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Thermally-activated magnetization switching.

Measurements of coercive field, retention time, Δ

Spin and Charge Transport

Abstract:

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
The size on the nucleation domain can be measured with a high precision of smaller than 1 % without using any microscope only from Hall measurements ( See details here)
I have developed this measurement method in 2017-2018

Features of thermally activated switching of the magnetization are described. A new high-precision measurement method of the coercive field, retention time, Δ and the size of a switching nucleation domain is described below. The method is based on the classical Néel model of the thermally-activated magnetization switching. Conditions, properties and requirements for a magnetic data storage and for a recording method of a magnetic memory are studied.


paper on this topic is here: V. Zayets, "Thermally activated magnetization reversal in a FeCoB nanomagnet. High-precision measurement method of coercive field, delta, retention time and size of nucleation domain" arXiv:1908.08435 (2019)

Note: experimental method, which is described below, allows to measure the coercive field with a precision better than 1 Oe, retention time (data storage time) in rage from a few minutes to a few billions years, the size of nucleation domain with a precision of a few nanometers, delta Δ with a precision of a few percents.
Note. I have develop this experimental method in 2017-2018

Content

click on the chapter for the shortcut

(2). Measurement method

(2a) In short. Measurement of coercive field Hc
(2b) In short. Measurement of retention time τretention
(2c) In short. Measurement of parameter Δ
(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 tswitch 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 Ebarrier between two stable magnetization states
(6.2). Step 2. Arrhenius low & transition state theory (TST)

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 Meff. 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 Hanis, effective magnetization Meff

(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

(15) Questions & Answers

 

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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 method except the thermally activated switching used as a recording method of a magnetic memory?

A. Yes. When the duration of the recording pulse is about 1/fFMR , where fFMR is the frequency of the ferromagnetic resonance (FMR), the resonance switching dominates over the thermally activated switching. The the resonance switching is used for a high-speed low-power recording methods.

 

 

 

 

 

 

 



(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 dependent switching mechanisms?

A. It is because of the size dependence of the magnetostatic interaction (or magnetic-dipole interaction). Let us consider the simplest case when only two magnetic domains exist. The energy of the magnetostatic interaction is the smallest when sizes of domains of opposite magnetization are the same. Also, the larger the domain size is, the larger the absolute value of the magneto-static energy is. The balance between the magnetostatic interaction and the exchange interaction determines the size of the static domains. The energy of exchange interaction is the lowest, when the magnetization is aligned in one direction over whole sample. In case of two domains the energy of exchange interaction increases comparing to no-domain case. This increase is called the energy of domain wall.

When the area of nanomagnet is reduced, the magnetostatic interaction decreases, but the exchange interaction remains the same. In case of a small area, the magnetostatic interaction cannot compensate the the exchange interaction and the magnetization aligns in one direction without any static domains.

The single-domain state occurs when the energy to form a domain is larger the reduction of the magnetostatic energy due to formation of two domains of opposite magnetization.

What is the difference between nucleation and static domains? Can abrupt change of magnetization occur in the case of 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 static domain, why the domain wall constantly moves with an increase of magnetic field, but why the domain wall does not move abruptly, why it always is stopped?

A. It is because at any applied external magnetic field there is a balance between magnetostatic and exchanges energies.

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 · EMS), the energy of domain wall is proportional to length of the domain wall (~R· EEX) and the energy of magnetic interaction with magnetic field is is roughly proportional to the area of domain (~R2 · EH· 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 · EMS- R· EEX - R2 · EH· H =constant

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

Since the energy barrier becomes smaller for smaller-size nucleation domain, why the nucleation domain has a finite size and does not become infinitely small? Why the magnetization reversal mechanism of smallest nanomagnets is single-domain type?

A. It is because the energy of domain wall of the nucleation domain. The creation of the nucleation domains results in the decrease of the energy due magnetization of the bulk of domain becomes parallel to the applied magnetic field and the energy increase due to creation of the domain wall. There is an optimum size of the nucleation domain when the total energy for the creation of the nucleation domain is smallest.


(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 Hc (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 magnetization does 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.
Click on image to enlarge it

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 Hc, 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 Hc

 

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 Hc , retention time τreten, 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 τreten, 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 tswitch as

where the magnetization Meff 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 τretention is the magnetization switching time, when the magnetic field is not applied H=0.

Linear dependence of log(tswitch) 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 Ebarrier and the rate of interaction finter 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 Hc and retention time τretention or Meff 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 Hanis 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 Hc

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 Hc is the field, at which measured line of log (tswitch ) vs H crosses the x-axis gives the coercive field Hc.

