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

Measurements of coercive field, retention time, Δ Spin and Charge TransportAbstract:
A new measurement method of a highprecision 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 20172018Method of a high precision measurements of the coercive fieldIt 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 wellfitted within this precisionThis 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 magnetizationreversal time τ
What is measured? The relaxation or magnetizationreversal time τ, the retention time τ_{reten}, the effective magnetization M_{eff} details are herestep 2: Rough measure of switching fieldWhat is measured? A rough value of the coercive field H_{c}
step 3: Simultaneous fittingWhat is measured? A precise value of the coercive field H_{c}
Coercive field
The magnetic field, at which magnetization is switched between its two stable magnetization directions.
Thermo activated nature of the magnetization switching
fact 1: Since the magnetization switching is a thermally activated process, the distribution of switching fields is described by the thermal statistics Average of the switching field is defined as the coercive field. fact 2: The longer the scan of magnetic field is, the shorter the coercive field is. Figure 3 explains this fact. When a magnetic field is applied opposite to the magnetization direction, for a while the magnetization is wiggling around its direction. next, it turns along magnetic field. The average magnetizationreversal time is fixed and it is proportional to the applied magnetic field.
Relaxation or magnetizationreversal time_{} τ
From Néel model, the magnetizationreversal time τ is given as:
where τ_{reten} is the retention time, H is the external magnetic field and M_{eff} is the effective magnetization
Néel model
L. Néel, Adv. Phys., 1955In 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 E_{barrier } between states. The probability of a thermalfluctuation is described by the Boltzmann distribution 1.energy barrier E_{barrier }The angledependent 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. E_{PMA} (or K_{eff}) is the the energy of the perpendicular magnetic anisotropy. M is the effective magnetization, H is the magnetic field. Maximum of energy is at θ _{max} with corresponded energy of Maximum of energy is at θ _{min}=0 and 180 degrees
Therefore, the barrier height is The PMA energy E_{barrier }can be express as where E_{barrier }is the anisotropy field (See here). Substituting Eq. (1.7) into Eq.(1.6) gives 2. Realistic case H_{c }<< H_{anisotrp }
As July 2018, in all my measurements of magnetic nano wires it always the coercive field H_{c} is about 24 % of the anisotropy field H_{anisotrp}. Therefore, H_{c }/ H_{anisotrp }~ 0.020.04 at measurement time of 1 s. 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 f_{0} is is the socalled attempt frequency associated with the frequency of the gyromagnetic precession; E_{barrier} is the energy barrier between two states when the magnetization is along and opposite to the external magnetic field. The Arrhenius low has its origins in the 1880s when Arrhenius proposed, from an analysis of experimental data, that the rate coefficient in a chemical reaction should obey the law where ΔV denotes the threshold energy for activation of the chemical reaction, f_{0} is the attempting frequency. Substituting Eq. (1.8) into Eq.(1.11) gives In the case when H_{c }<< H_{anisotrp }, Eq(1.12) is simplified to the retention time τ_{reten} is the average time of magnetization reversal without any external magnetic field. From Eq.(1.13), the t_{reten} can be calculated as From Eqs.(1.13 and 1.14) , the magnetization reversal time τ is given as:
4. Néel  Brown modelW.F.Brown (1963), W. T. Coffey, Y. P. Kalmykov (2012)In Néel  Brown model, it is assumed that the magnetization switching occurs not because an energy of a thermal fluctuation exceed E_{barrier}, but because a more complex magnetization dynamic described by the LandauLifshitz equations.
resonance magnetization switching It is the case when the Néel  Brown model should be used instead of the Néel model!When frequency of magnetic field, or electrical current or electrical field is close to the frequency of the ferromagnetic resonance (FMR), the magnetization switching may occur at the energy much smaller than the barrier energy E_{barrier}. Example 1: Microwave assistant magnetic recording (MAMR) to hard disk. When data density in hard disk becomes very high, the required magnetic field to record one data because unacceptably large. In order to solve this problem the MAMR is used. In the case of MAMR, a weak microwave radiation, which frequency is close to the FMR of harddisk media, excites magnetization precession. After that, the magnetic field of recording head reverses the magnetization and records a data bit. The required recording magnetic field is substantially small than in the case without the microwave radiation. Example 2: Data recording of magnetic random access memory (MRAM) using the VCMA effect The VCMA effect is weak effect. At present, it is hard to hard to reverse magnetization by the VCMA effect using DC gate voltage. However, when a pulse of interval close to the reverse of the FMR frequency, the magnetization may be reversed even by small amplitude pulse (Shiota 2012). In this case, the pulse energy is substantially smaller than E_{barrier}.
