposite to the magnetization direction,at first 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 (blue line) moves expanding the nucleation domain over the whole sample. 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. Magnetization vs magnetic field (schematic loop) The magnetization smoothly increases (decreases) without any steps vs H. Since it requires a magnetic energy the pinning of domain wall., the magnetization M is different for the H scan from higher to lower value and the reversed H scan measured Hall angle vs H for Co(0.6):Pt(10) nanomagnet. H1=0.8 kG is field of nucleation of a static domain. Till field of H2=3.2 kG, static domains expand. For H<H1, there is no static domain and magnetization is directed opposite to H. At H=H1 the static domain is nucleated. From H1 to H2, the volume of static domains expands until the magnetization of whole nanomagnet becomes || to H. See similar loop for Ta(5):FeB(0.4) nanomagnet Hall angle vs H for FeB(1):Ta(3) 3000 nm x 3000 nm nanomagnet. The loop of a rectangular shape. The magnetization switching is sharp and step-like. At switching field of ~0.2 kG, the nucleation domain is created and its wall quickly expand over the whole nanomagnet. There is no static domains at any H. Hall angle vs H for FeB(1):Ta(4) 70 nm x 40 nm nanomagnet. The loop of a rectangular shape. The magnetization switching is sharp and step-like. At switching field of ~0.25 kG, the magnetization of the whole nanomagnet coherently rotates to

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3 mechanisms of magnetization reversal

(magnetization reversal) Mechanism 1: Domain wall movement of static domains

samples: large-size 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: moderate-size 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 rotation

samples: smallest-size 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.

(magnetization reversal) mechanism 1a: movement of wall of static-domains
(left) hysteresis loop of nanomagnet (magnetic film) with static domains. (right) top view of a domain structure of the magnetic film. The whiter areas show area, where magnetization direction is to the left. The darker areas show area, where magnetization direction is to the right. Blue arrow shows the direction and magnitude of external magnetic field. Green arrow shows the direction and magnitude of the total magnetization of the film.
As magnetic field H increases, the domain walls move and domains of magnetization direction parallel to H increase until they cover all film. Because of domain pinning, the extension of the domain requires the energy. It is reason why the domain area extends gradually with increase of magnetic field and why the area is different parts of increase and decrease of H (the reason for the loop)
(magnetization reversal) mechanism 1b: nucleation of a static domain following movement of wall (under increase of magnetic field)
(left) measured Hall angle vs H for Co(0.6):Pt(10) ; (right) top view of a domain structure of the magnetic film.
There is no static domain at H=0. Static domains are nucleated at at a weak magnetic field. Next, static domain expand their volume as H increases. Above H2 the magnetization is parallel to H.
difference between magnetization-reversal mechanism 1b & 1a: (mechanism 1a) domain walls move at both the positive and negative magnetic field, because magneto-static energy of domains is high. (mechanism 1b) static domains are nucleated & domain walls move only when magnetic field is opposite to the magnetization, because magneto-static energy of domains is low.
mechanism 1b-> a thinner film. mechanism 1a-> a thicker film.
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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. magnetic-dipole interaction). There are two interactions between spins of different electrons. The short-range very-strong exchange interaction and the long-range moderate (low)-strength magnetic-dipole 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 domain-wall) 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 single-domain 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 non-zero- 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 pⁿ. Therefore, the probability is higher to reverse a smaller amount of the spins.

Can only one spin be thermally-reversed? 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 large-area 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 (~R² · 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 (~R² · 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: R² · E_MS- R· E_EX - R² · 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 (domain-wall 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 single-domain 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 single-domain state?

A. It is because the energy barrier between single-domain and nucleation domain states is smaller than the energy barrier between two single-domain 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 domains

samples: large-size 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

Excellent 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)

Fig. 1 (part) from Cao et. al 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 defect-free 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 due to domain- wall movement of static domains. Two mechanisms.