The definition of coercive field Hc :

(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 Hc

The measured dependence of log(tswitch/1 sec) vs H is a line. The coercive field Hc corresponds to magnetic field, at which tswitch=1 sec or log(tswitch/1 sec)=0. A linear fitting experimental data gives the Hc with a very high precision.


Why a measurement of Hc from a hysteresis loop should be avoided and should be used only as a rough estimate of Hc

(reason 1) Each repeated measurement dives a slightly different value of Hc (See Fig.8)

(reason 2) A measurement at scanning rate of the magnetic field gives different value Hc(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 Hc, it is better to show the measurement time!
All my measurements of Hc (all data of Hc 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 Hc , when the measurement time is not indicated? Since coercive field Hc 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 τretention<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)
click on image to enlarge it

In short. Measurement of the retention time τretention

The precise value of τretention is measured as a point where the line of log (tswitch ) vs H crosses the y-axis.

The definition of retention time τretention:

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.

τretention is referred to the maximum data storage time of a magnetic memory.

Using the described method, the τretention 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 tswitch is measured under external magnetic field. In my experimental setup I can measure tswitch in the range from 0.3 s to a few minutes. I adjust the external magnetic field for to be in this range. Since tswitch increases exponentially when H decreases, the tswitch can easily reach a million or billion years when extrapolated to H=0. It depends of the slope of Fig.10 and Hc

High-precision measurement of τretention

The measured dependence of log(tswitch/1 sec) vs H is a line. The retention time τretention 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 τretention with a very high precision.

The Néel model calculates the magnetization switching time tswitch 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 τretention 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
click on image to enlarge it

In-short. Measurement of the size of nucleation domain for magnetization reversal

The precise value of the size of nucleation domain is measured from the slope of the line of log (tswitch ) vs H and the magnetization of the nanomagnet, which 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.

How is it possible to measure the size of the nucleation domain from a magnetization switching experiment?

According to the Néel model, the magnetization switching occurs when the energy of the nucleation domain becomes comparable with the thermal energy.

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 (tswitch ) is linearly proportional to the external magnetic field (See here):

where the slope of this dependence is calculated as

the magnetization Meff 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 Mmagnet per volume Vmagnet 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

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

Static magnetic domain vs switching magnetic domain

They are two very different objects!

(Static magnetic domains): The domains or regions of the different (often opposite) magnetization directions.

It is the feature of a nanomagnet or a magnetic film of a larger area. The static magnetic domains usually exists in the absence of the external field. However, sometimes some external magnetic field is required in order to the static magnetic domain. The static magnetic domains are formed to make a balance between the exchange force, which is trying to align all magnetization of all regions to be in one direction, and magnetitic dipole force, which is try to align the magnetization in neighbor regions to be antiparallel.

Note: The FeB and FeCoB nanomagnets of size of a few micrometers and smaller do not have any static domains. The magnetization of the whole nanomagnet is in one direction.

Note: A nanomagnet free of static magnetic domains have a rectangular- shape hysteresis loop (See Fig.1). The hysteresis loop of a nanomagnet with static domains is not rectangular shape. It is either of shape shown in Fig.2 or with some steps.

Magnetization switching mechanism of a nanomagnet with static domains: domain wall expansion of static magnetic domains.

(switching magnetic domain): The region, which the coherently rotates during of the first step of the magnetization reversal.

The switching domain exists in a 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
click on image to enlarge it

In-short. Measurement of the parameter Δ

The precise value of the Δ is measured from the slope of the line of log (tswitch ) vs H and the anisotropy field Hanis 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 = Ebarrier/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 Meff 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 Hanis is the anisotropy field. The Hanis 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 tswitch 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
Click on image to enlarge it.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Néel model of thermally activated magnetization switching

L. Néel, Adv. Phys., 1955

Just a fact: The simple Néel model describes all features of the thermally-activated switching in nearly all cases except only a few exceptions (e.g. the resonance switching, the strong domain pinning)

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 Ebarrier 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 perfect description of the thermally -activated magnetization switching and a perfect fit of the experimental data!!!


Calculations steps the Néel model

(step 1) Calculation of the energy barrier Ebarrier

(step 2) Calculation of the switching probability

(step 3) Calculation of the magnetization switching time


( Néel model. Step 1) . Calculation of energy barrier Ebarrier 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, EPMA 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 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 Ebarrier can be express as

where Ebarrier is the anisotropy field (See here). Substituting Eq. (1.7) into Eq.(1.6) gives

Realistic case Hc << Hanisotropy

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 Hc is about 2-4 % of the anisotropy field Hanisotrp. Therefore, Hc / Hanisotrp ~ 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 Hc << Hanisotropy 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 Ebarrier 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 Hc << Hanisotropy ?

