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Measurement of spin polarization

Spin and Charge Transport

Abstract:

An experimental method to measure the spin polarization of a ferromagnetic nanomagnet using Anomalous Hall effect (AHE) is described. A high precision, repeatability and simplicity are the features of this method.

Paper on this topic is here and more details about spin polarization of the electron gas is here
The method is based on calculations of the spin pumping in a magnetic field described here

Measurement of spin polarization from AHE

Spin polarization increases in a magnetic field due to precession damping Increase of spin polarization vs H increase makes the loop non-linear
Fig 1(a) Schematic diagram of measured hysteresis loop of Hall angle αHall in a ferromagnetic metal (solid blue line). The green dash line shows the imaginary case if spin polarization would not depend on magnetic field. The red dot line shows the case when spin polarization of metal is close to 100 %.Click on image to enlarge it. Fig. 1(b)Spin precession and precession damping in a magnetic field Hext . During the precession the spin aligns itself along the direction of the magnetic field. Click on the image to enlarge it. The spin polarization is evaluated from non-linear dependence of Hall angle αHall on H. The green dash line shows the imaginary case if spin polarization would not depend on magnetic field. The red dot line shows the case when spin polarization of metal is close to 100 %.Click on image to enlarge it.

 

Merits of this measurement method:

Merit 1: Simplicity of measurements

Merit 2: ability for measurement of spin-polarization even in a nano- sized object;

 

Main idea:

The spin polarization is evaluated from the measured dependency of the Hall angle on an applied perpendicular magnetic field

 

Experimental Fact 1: Hall rotation angle of Anomalous Hall effect increases under applied external magnetic field (See Fig.3 below)

Formulas:

Measurement of spin polarization

Measured spin polarization as a function of external magnetic field

Volt 54B ud40 Ta(2.5)/FeB(1.1)/ MgO(6)/ nanowire width 1000 nm nanomagnet length 500 nm. Fit by Eq.(1.8) gives (region of nanomagnet):sp0=81.2% ; Hpump=0.425 kG ;

Click on image to enlarge it.

Spin polarization sp in a magnetic field is calculated as

where sp0 is the spin polarization in absence of an external magnetic field, Hpump is the pumping magnetic field (a material parameter)

The measured Hall angle αHall is the sum of the Hall angle αOHE of the ordinary Hall effect and the Hall angle αAHE of the anomalous Hall effect. It can be calculated as

where αAHE is the Hall angle in the absence of an external magnetic field, βOHE is the Hall coefficient; H is the magnetic field applied perpendicularly to the film.

 

 

Measurement of spin polarization

FeB
Hall angle αAHE 1st derivative 2nd derivative
sp0=81.7% ; Hpump=0.834 kG ; Ta (2.5 nm):FeB(1 nm):MgO; sample:
Hall rotation angle αHall as a function of applied magnetic field H ; 1st and 2nd derivatives of αHall normalized to its value at H=0. Black triangles show the measured data. The red line shows the fitting by Eq.(1.12)
Click on image to enlarge it.
Can experimentally observed increases of the Hall angle be due to the ordinary Hall effect?

A. The contribution of the ordinary Hall effect (OHE) depends linearly on the magnetic. The experimentally measured dependence has two contributions the linear contribution due to OHE and non- linear contribution due to AHE

Which parameters influence the Hall angle of Anomalous Hall effect (AHE) αAHE ?

A. The is linearly proportional to spin polarization (sp) of electron gas, the magnetization and the strength of the spin-orbit (SO) interaction (See details here). The strength of the SO interaction mainly depends on the ratio of the holes and electrons in a metal. Therefore, it can be assumed that it does not change in an external magnetic field. In this case the Hall angle αAHE can be calculated as

where σxx, σxy are diagonal and off-diagonal components of the conductivity tensor; a is the proportionality constant;M is out-of-plane component of magnetization

 

Parameters, which makes Hall angle αAHE to be dependent on the magnetic field:

From Eq.(1.11), three contributions can be identified.

(1) major contribution: an increase of spin polarization due to increase of spin pumping

The spin polarization increases in a magnetic field due alignment of spins of conduction electrons along the magnetic field (See Fig.1b).

(2) minor contribution: an increase of spin polarization due to decrease of spin relaxation

An external magnetic field may suppress some some spin relaxation mechanisms (See below for details). The reduction of the spin relaxation enlarges the spin polarization (See here for details)

(2) minor & major contribution: a change of magnetization M

An external magnetic field may suppress some some spin relaxation mechanisms (See below for details).

Experiment

Measurement of spin polarization

Fig.2. A Fe nanomagnet connected to the Hall probe

The Hall voltage increases under external magnetic field!! From fitting of the dependence of Hall voltage on magnetic field the spin polarization is evaluated

Click on image to enlarge it.

The spin polarization of a nanomagnet was evaluated using the Anomalous Hall Effect (AHE).

The FeB, FeCoB and FeTbB films were grown on a Si/SiO2 substrate by sputtering. A Ta layer was used as non-magnetic adhesion layer. A nanowire of different width between 100 and 3000 nm with a Hall probe was fabricated by the argon milling. The width of the Hall probe is 50 nm. The FeB and FeCoB layers were etched out from top of the nanowire except a small region of the nanomagnet, which was aligned to the Hall probe. The nanomagnets of different lengths between 100 nm to 3000 nm were fabricated.

When it is not mentioned, the Hall angle is measured at current density of 5 mA/mm2. The aAHE in the ferromagnetic metal was evaluated as

where  sferro , snonMag  are conductivities of ferromagnetic and non-magnetic metals;  tferro , tnonMag are their thicknesses, VHall is the measured Hall voltage, I is the bias current and R,L,w are the resistance, length and width of the nanowire, correspondingly.