Mechanism 1: Magneto-static Mechanism 2: Thermo- activation Total hysteresis loop as a sum of mechanism 1 and 2 It is the case of no defects. Domain wall moves smoothly and continuously. There are defect or boarder irregularity, which the domain wall is not able overcome without a therm fluctuation. The thermo fluctuation switches the magnetization between two stable states The domain wall moves until the defect or boarder irregularity. A thermo fluctuation makes to overcome the movement obstacle. loop does not depend on measurement time measurement loop depends on measurement time. A probability of a thermo fluctuation is higher for a longer time. measurement loop depends on measurement time. A probability of a thermo fluctuation is higher for a longer time. Fully reversible . No coercive loop. It returns to exactly the same point when when H scanned back. Non- reversible. There is a coercive loop. It returns to a different point when when H scanned back. Non- reversible. There is a coercive loop. It returns to a different point when when H scanned back.

Coercive loop of a nanomagnet with static domains consists of two distinguished parts. 1st part corresponds to magneto-static mechanism of domain- wall movement. 2nd part is domain- wall movement through an energy barrier of defect assisted by a thermo fluctuation.

Additional loop features may exist when there is no static domain without magnetic field and static domains are nucleated under magnetic field applied opposite to the magnetization (See Fig. 1b above)

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Magnetization reversal type: domain wall movement of static domains. Two mechanisms of movement of domain wall.

(mechanism 1: reciprocal) :Magneto-static

Domain wall moves in order to decrease the magneto-static 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 thermo-activation mechanism. It means it assisted by a thermo fluctuation. The change of the total magnetization is step-like 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 domains

The 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 step-like features.

( type 2): Nucleation domains

A 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 step-like features.

Below the features of thermally-activated 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 moderate-size and large-size nanomagnet, which is important for the memory applications

Hysteresis loop of a nanomagnet

Fig1 Schematic diagram of a hysteresis loop of a nanomagnet. The magnetization M of a ferromagnetic nanomagnet as a function of applied external magnetic field H. The filled arrow shows the direction of the magnetic field H. The unfilled arrow shows the magnetization direction. The nanomagnet has only two "up" and "down" stable magnetization directions along the easy axes. The magnetization switching between the two stable states is sharp and it occurs at magnetic field H defined as the coercive field Hc.

Hysteresis loop of a nanomagnet is of a rectangular shape

Click on image to enlarge it

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 thermally-activated magnetization switching. The properties of the thermally-activated 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.

Two methods to measure Hc: switching time or switching field

measurement parameter: switching field measurement parameter: switching time fixed parameter: switching time. fixed parameter: switching field. classical rough measurement: from hysteresis loop precise measurement: from switching time External magnetic field is scanned until the magnetization is reversed. The measurement time is a constant due to the fixed field scanning rate In this measurement, the waiting time until magnetization is reversed is measured. The external magnetic field is fixed switching method: lowering Ebarrier switching method: waiting a longer time for a thermal fluctuation of a higher energy

There is an energy barrier Ebarrier between up- and down-magnetization states. A thermal fluctuation assists the to overcome the barrier and to reverse direction

Two method to overcome Ebarrier : (1) make Ebarrier lower. (2) wait for a thermal fluctuation of a higher energy.

(method 1 to reverse magnetization): increase magnetic field. It makes the energy barrier between two states smaller.

(method 2 to reverse magnetization): wait a longer time. It the probability of a required higher-energy thermal fluctuation higher.

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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 higher-energy 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.

Hysteresis loop of a multi particle system or a magnetic film with static domains

Fig. 2 The hysteresis loop is not of the rectangular shape as in the case of nanomagnet (Fig.1) . Fig is from here

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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 switching

The 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

(single-shot measurement) Don't do a single-shot measurement for parameters of thermally- activated magnetization switching like H_c

Magnetization switching -> thermally- activated process	Coercive field -> dependance on measurement time
Fig.8. Each time the switching of magnetization occurs at slightly different magnetic field. Click on image to enlarge it	Fig.9 The longer the scan of magnetic is, the shorter the coercive field is.

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 magnetization-reversal time is fixed and it is proportional to the applied magnetic field.