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 ~ Hanisotropy , 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 << Hanisotropy. The Hc 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 tswitch 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; Ebarrier 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 Hc << Hanisotrp , Eq(1.12) is simplified to

the retention time τretention is the average time of magnetization reversal without any external magnetic field. From Eq.(1.13), the τretention 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( tswitch ) 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( 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)


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 Hc 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 Hc and Δ :

(method 1): graded magnetic field

(method 2): graded pulsed magnetic field


Common errors of the alternative measurements methods of Hc 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 Ebarrier 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 τretention 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 Hanis (See Eq.(a7)) and the volume of the nucleation domain is proportional to the magnetization M of the nanomagnet (See Eq.(a3)). Both Hanis 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. Hanis 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 tswitch 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 Ebarrier 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 finter 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 Pnot that the magnetization is not reversed, than for the probability Prever that the magnetization is reversed. Of course, Prever+Pnot=1

The probability Prever(t,t+dt) of the magnetization reversal in a small time interval between t and t+dt is calculated as

where

The probability Pnot(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 Pnot(t0 ,t+dt) is calculated as

Eq.(5.6) can be simplified as

The function Pnot(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 Pnot(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 Ebarrier 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 Pnot(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 tswitch

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 dpswitch 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 tswitch 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 tswitch

This measurement should be used in case when a time of measurement of magnetization direction is comparable with tswitch (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 tswitch shorter than 5 second using the Hall setup, the pulsed magnetic field should be used.

In order to measure tswitch 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<< Hanisotrp, substituting Eq(1.15) into Eqns.(2.1-2.2) gives

 

To see how to obtain Eqs, 2.1,2.2, 2.7, 2.8 click here to expand

It is easier and simpler to calculate the probability of not switching Pnot by time t rather than the probability of switching Pswitch by time t. The relation between Pnot and Pswitch is straightforward Pnot + Pswitch=1

The non- switching of the magnetization in time interval interval [0,t+dt] means that the magnetization is not switched in both time intervals [0,t+dt] and [t,t+dt]. The probability of non-switching in time interval [0,t+dt] equals to the product of probabilities of non-switching in time intervals [0,t+dt] and [t,t+dt]:

Probabilities of switching in a small time interval [t,t+dt] is linearly proportional to dt

where 1/τ is the coefficient of the proportionality.

From Eq.(2.41) we have

Substituting Eq.(2.42) into Eq.(2.40) gives

From time t to t+dt the value of the non-switching probability change on

Substituting Eq.(2.40a) into Eq.(2.43) gives

Case 1. External magnetic field is constant

in this τ is constant. The integration of Eq.(2.44) gives

 

Case 2. External applied magnetic field changes in time

Substituting Eq.(1.8) into Eq.(2.47) gives

where

 

Case 3. External magnetic field linearly ramped in time

For example, if time dependence of magnetic field is described as

Using integral

and substituting Eq.(2.6) into Eq.(2.48) and integrating gives

 

Case 4. External applied magnetic field linearly ramped in time and it is H(t)<<Hanisotropy (realistic case)

In the case of small field H(t)<<Hanisotropy, the following approximation can be used

substituting Eq(2.49a) into (2.48) gives

In the case of linear ramping

integration of (2.49c) gives

where

Meff is the effective magnetization or the magnetization of a nucleation domain

 

 

 

 

Case when External magnetic field linearly ramped

In this case time dependence of magnetic field is described as

See X. Feng and P. B. Visscher, Journal of Applied Physics 95, 7043 (2004)

 

Alternative measurement method of Hc. 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 Hc. 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<< Hanisotrp 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 Pnon-switch([0,H]), that the magnetization is not switched, when magnetic field increases from o to H, and the probability Pswitch([0,H)), that the magnetization is switched, as

The probability Pswitch([H,H+dH)) that the magnetization is switched at the magnetic field between H and H+dH is proportional to

where A is the proportionality constant, which can from normalization condition

and it gives the switching probabilities as

The average switching magnetic field is calculated as

The mean deviation is calculated as

to see how to obtain Eqs. (4.8) and (4.9) , click to expand

 

In normalization

the following integral was used

to obtain the distribution

average switching magnetic field

The average field is defined as

Substituting Eq.(4.7) gives

We used the value of the integral

Eq(4.8c) gives

mean deviation

The mean deviation is defines as

Substituting Eqs. (4.7) and (4.8) gives

or

 

 

 


 

 



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 Ebarrier between two stable magnetization states to the thermal energy. It characterized how much bigger Ebarrier should be than the thermal energy to avoid an undesirable magnetization reversal due to a thermal fluctuation.