Details of the Néel  Brown modelIn the Néel  Brown model, the random magnetic field is assumed to act on the magnetization. The magnetization switching conditions are derived from a solution LandauLifshitz equations for the magnetization affected by the random magnetic field. What is the physical meaning of the random magnetic field of the Néel  Brown model?In the model the random magnetic field is a pure mathematical tool. However, the physical meaning of this fields is associated with the interaction of the magnetization with magnons and the electron scattering (spd interaction) between localized d states and states of spinunpolarized conduction electrons.
Which the Néel model or Néel  Brown model is correct?As July 2018, all my experimental measurements fit to the Néel model extremely well (See for example here). Even though I have used the magnetic and electrical pulses at a frequency much smaller than the FMR frequency of my studied samples. The the Néel model is simple and intuitive. The mathematical description of this model is relatively simple. It is based on two simple facts: (1) there is the energy barrier E_{barrier} between two stable magnetization states and (2) the energy of a thermal fluctuation should be larger than E_{barrier}. In contrast, the Néel  Brown model is more complex and less intuitive. The Néel  Brown model should be used only in the case when the Néel model clearly fails to describe the experiment.
Switching probabilityThe probability that the magnetization is not switched by the time t is described as where τ is the relaxation time Correspondingly, the probability that the magnetization is not switched by the time t is described as In the case when H<< H_{anisotrp}, substituting Eq(1.15) into Eqns.(2.12.2) gives
To see how to obtain Eqs, 2.1,2.2, 2.7, 2.8 click here to expand
It is easier and simpler to calculate the probability of not switching P_{not} by time t rather than the probability of switching P_{switch} by time t. The relation between P_{not} and P_{switch} is straightforward P_{not} + P_{switch}=1 The non switching of the magnetization in time interval interval [0,t+dt] means that the magnetization is not switched in both time intervals [0,t+dt] and [t,t+dt]. The probability of nonswitching in time interval [0,t+dt] equals to the product of probabilities of nonswitching in time intervals [0,t+dt] and [t,t+dt]: Probabilities of switching in a small time interval [t,t+dt] is linearly proportional to dt where 1/τ is the coefficient of the proportionality. From Eq.(2.41) we have Substituting Eq.(2.42) into Eq.(2.40) gives From time t to t+dt the value of the nonswitching probability change on Substituting Eq.(2.40a) into Eq.(2.43) gives Case 1. External magnetic field is constant in this τ is constant. The integration of Eq.(2.44) gives
Case 2. External applied magnetic field changes in time Substituting Eq.(1.8) into Eq.(2.47) gives where
Case 3. External magnetic field linearly ramped in time For example, if time dependence of magnetic field is described as Using integral and substituting Eq.(2.6) into Eq.(2.48) and integrating gives
Case 4. External applied magnetic field linearly ramped in time and it is H(t)<<H_{anisotropy} (realistic case) In the case of small field H(t)<<H_{anisotropy}, the following approximation can be used substituting Eq(2.49a) into (2.48) gives In the case of linear ramping integration of (2.49c) gives where M_{eff} is the effective magnetization or the magnetization of a nucleation domain
Case when External magnetic field linearly ramped In this case time dependence of magnetic field is described as in the case of small magnetic field H<< H_{anisotrp}, notswitch probability is in the case of larger magnetic field H~ H_{anisotrp} Similar equation is obtained here X. Feng and P. B. Visscher, Journal of Applied Physics 95, 7043 (2004)The probability of magnetization switching P_{switch} is related to probability of nonswitching P_{non} as P_{switch}=1P_{non}Note: Equation (2.49d) or (2.4) is used to evaluate the effective magnetization M_{eff}. The measured distribution of switching probabilities are fitted by Eq. (2.49d) or (2.4) with M_{eff} as a free parameter. Combining with additional measurements of the anisotropy field H_{anisotrp} (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<< H_{anisotrp} 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
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 magnetizationreversal time is calculated as Distribution as function of magnetic field
Below I calculate the dependence of the switching probability on magnetic field. Below only the realistic case of H<< H_{anisotrp} is calculated. The switching probability in this case is described by Eqs. (2.3),(2.4) As was show above, the coercive field (switching field) depends on the measurement time τ (see Fig.11). Let us refer to the coercive field as the switching field at measurement time of 1 second τ=1. Than, from Eq.(1.15) Using Eq.(4.1), Eq.(1.15) is simplified to Substituting Eq.(4.3) into Eqs.(2.1) and (2.2) gives the probability P_{nonswitch}([0,H]), that the magnetization is not switched, when magnetic field increases from o to H, and the probability P_{switch}([0,H)), that the magnetization is switched, as The probability P_{switch}([H,H+dH)) that the magnetization is switched at the magnetic field between H and H+dH is proportional to where A is the proportionality constant, which can from normalization condition and it gives the switching probabilities as The average switching magnetic field is calculated as The mean deviation is calculated as to see how to obtain Eqs. (4.8) and (4.9) , click to expand
In normalization the following integral was used to obtain the distribution average switching magnetic fieldThe average field is defined as Substituting Eq.(4.7) gives We used the value of the integral Eq(4.8c) gives mean deviationThe mean deviation is defines as Substituting Eqs. (4.7) and (4.8) gives or
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 H_{c} with a very high precision?Two independent experiments of Fig.11 and Fig.12 give H_{c}. Fitting both measurements simultaneously give H_{c} with a high precision. The measurement precision of 1 Oe is achievable with a moderate number of measurements (~6080) and a short time (24 hours). A repeated measurement after a 1month interval is well fit within the precision.