Method of a high precision measurements of parameters of the thermally-activated switching

magnetization-reversal time tswitch magnetization-reversal time tswitch vs magnetic field Néel model predicts: 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 magnetization-reversal time is fixed and it is proportional to the applied magnetic field. It is also called the relaxation time, . Click on image to enlarge it Fig 10 The crossing of the line with the y-axis gives the retention time τretention . The τretention is the maximum storage time of a data in a nanomagnet. The crossing of the line with the x-axis gives the coercive field Hc. The slope gives the effective magnetization of the nucleation domain M for magnetization switching. The experimental measurements perfectly fit the linear dependence predicted by the Néel model.

From this measurement, all parameters of thermally -activated switching ( Hc, τreten , Δ and size of a nucleation domain) are evaluated with a

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 1-4 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 multi-domain switching) or the total magnetization of the nanomagnet(case of single-domain 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 thermally-activated switching can be evaluated

How many there are free parameters thermally-activated 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 magnetization-reversing 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 thermally-activated magnetization switching. Additionally, there are parameters, which are related to the magneto-static 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 thermally-activated 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

Measurement of Coercive field Hc

high- precision measurement rough (low precision) measurement Hysteresis loop Fig.11 Hc is measured from from the linear dependence of log (tswitch) vs magnetic field (see Fig.10). The crossing of the line with the x-axis gives the coercive field Hc. Measurement precision of this method is 0.1 Oe and better From the hysteresis loop. Hc is measured as a magnetic field, at which magnetization is switched. Measurement precision is very poor. It is about 1-10 % from Hc The coercive field Hc is the magnetic field, at which magnetization is switched along two opposite direction of a magnetic field, . Click on image to enlarge it

click on image to enlarge it

The precise value of H_c is the field, at which measured line of log (t_switch ) vs H crosses the x-axis 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.

High-precision 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)

Measurement of retention time τretention

magnetization reversal due to a thermal fluctuation high-precision measurement of τretention telegraphic noise Even without any external influence, the magnetization reverses its direction with average time of τretention due to a thermal fluctuation. Depending on the nanomagnet, τretention can be a minute or a billion years fig 12. τretention is measured from the linear dependence of log (tswitch ) vs magnetic field (see Fig.10). The crossing of the line with the y-axis (H=0) gives the coercive field τretention. Even in the case of τretention=1 billion years, the measurement precision of τretention is better than 0.1 %. The magnetization of the nanomagnet is randomly switched between its two stable directions (e.g. up and down) with an average time of τretention. The behavior is called telegraphic noise. The telegraphic noise is very obvious and easily measurable in the case of soft nanomagnet (e.g. τretention=1 second)

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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 y-axis.

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

High-precision 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.

Measurement of the size of nucleation domain

multi-domain magnetization reversal mechanism high-precision measurement of size of nucleation domain When a magnetic field is applied opposite to the magnetization direction,at first the domain wall (blue line) is formed. Within this domain the magnetization is reversed by a thermal activation (Neel mechanism). Next, the domain wall moves along the nanowire. When it stops, only a small domain remains. Its magnetization is reversed by a thermal-activation as well. Fig. 13 The size of the nucleation domain is evaluated from slope of the linear dependence of log( tswitch ) vs magnetic field (see Fig.10) and the magnetization of the nanomagnet M, which is measured by a magnetometer.

Measurement of the size of nucleation domain consists of two measurements: (switching measurement): slope of log( tswitch ). (magnetostatic measurement) magnetization of ferromagnetic metal of nanomagnet

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In-short. 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 multi-domain 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 size

If 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 pⁿ. 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 g-factor 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 multi-domain switching) or the total magnetization of the nanomagnet(case of single-domain 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 spin-polarized 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

Hysteresis loop of a nanomagnet

without static magnetic domains with static magnetic domains rectangular shape non- rectangular shape magnetization reversal mechanism: (step 1) nucleation of switching domain; (step 2) expansion of domain wall of switching domain over the whole nanomagnet magnetization reversal mechanism: domain wall expansion of static magnetic domains

Different magnetization switching mechanisms in a nanomagnet with and without static domains is the reason of different shape of the hysteresis loop.

Click on image to enlarge it

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 moderate-size 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 reasonably-good quality is 45-55 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 non-rectangular shape?