According to Neel model, the energy barrier Ebarrier between two stable magnetization states in absence of magnetic field equals to EPMA. Therefore

The retention time can be calculated as

or

There are three possible techniques to measure Δ. Each technique requires additional measurement of Hanisotropy (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 Meff

On log scale, τ is linearly proportional to the applied magnetic field(Fig.11). The slope of the fitting lines is proportional to Meff and the horizontal offset is proportional to τretention

step 2 : Measuring the anisotropy field Hanisotropy

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 EPMA is calculated from Hanisotropy as (See here)

where Meff is the total magnetization in the case of a single-domain magnetization reversal or the effective magnetization Meff 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 Hanisotropy and Δ only by fitting the distribution of magnetization switching probability with two independent parameters Hanisotropy 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 Hanisotropy or Meff. It is important that Hanisotropy should be measured independently. Otherwise, the fitting gives incorrect result. The measurement of Hanisotropy 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 Ebarrier 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 Meff. 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 Meff 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 Ebarrier 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)

Meff equals to the saturation magnetization Msat of material multiplied by the sized of the nanomagnet

 

Meff 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, Meff 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 Meff is equal to the saturation magnetization M multiply to volume of the nanomagnet

multi-domain switching

It is the case when the effective magnetization Meff 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

 

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 Meff divided per the saturation magnetization M.

The saturation magnetization M is measured by SQUID magnetometer before nano fabrication and The the effective magnetization Meff is measured by this method after micro fabrication.

Note: M is the magnetization per unit of volume; Meff 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 Meff 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

 

 

part where anisotropy field, kG retention time 10^ s Meff E24 T m3 coercive field, Oe Δ nucleation domain size, nm Hall angle, mdeg
MgO is not etched 8.932 32.82 4.3 394.6 371.5 55.4 284.8
MgO is etched 7.25 21.66 3.88 288.8 271.8 52.6 33.8

 

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 Hanis, effective magnetization Meff

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= EPMA /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 EPMA 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 τup-to-down and spin-down to spin-up states τdown-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 τup-to-down and from spin-down τdown-to-up to up state should be the same

Effects, which make τup-to-down and τdown-to-up different:

-applying an in-plane magnetic field

-effect of spin-orbit torque

Effects, which keep τup-to-down and τdown-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 Ebarrier, 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 Ebarrier.

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 Ebarrier.

 

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 Ebarrier between two stable magnetization states and (2) the energy of a thermal fluctuation should be larger than Ebarrier.

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 Ha, and the FORC is defined as the magnetization curve that results when the applied field is increased from Ha back to saturation. This measurement procedure is repeated for different values of Ha to obtain a suite of FORCs (Figure l b). The magnetization at the applied field Hb on the FORC with reversal point Ha is denoted by M(Ha, Hi), where Hi, > Ha (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)

 


 

 

 

 

 

 


Known tricks and methods of fake and highlight research

 

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

E.g. tracing the change Hc as function of a gate voltage or bias current or some another parameter of a weak influence on Hc

(Where is the trick?) Measurement precision of Hc from coercive loop is very poor. Also, the width of the coercive is slightly different for each measurement. the distribution o

(trick 2 ) Measuring properties of magnetization switching without a measurement of retention time

E.g.

(Where is the trick?) Measurement

 

(trick 3 ) Reporting an "efficient" magnetization switching under bias in-plane magnetic field

E.g. tracing the change Hc as function of a gate voltage or bias current or some another parameter of a weak influence on Hc

(Where is the trick?) Measurement

(trick 4 ) Reporting an "efficient" magnetization switching under bias in-plane magnetic field

E.g. tracing the change Hc as function of a gate voltage or bias current or some another parameter of a weak influence on Hc

(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?

(Hc of film vs nanomagnet) The coercive loops are very different. The coercive field Hc of continuous film is substantially smaller than Hc 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 Hc. 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 Hc is larger for a nanomagnet.

Since the coercive loop is very different of continuous film and a nanomagnet, should I measure of Hc for a continuous film before nanofabrication or it is useless?

(film quality vs Hc) A measurement of coercive field of continuous film is very useful to estimate the quality of the grown film. The smaller Hc is, the better quality is. The smaller Hc means that there are 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


I am strongly against a fake and "highlight" research

 

 

 

I truly appreciate your comments, feedbacks and questions

I will try to answer your questions as soon as possible

 

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