Measuring delta Δ
The Δ is parameter, which characterized the magnetic stability of nanomagnet. It is defined as a ratio of the energy barrier E_{barrier} between two stable magnetization states to the thermal energy. It characterized how much bigger E_{barrier} should be than the thermal energy to avoid an undesirable magnetization reversal due to a thermos fluctuation.According to Neel model, the energy barrier E_{barrier} between two stable magnetization states in absence of magnetic field equals to E_{PMA}. Therefore
The retention time can be calculated as or There are three possible techniques to measure Δ. Each technique requires additional measurement of H_{anisotropy} (see here the measurement details). Technique 1. Using linearlyramped magnetic fieldlow precision & moderate measurement timeThe Δ is evaluated by fitting the distribution of magnetization switching probability Eq.(2. 49b) or (2.8). Technique 2. Using magnetic pulses of gradually increases amplitude (Fig.12)low precision & long measurement timeThe Δ is evaluated from the width of the distribution of magnetization switching probabilities Eq.(4.2) and Eq.(4.7).(See Fig.12) Technique 3. From magnetization switching time τ (Fig.11)high precision & long measurement timeThe Δ is evaluated by from measurements of the dependence magnetizationreversal time vs magnetic field(See here ) This highprecision measurement of Δ requires 3 steps step 1 : Measuring the effective magnetization M_{eff}_{} On log scale, τ is linearly proportional to the applied magnetic field(Fig.11). The slope of the fitting lines is proportional to M_{eff} and the horizontal offset is proportional to τ_{retention }
step 2 : Measuring the anisotropy field H_{anisotropy}_{} Method to measure anisotropy field is described here It is a relatively easy to measure the anisotropy field (See here) . Even though it often requires a relatively large inplane magnetic field. Without magnetic field the magnetization is perpendicularlytoplane. Under magnetic field, the magnetization turns toward magnetic field. The field, at which the magnetization turns completely inplane, is called the anisotropy field. The E_{PMA} is calculated from H_{anisotropy} as (See here) where M_{eff} is the total magnetization in the case of a singledomain magnetization reversal or the effective magnetization M_{eff} in the case of multidomain magnetization reversal step 3 : Calculating Δ_{} The Δ can be simply calculated as: Important: Some researchers are trying to find both H_{anisotropy} and Δ only by fitting the distribution of magnetization switching probability with two independent parameters H_{anisotropy}_{} and Δ. It is incorrect and leads to incorrect result. From the Neel model, the distribution has only one independent parameter, which is the ratio of Δ and H_{anisotropy}_{} or M_{eff}. It is important that H_{anisotropy}_{} should be measured independently. Otherwise, the fitting gives incorrect result. The measurement of H_{anisotropy} is relatively simple (See here)Additional method to measure ΔFrom nonlinear dependence of switching time on magnetic fieldit requires a high magnetic field and measurement of a shorter switching time!Any method related to Neel model is based on only one important parameter, which is the barrier height E_{barrier} between two stable magnetic states: There are three component, which are proportional to magnetic field H in power 0, 1,2. The Δ can evaluated from independent measurements of the 2d and 3d components.
What is effective magnetization M_{eff}
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 M_{eff} is magnetization of first magnetic domain, which triggers the magnetization reversal. The reason, why magnetization switching occurs by this mechanism: A thermal activation energy to reverse magnetization of a small domain is much smaller, than the energy to reverse magnetization of the whole film. How small is the size of the nucleation domain The size of the nucleation domain is determined by trade of between the exchange energy and the barrier energy E_{barrier}_{ }for the magnetization switching. The stronger exchange interaction is, the larger size of the nucleation domain become. The size of the nucleation domain is evaluated from measurements of Fig.11
Singledomain magnetization switching In this case the magnetization of whole nanomagnet is reversed coherently. This switching occurs only in a nanomagnet of very small size (diameter ~1040 nm) M_{eff} equals to the saturation magnetization M_{sat} of material multiplied by the sized of the nanomagnet
M_{eff} gives the magnetization of the initial domain, which is first switching during magnetization (case of multidomain switching). In the case of singledomain reversal, M_{eff} equals to product of the saturation magnetization and the volume of nanomagnet.