Measurement of parameter Δ

measurement of Δ. step 1. Measurement of slope of log( tswitch ) vs H measurement of Δ. step 2. Measurement of Hanis Measurement step 1: Measurement of slope of the linear dependence of log( tswitch ) vs magnetic field (see Fig.10) Measurement step 2: Measurement of anisotropy field Hanis. In-plane magnetic field is applied to nanomagnet and the in-plane component of the magnetization is measured. Hanis is the magnetic field, at which magnetization turns fully in-plane.

high-precision measurement of the Δ consists of two measurements: (switching measurement): slope of log( tswitch ). (magnetostatic measurement) anisotropy field Hanis

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In-short. Measurement of the parameter Δ

The 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 multi-domain switching, the magnetic energy is the energy of the nucleation domain.

In the case of single-domain 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 multi-domain switching) or .the total magnetization of the nanomagnet(case of single-domain 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

Transformation of Hysteresis loop

temperature SOT effect in-plane magnetic field VCMA effect Width and height of the loop decrease. There is no shift of loop position and both magnetization switching (spin-up-to-down and spin-down-to-up) occur at same absolute value of magnetic field. The position of loop is shifted due to current, but width and height of the loop are constant.(Details of SOT effect are here) Under in-plane magnetic field, width of the loop decreases. Additionally, the loop position is shifted. Under a negative gate voltage, the width and height of the loop increase. Under a positive gate voltage, the width and height of the loop decrease. There is no shift of loop position and both magnetization switching (spin-up-to-down and spin-down-to-up) occur at same absolute value of magnetic field. (Details of VCMA effects are here)

The x-axis is applied out-plane magnetic field. Click on image to enlarge it.

In short. Magnetization-reversal time t_switch influenced by different effects

Switching time influenced by different effects

in-plane magnetic field temperature Voltage- controlled magnetic anisotropy, VCMA effect spin-orbit torque, SOT effect   switching from spin-down to spin-up switching from spin-up to spin-down   When temperature rises, both the slope and offset decreases. This means that both the retention time and the effective magnetization Meff decreases. Change of nanowire resistance in % is also shown. Under gate voltage, the line is shifted, but remained parallel. It means that the gate voltage modulates the retention time, but it does not affect Meff. Note: The retention linearly depends on the gate voltage (Details of VCMA effects are here) For spin-down-to-up magnetization switching, switching time increases under a negative current and decreases under a positive current. The dependence is opposite for spin-down-to-up switching. The switching time increases under a positive current and decreases under a negative current. (Details of SOT effect are here)   Sample Volt34Free82: Ta(2):FeB(1.4):MgO   Sample ud30 Volt53B Ta(2.5):FeCoB(1):MgO Nanowire width: 1000 nm, length 200 nm. Measurements date is 10. 2018

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Classical or Néel model of thermally- activated magnetization switching

L. Néel, Adv. Phys., 1955

Just a fact: The classical Néel model describes the thermally- activated magnetization reversal as a thermally-assisted jump or tunneling of an electron through an energy barrier, which separates the spin-up and spin-down stable energy states.

Results of Néel model
(main result): magnetization switching time tswitch vs magnetic field H
logarithm of the magnetization switching time log( tswitch ) vs H is linearly proportional to external magnetic field H. The proportionality coefficient is the magnetization M of a nucleation domain for switching (multi-domain switching) or the magnetization M of nanomagnet (single-domain switching)
(result 2): Energy barrier Ebarrier between two stable states of a nanomagnet is linearly proportional to external magnetic field H
Click on image to enlarge it

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 thermal-fluctuation 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 non-zero 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 Landau-Lifshitz (LL) equations, and a model of the thermally-activated 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 states

The angle-dependent 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.
Below the case when a magnetic field is applied along the magnetic easy axis (perpendicularly to the film) ( φ=0) is calculated. The maximums of energies can be found from the condition

