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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

A new measurement method of a high-precision measurement coercive field, retention time, Δ is described below. The method is based on the Néel model of the magnetization switching.


Note: experimental method, which is described below, allows to measure the coercive field with a precision better than 1 Oe.
Note. I have develop this experimental method in 2017-2018

Method of a high precision measurements of the coercive field

It gives a measurement precision of the coercive field at least 1 Oe (often 0.1 Oe). I have verified that a repeated measurements within a month or longer are well-fitted within this precision

This method also measures effective magnetization and retention time. Combing this measurements with additional measurement of anisotropy field gives delta Δ.

How to measure coercive field with a very high precision?

1) A pulsed magnetic field should be used

2) A substantial number of measurement should be used in order to accumulate a sufficient statistics.

 

step 1: Measure of magnetization-reversal time τ

High precision measurements of the coercive field

step 1: Measure of magnetization-reversal time step 2: Rough measure of switching field
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. 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.
click on image to enlarge it

What is measured?

The relaxation or magnetization-reversal time τ, the retention time τreten, the effective magnetization Meff

details are here

step 2: Rough measure of switching field

What is measured?

A rough value of the coercive field Hc

 

step 3: Simultaneous fitting

What is measured?

A precise value of the coercive field Hc


 

Coercive field

Hysteresis loop. Coercive field 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

 

The magnetic field, at which magnetization is switched between its two stable magnetization directions.

 

 

 

 

 

 

 

 

Thermo- activated nature of the magnetization switching

Magnetization switching -> thermally- activated process

Coercive field -> dependance on measurement time

Each time the switching of magnetization occurs at slightly different magnetic field. Click on image to enlarge it 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.

 

 

Relaxation or magnetization-reversal time τ

 

 

From Néel model, the magnetization-reversal time τ is given as:

magnetization-reversal time τ

magnetization-reversal time τ vs magnetic field
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 The slope of the lines is proportional to Meff and the horizontal offset is proportional to τretention and consequently to EPMA

where τreten is the retention time, H is the external magnetic field and Meff is the effective magnetization

 

 

 

 

 

 

 

 

 


Néel model

Results of Néel model

Ebarrier is barrier height between states of magnetization along and opposite to the magnetic field

tswitching is magnetization magnetization-reversal time

L. Néel, Adv. Phys., 1955

In the Néel model it is assumed 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

1.energy barrier Ebarrier

The angle-dependent part of the energy E of the uniaxial magnetic anisotropy can be written as

θ and φ denote the angles subtended by the magnetization and field, respectively, and the film normal. EPMA (or Keff) is the the energy of the perpendicular magnetic anisotropy. M is the effective magnetization, H is the magnetic field.
In case when φ=0, the maximums of energies can be found from

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

2. Realistic case Hc << Hanisotrp

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)

From fitting these experimental data, retention time τretention , effective magnetization Meff and coercive field Hc can be obtained with a high precision

The slope of the lines is proportional to Meff and the horizontal offset is proportional to τretention and consequently to EPMA

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

As July 2018, in all my measurements of magnetic nano wires 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. See for example, the VCMA measurements or Spin hall effect measurements.

Using this assumption, the Eq.(1.8) is simplified as

Note: Magnetization switching occurs when the barrier height is reduced only by about 6 %at measurement time of 1 second

3. Arrhenius low & transition state theory (TST)

Néel suggested that the average magnetization reversal time 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.

Substituting Eq. (1.8) into Eq.(1.11) gives

In the case when Hc << Hanisotrp , Eq(1.12) is simplified to

the retention time τreten is the average time of magnetization reversal without any external magnetic field. From Eq.(1.13), the treten can be calculated as

From Eqs.(1.13 and 1.14) , the magnetization reversal time τ is given as:

 

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.


 

Switching probability

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

in the case of small magnetic field H<< Hanisotrp, not-switch probability is

in the case of larger magnetic field H~ Hanisotrp

Similar equation is obtained here X. Feng and P. B. Visscher, Journal of Applied Physics 95, 7043 (2004)
The probability of magnetization switching Pswitch is related to probability of non-switching Pnon as Pswitch=1-Pnon

Note: Equation (2.49d) or (2.4) is used to evaluate the effective magnetization Meff. The measured distribution of switching probabilities are fitted by Eq. (2.49d) or (2.4) with Meff as a free parameter. Combining with additional measurements of the anisotropy field Hanisotrp (see here), the delta Δ can be evaluated.

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

Often Eq.(2.8) instead of Eq. (2.4) is fitted in order to evaluate the delta Δ. Since the condition H<< Hanisotrp is near always satisfied, it is unnecessary complication. The fitting of Eq.(2.4) is much easier !!!.

 


 

Distribution of switching probabilities. Theory and measurements

 

Distribution as function of time

Fig.11. step 1. Measurement of magnetization-reversal time (relaxation time) τ

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 gives the relaxation time τ, retention time, τreten ,effective magnetization Meff and coercive field Hc

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

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

Distribution as function of magnetic field

Fig.12. step 2. Rough measure of 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. 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

 

 

For fitting data of Fig.12, which fitting is better the direct fit or from average?

When the number of measurement is small (<50), the fitting from average gives better precision. Otherwise, the direct fitting is better.

Note: As seen from Fig.12, the fitting from average and the direct fitting and experimental data are very close. It proves that Eqs.(4.7) and (4.8) describes the experimental data very well


 

How to obtain Hc with a very high precision?

Two independent experiments of Fig.11 and Fig.12 give Hc. Fitting both measurements simultaneously give Hc with a high precision. The measurement precision of 1 Oe is achievable with a moderate number of measurements (~60-80) and a short time (2-4 hours).

A repeated measurement after a 1-month interval is well fit within the precision.

 



Measuring delta Δ

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 thermos 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).

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).

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)

Technique 3. From magnetization switching time τ (Fig.11)

high precision & long 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

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

In this case, the magnetization reversal is not coherent. At first, the magnetization is reversed in a small domain. Next, next the domain wall moves and expands. As a result, the magnetization of whole film 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


Measuring the size of the nucleation domain

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 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 were independently at different temperatures. Both data measured for switching spin- up to -down and for switching spin -down to -up are shown.
Click on image to enlarge it
 

 

 

Both the delta and retention time characterize the probability of the magnetization switching in absence of magnetic field.

The relation between them is (See here)

or

Experimentally the retention time and the delta 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.

 

 

 

 

 

 


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.

 

Parameters of hysteresis loop, which may be changed:

1. Hall angle (height of the loop)

2. Coercive field (half of width of the loop).

3. Difference between switching fields from spin-up to spin-down state and from spin-down to spin-up state

 

 

 

 

 

 


Switching time 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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