Singledomain switching and multidomain switchingThis method can unambiguously measure for a tested device whether magnetization switching is singledomain or multidomain. singledomain switching It is the case when the effective magnetization M_{eff}_{} is equal to the saturation magnetization M multiply to volume of the nanomagnet multidomain switching It is the case when the effective magnetization M_{eff}_{} is smaller than the saturation magnetization M multiply to volume of the nanomagnet Measuring the size of the nucleation domain
Estimated measurement precision of the nucleation domain size is better than 1 %.The size on the nucleation domain is measured without using any microscope. Only data of Hall measurements are used!!!How to measure?The size of nucleation domain equals the effective magnetization M_{eff} divided per the saturation magnetization M. The saturation magnetization M is measured by SQUID magnetometer before nano fabrication and The the effective magnetization M_{eff} is measured by this method after micro fabrication. Note: M is the magnetization per unit of volume; M_{eff} is the total magnetization of the nucleation domain.Note: The area of the domain measured. The size of domain is estimated as a square room from domain volume assuming a domain of a square shape.Note:I have developed this method in 20172018.
Size of the nucleation domain in different materialsFeB and FeCoB amorphous nanomagnet + anneal and partial recrystallization As can be seen from the right picture, the size of the nucleation domain varies from 30 nm to 60 nm. However, there are nanomagnets with a longer domain size. As Nov. 2018, I have studies more than 100 nanomagnets from 25 FeB, FeBCo, (FeB/W)n samplesCo singlecrystal nanomagnetThe variation is narrow: from 40 nm to 50 nm. H As Nov. 2018, I have studies only 6 nanomagnets from 2 samples
Influence of MgO/FeCoB on magnetic properties of FeCoB nanowire
Experiment to test the influence of the interface on magnetic properties of nanowire. The key feature of this experiment: MgO is removed from a half of nanowire and the magnetic properties of both parts are measured. Since all magnetic properties of both are identical except Similar slope of lines of right figure indicate that the effective magnetization M_{eff} is nearly the same for both parts. It means that the etching was stopped just after MgO and FeCoB was not etched Effect of Removal of MgO: 1) Anisotropic field: decreases
Note: The Hall angle and Hall resistance in FeCoB nanowire greatly depend on the proximity of MgO gate
Relation between delta and retention time
Both the delta and retention time characterize the probability of the magnetization switching in absence of 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 fAll my experimental data (by Nov. 2018) show that attempt frequency f is nearly the same for all samples made of the same ferromagnetic metal. However, it is different for different metals. For example, there is a two order difference between f in samples made of an amorphous FeB and made of a single crystal Co.
Temperature dependence of delta Δ, anisotropy field H_{anis}, effective magnetization M_{eff}
When temperature rises, all the Hall angle , coercive field, anisotropy field, effective magnetization, saturation magnetization, retention time and nucleation domain size decrease Q. By definition the Δ= E_{PMA} /kT. Does it mean that the temperature dependence of Δ following the law 1/T?A. No. The decrease of the Δ with a temperature rise is more sharp. It is because E_{PMA} substantially decreases with a rising of temperature. Additionally, the size of a nucleation domain for magnetization reversal may change with temperature. That also affects the temperature dependance of Δ.
Transformation of Hysteresis loop under different effects
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 spinup to spindown state and from spindown to spinup state
Switching time influenced by different effects
Difference of switching fields between spinup to down τ_{uptodown} and spindown _{ }to spinup states τ_{downtoup}
In principle, the magnetization switching times for switching from spinup to down state τ_{uptodown} and from spindown τ_{downtoup} to up state should be the same Effects, which make τ_{uptodown} and τ_{downtoup} different:applying an inplane magnetic field Effects, which keep τ_{uptodown} and τ_{downtoup} equal: change of temperature
Note: Any possible systematical error in measurements of relative position of switching times was eliminated by a precise calibration by a Hall measurement for a nonmagnetic metal.A problem of magnetic random access memory (MRAM). Measurements and solutions
A serious problem of the MRAM is a wide variation of magnetic properties from a cell to cell The major variation is due to the variation of the domain nucleation size, which may be influence by technologydependent factors.

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