Maximum of energy is at θ _max

with corresponded energy of

Maximum of energy is at θ _min=0 and 180 degrees

Magnetic energy vs angle between magnetization and external magnetic field Magnetic energy vs angle between magnetization and external magnetic field Energy barrier Ebarrier vs magnitude of external magnetic field H Energy barrier Ebarrier vs magnitude of external magnetic field H Magnetic energy depends on the angle between the magnetization and external magnetic field. There are two energy minimums corresponding to the magnetization is parallel and anti parallel to the magnetic field The energy of the uniaxial magnetic anisotropy E (Eq. 1.1) as function of magnetization angle with respect to the applied external magnetic field. There are two minimums, which correspond to the magnetization be parallel or anti parallel to external magnetic field. There is an energy barrier Ebarrier between these two stable states. The ratio of magnitude of external magnetic field H to the anisotropy field Hanis equals 5% The energy of the uniaxial magnetic anisotropy E (Eq. 1.1) as function of magnetization angle with respect to the applied external magnetic field. Animated parameter is the ratio H/Hanis of magnitude of external magnetic field H to the anisotropy field Hanis. The energy barrier Ebarrier as function of ratio of applied magnetic field to the anisotropy field Hanis. Interesting fact: The external magnetic field reduces the energy barrier Ebarrier only by ~6 % in order to reduce the magnetization direction. The magnetization is reversed under the coercive field Hc. From experimental measurements (see below) Hc~0.02-0.05 Hanis.

Néel model: Magnetization reversal: energy of a thermal fluctuation >Ebarrier

Néel-Brown model: Magnetization reversal: energy of a thermal fluctuation can be < Ebarrier

click on image to enlarge it

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

magnetization-reversal time τ vs magnetic field (Eq.(1.15))

200-nm wide 25-um long Ta(8)/FeB(0.9)/MgO(7.1)/Ta(1)/Ru(5) nanowire 400-nm wide 25-um long W(3):[FeB(0.68)/W(0.25)]x4/ FeB(0.68)/ MgO(5.5)/Ta(1)/Ru(5) nanowire

All data fit very well to Néel model and Néel formula Eq.(1.15)

The crossing of the line with the y-axis gives retention time τretention

The crossing of the line with the x-axis gives coercive field Hc

The slope of the line is proportional to the Meff (magnetization of a nucleation domain for switching (multi-domain switching) or magnetization of nanomagnet (single-domain switching))

Data measured: jan. 2018. Click on image to enlarge it

In all my measurements of magnetic nanomagnet it always the coercive field H_c is about 2-4 % of the anisotropy field H_anisotrp. Therefore, H_c / H_anisotrp ~ 0.02-0.04 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 (multi-domain switching) or the magnetization of nanomagnet (single-domain 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 (multi-domain switching) or the magnetization M of nanomagnet (single-domain 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 second

What 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 practically-measurable 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 f₀ is is the so-called 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, f₀ is the attempting frequency.

Main result of Néel model
logarithm of the magnetization switching time log( tswitch ) vs H is linearly proportional to external magnetic field H. The proportionality coefficient is the magnetization M of a nucleation domain for switching (multi-domain switching) or the magnetization M of nanomagnet (single-domain switching)
Click on image to enlarge it

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 reversal

The 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 non-resonance 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 (multi-domain switching) or the magnetization M of nanomagnet (single-domain switching)

The part below is designed for an advanced reader interested in complex details

Alternative method to measure Hc and Δ

method 1: graded magnetic field method 2: graded pulsed magnetic field In this method, the external magnetic field is changed with a constant linear rate. The field, at which the magnetization is switched, is monitored. Hc and Δ are evaluated from the distribution of the switching field. Pulses of magnetic field of an increasing amplitude is applied until the magnetization is reversed In the interval between a state of the magnetization is checked. When the magnetization is reversed, the number of applied pulses gives the magnetization-reversal time.

this methods are not recommended, because the probability of a systematic error for these methods is very high

Click on image to enlarge it

Alternative methods to measure H_c and Δ

(not recommended)

The probability of a systematic error for each of below-described 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 thermally-activated 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 thermally-activated magnetization switching

The 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 magneto-static 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 magneto-static experiment without the use of any thermally-activated switching measurements. The magnetization M of a ferromagnetic metal can be measured by a magnetometer. H_anis can be measured by applying an in-plane magnetic field and monitoring the in-plane 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 GHz

It 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 model

Proof of the validity of the Arrhenius low (Eq.1.10) for the magnetization reversal

it 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 particles

The Néel model assumes that the magnetization reversal occurs only when the spin of the nanomagnet interacts with a "non-zero-spin" 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 n₀ 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 reversal

Note: 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=1

The 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 non-reversal of the magnetization in the time interval from t₀ to t. From Eq. (5.7), the function P_not(t) satisfies the following differential equation:

(step 3): Solution of the differential equation

In the case when the external magnetic field H is a constant and time-independent, the energy barrier E_barrier and τ are time-independent 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 non-reversal of the magnetization in the time interval from t₀ 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 t₀=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

pulse magnetic field to measure tswitch

Pulses of magnetic field of a constant amplitude is applied until the magnetization is reversed In the interval between a state of the magnetization is checked. When the magnetization is reversed, the number of applied pulses gives the magnetization-reversal time. Measurement of τ as function of magnetic field. Every sample shows a perfect line indicating a perfect match with Néel model. Sample:200-nm wide 25-um long Ta(8)/FeB(0.9)/MgO(7.1)/Ta(1)/Ru(5) nanowire

This measurement should be used in case when a time of measurement of magnetization direction is comparable with tswitch (e.g. Hall measurement)

Data measured: jan. 2018. Click on image to enlarge it

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, magneto-optical 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 probability

This part is designed for an advanced reader only

It is a ration of the magnetization switching events to the sum of switching and non-switching 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.(2.1-2.2) gives

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 measurements

This part is designed for an advanced reader only

Distribution as function of time

below 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 magnetization-reversal time is calculated as

Alternative measurement method of H_c. Distribution as function of magnetic field

This part is designed for an advanced reader only

Alternative measurements of Hc. Rough measure of switching field

Switching probability as a function of magnetic field. Eq.(4.4) Switching distribution as a function of magnetic field. It shows the proportional amount of switching within range of magnetic field. Eq.(4.7)

This measurement gives coercive field Hc

Data measured: jan. 2018. Sample:200-nm wide 25-um long Ta(8)/FeB(0.9)/MgO(7.1)/Ta(1)/Ru(5) nanowire. Number of measurements is 80. Two distributions, which is obtained from a direct fit by Eq.(4.4) or from average switching field Eq.(4.8), are shown. Click on image to enlarge it

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)

where

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_non-switch([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

Measurement of delta Δ

This part is designed for an advanced reader only

Fig.15. Measurement of anisotropy field Hanisotropy

The arrow shows the direction and magnitude of the applied in-plane magnetic field. The ball shows the magnetization direction. Without magnetic field the magnetization is perpendicularly-to-plane. Under magnetic field, the magnetization turns toward magnetic field. The field, at which the magnetization turns completely in-plane, is called the anisotropy field. The dots of the right graph shows experimental data. Measurement date: May 2018.

Click on image to enlarge it

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 linearly-ramped magnetic field

low precision & moderate measurement time

The 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 time

The 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 time

The Δ is evaluated by from measurements of the dependence magnetization-reversal time vs magnetic field(See here )

This high-precision 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 in-plane magnetic field. Without magnetic field the magnetization is perpendicularly-to-plane. Under magnetic field, the magnetization turns toward magnetic field. The field, at which the magnetization turns completely in-plane, is called the anisotropy field. The E_PMA is calculated from H_anisotropy as (See here)

where M_eff is the total magnetization in the case of a single-domain magnetization reversal or the effective magnetization M_eff in the case of multi-domain 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 non-linear dependence of switching time on magnetic field

it 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 switching

This part is designed for an advanced reader only

Effective magnetization Meff. Magnetization reversal in a ferromagnetic nanowire

When a magnetic field is applied opposite to the magnetization direction,at first the domain wall (blue line) is formed. Within this domain the magnetization is reversed by a thermal activation (Neel mechanism). Next, the domain wall moves along the nanowire. When it stops, only a small domain remains. Its magnetization is reversed by a thermal-activation as well.

Click on image to enlarge it

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

Single-domain 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 ~10-40 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 multi-domain switching). In the case of single-domain reversal, M_eff equals to product of the saturation magnetization and the volume of nanomagnet.

Single-domain switching and multi-domain switching

This method can unambiguously measure for a tested device whether magnetization switching is single-domain or multi-domain.

single-domain switching

It is the case when the effective magnetization M_eff is equal to the saturation magnetization M multiply to volume of the nanomagnet

multi-domain 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

Distribution of sizes of nucleation domain in FeB nanomagnets Dependence of delta Δ on the size of nucleation domain in FeB nanomagnet Data of more than 100 measured samples are shown Roughly, The delta is linearly proportional to the domain size in the case of near similar films

Samples of different width and length are shown. The widths of FeB nanomagnets are from 100 to 3000 nm. Lengths of nanomagnets is from 100 to 20000 nm. Samples with Ta and W buffer are shown. FeB and FeB multilayers of different thicknesses are shown.

Click on image to enlarge it. Zayets 2018.

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 2017-2018.

Size of the nucleation domain in different materials

FeB 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 samples

Co single-crystal nanomagnet

The 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

Influence of interface on effective magnetization Meff

Ta:FeB:MgO nanowire. Front half of the nanowire: the MgO and some FeB were etched and SiO2 was deposited instead. Back part of nanowire: FeB:MgO remains. Magnetization switching properties were measured for each part of nanowire. In both cases, the magnetization switching is multi domain type. Coercive loop of the Hall angle. Loop for part with MgO is shown in black. Loop for part without MgO is shown in red. Both loops were measured at the same time for a single scan of magnetic field. A weak exchange coupling between parts is noticeable from the loop shape. Magnetization switching time as function of perpendicular magnetic field. Measurements at part, where MgO is not etched, shown in black. Measurements at part, where MgO is etched, shown in red.

Sample: Ta(5):FeCoB ( 1 nm, x=0.3):MgO(7) Volt58A (L58B); nanowire width is 3000 nm, nanowire length is 25 um, length of etched section is 3 um

Click on image to enlarge it

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
2) Magnetization:                            no change
3) Coercive field                            decreases
4) Retention time:                          decreases
5)  Effective magnetization  :         no change
6) Delta:                                          decreases
7) nucleation domain size:              no change
8) Hall angle:                                  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

delta Δ vs retention time τretention

Sample R68B Volt58B Ta(5):FeCoB(1):MgO Sample Volt34Free82: Ta(2):FeB(1.4):MgO

Δ vs retention time τretention (logarithmic scale). Each point corresponds to a different temperature of nanomagnet. When the temperature increases, both the Δ and the retention time τretention decrease.

The experimental data proves the validity of Eq.(2.60).

Click on image to enlarge it

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 f

All 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

temperature dependence of delta Δ	temperature dependence of Hall Angle	temperature dependence of anisotropy field
Δ rapidly decreases with increases of temperature. The temperature dependence of EPMA mainly defines temperature dependence of Δ.	Hall angle always decreases with increase of temperature. For temperatures far from Curie T, the decrease is nearly linear.	Hanis rapidly decreases with increases of temperature. It is similar to T dependence of Δ
Click on image to enlarge it	Sample: Ta(5):FeCoB ( 1 nm, x=0.3):MgO(7) Volt58A (L58B)	Sample: Volt58A (L58B)

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 spin-up to down τ_up-to-down and spin-down to spin-up states τ_down-to-up

Magnetization switching time for spin-up to down and spin-up to down switching	Hysteresis loop
Usually, the lines are parallel with a slight offset. In nanowire with a higher density of defects (domain nucleation centers) the lines may have a slightly different slope and a larger offset	At a negative magnetic field the magnetization is switched from spin-up to spin- down state. At a positive magnetic field the magnetization is switched from spin-down to spin- up state. Click on image to enlarge it

In principle, the magnetization switching times for switching from spin-up to down state τ_up-to-down and from spin-down τ_down-to-up to up state should be the same

Effects, which make τ_up-to-down and τ_down-to-up different:

-applying an in-plane magnetic field

-effect of spin-orbit torque

Effects, which keep τ_up-to-down and τ_down-to-up equal:

- VCMA effect

- 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 non-magnetic 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 technology-dependent factors.

4. Néel - Brown model

W.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 Landau-Lifshitz 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 hard-disk 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 model

In the Néel - Brown model, the random magnetic field is assumed to act on the magnetization. The magnetization switching conditions are derived from a solution Landau-Lifshitz 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 (sp-d interaction) between localized d- states and states of spin-unpolarized 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

FARC measurements

Fig. 1(a). One scan of a FARS measurement Fig.1(b). Set of FARC measurement Fig.1 (d) Result of FARC measurement

figures from A.P. Roberts, et.al. J. Geophys. Res. (2000)

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. First-order 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 submicron-scale 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 first-order 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 state

It 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 submicron-scale 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

Systematic measurements in FeCoB nanomagnets of a different structure and composition
dependence on anisotropy field Hani
coercive field Hc   size of a nucleation domain   retention time   Samples. Thickness in nm shown in blankets       There is no clear dependence of the coercive field Hc on the anisotropy field. For FeCoB nanomagnet of the smallest Hani~3 kG and of the largest Hani~9 kG, there are nanomagnets of the smallest Hc~100 G and of the largest Hc ~600 G.   There is no clear dependence of the size of the nucleation domain on the anisotropy field Hani. The domain size varies from ~30 nm to ~80 nm independently of Hani.   There is no clear dependence of the retention time on the anisotropy field Hani. The retention time varies from ~1E7 seconds to ~1E50 seconds independently of Hani.
dependence on internal magnetic field Hint
coercive field Hc   size of a nucleation domain   retention time   Samples. Thickness in nm shown in blankets       There is a weak increase of the coercive field Hc with an increase the internal magnetic field. Due to a variation of domain size, the dependence is sparse.   There is a weak increase of the size of the nucleation domain with an increase the internal magnetic field Hint. At a smaller Hint, the domain size is ~30 nm. At a larger Hint, the domain size is ~80 nm.   Retention time increase with an increase the internal magnetic field Hint. At a smaller Hint, the retention time is ~1E7 seconds. At a smaller Hint, the retention time is ~1E50 seconds.
Dots of the same color and shape correspond to nanomagnet fabricated at different places of the same wafer. Stars show multilayer nanomagnets, which contain several ferromagnetic layers.
click on image to enlarge it

Known tricks and methods of fake and highlight research

(trick 1 ) Tracing a small change of coercivity field H_c using a coercivity loop

E.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 time

Many "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 non-switching 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 in-plane magnetic field

Many "champion" data have been reported for magnetization switching of a nanomagnet, when a bias magnetic field is applied in-plane (along the hard axis of nanomagnet)

(Where is the trick?) As it is shown here, the magnetization under bias of in-plane 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 ) Reporting

E.g.

(Where is the trick?) Measurement

Questions & Answers

How 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 square-shaped 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 specially-made domain-nucleation 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 domain-wall motion, after creation of the nucleation domain the domain wall is moving fast and generally it cannot be pinned (except if there is a really-strong domain-pinning 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.

Coercive loop vs number of defects

Fig.34 Dependence of coercive loop on number of defects and imperfections in a continuous magnetic film

(blue line) is the coercive loop for the film with a large number of defects. The coercive field Hc is large. (red line) is the coercive loop for the same film without defects. There is almost no loop and Hc is almost zero

(note) In case of a nanomagnet, the coercive loop is of a rectangular shape. Similarly, the Hc strongly depends on the number of defects and imperfections in the nanomagnet.

click on image to enlarge it

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 single-domain 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 (~R²) 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?

Experimental proof on absence of static domains in a nanomagnet

Linear dependence of in-plane magnetization vs. in-plane magnetic field. In this measurement, an external magnetic field is applied in plane (along the magnetic hard axis). The magnetization turns towards the in-plane field. The dependence of the in-plane component of the magnetization vs H is a perfect line, which is only possible when the nanomagnet is in a single- domain state. The linear dependence proves that the magnetization turns as whole (as a mono domain) towards the magnetic field instead of a movement of domain wall across nanomagnet as it would be in case of existence of a static domain.

click on image to enlarge it

(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 in-plane field. The dependence of the in-plane 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 in-plane 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 single-domain 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 single-domain 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 (~R²) 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