Both the aAHE and aOHE reverse their sign, when M and H are reversed. In order to avoid a systematic error due to a possible misalignment of the Hall probe, the Hall angle was measured as


What is the spin polarization?

All conduction electrons in a ferromagnetic metal can be divided into groups of spin-polarized and spin-unpolarized electrons. The spin polarization sp of the electron gas is defined as a ratio of the number of spin-polarized electrons to the total number of the spin-polarized and spin-unpolarized electrons:

where nTIA and nTIS are the numbers of spin-polarized and spin-unpolarized electrons, respectively.

How to divide all conduction electrons into the group of spin-polarized and spin-unpolarized electrons?

Detailed explanation is here. Explanation in short:

In fact, all conduction electrons in a ferromagnetic metal are divided into 3 groups: of spin-polarized, spin-unpolarized electrons and spin-inactive electrons. In the group of the spin-polarized electrons,   the spins of all electrons are in the same direction. In the group of the spin-unpolarized electrons, the spins are distributed equally in all directions. Additionally, there are some electrons, which are "spin-inactive". A pair of these electrons with opposite spins occupies one quantum state. The occupation of quantum states by the electrons of both the spin-polarized and spin-unpolarized groups is one electron per a state. As a result, the spin of each state is 1/2 and the spin direction for each quantum state is defined. The spin direction represents the direction of the local breaking of the time-inverse symmetry for the state. When a quantum state is occupied by two conduction electrons of opposite spins, the spin of such quantum state is zero. As a result, the spin direction of this state cannot be defined and the electrons occupying this state are "spin-inactive". The electrons, which energy is substantially below the Fermi energy, mainly belong to this group. For example, nearly all of the “deep level” electrons belong to this group. In contrast, the energy of electrons of the groups of spin-polarized and spin-unpolarized electrons is distributed mainly nearly the Fermi energy. See details here.

 


Measurement methods of spin polarization

Measurement of spin polarization is important for spintronics
Click on image to enlarge it.

Measurement 1: From tunneling magneto-resistance (TMR) using Julliere formula

M. Julliere, Phys. Lett. A 54, 225 (1975).

The TMR ratio of a magnetic tunnel junction (MTJ) is proportional to the spin polarization of its electrodes. In the most simplified case, the TMR ratio

Estimated spin polarization for FeCoB: 70-90 %

merits: Simplicity of measurements

demerits: systematic error due to substantial limitations and approximations of Julliere formula (e.g. )

 

Measurement 2: from Andreev reflection

R. Meservey and P.M. Tedrow, Phys. Rep. 238, (1994). R.J. Soulen Jr RJ et al, Science 282(1998).

The spin polarization is evaluated from the tunneling properties of a superconductor-metal contact.

Estimated spin polarization for FeCoB: 30-50 %

merits: ???

demerits: (1) limitation of only a low temperature measurement; (2) a systematic error due to simplifications and approximations for calculations of the transport through a superconductor-metal contact; (3) clear under estimation of the value of the spin polarization;

 

Measurement 3: using Anomalous Hall effect (the method described below)

V.Zayets arxiv 1902.06451 (2019)

The spin polarization is evaluated from the measured dependency of the Hall angle on an applied perpendicular magnetic field

Estimated spin polarization for FeCoB: 30-50 %

merits: (1) Simplicity of measurements (2) ability for measurement of spin-polarization even in a nano-sized objects;

demerits: (1) the spin polarization can be measured only in sample with perpendicular magnetic anisotropy (PMA); (2) the magnetization of the sample should be well-fixed in one direction. ;

 

Measurement 4: using spin-dependent photoluminescence, spin-dependent electroluminescence and spin LED

C. Aku-Leh,et al .Phys. Rev. B 76, 155416 (2007). B.T. Jonker, Proc. IEEE 91, 727 (2003).

The spin polarization is evaluated from the amount of circular-polarized light emitted from a semiconductor, in which spin-polarized current is injected

Estimated spin polarization for FeCoB: 60-95 %

merits: (1) spatial distribution of spin polarization can be checked.

demerits: (1) incorrect description of spin injection can cause a systematic error; (2) incorrect description of complex features of spin-light interaction can cause a systematic error; (see here and here);


 

Measurement of spin polarization using Anomalous Hall effect

Measurement of spin polarization. Comparison of different materials

FeB
Hall angle αAHE 1st derivative 2nd derivative
sp0=81.7% ; Hpump=0.834 kG ; Ta (2.5 nm):FeB(1 nm):MgO; sample:
FeCoB
Hall angle αAHE 1st derivative 2nd derivative
sp0=89.4 % ; Hpump=1.098 kG ;Ta (5 nm):Fe0.4Co0.4B0.2 (1 nm):MgO ; sample:
Hall rotation angle αHall as a function of applied magnetic field H ; 1st and 2nd derivatives of αHall normalized to its value at H=0. Black triangles show the measured data. The red line shows the fitting by Eq.(1.12)
Click on image to enlarge it.

Main idea:

The spin polarization is evaluated from the measured dependency of the Hall angle on an applied perpendicular magnetic field

 

Why in one metal the spin polarization is smaller and in another metal is larger? What does influence the spin polarization?

The spin polarization sp of the electron gas is defined as a ratio of the number of spin-polarized electrons to the total number of the spin-polarized and spin-unpolarized electrons. The amount of electrons in each group is determined by a balance between the spin pumping and the spin relaxation. The spin pumping is the conversion of electrons from groups of spin-unpolarized electrons into the group of the spin-polarized electrons. The spin relaxation is the conversion in the opposite direction.

Detailed explanation about spin polarization is here.

 

Spin-pumping rate:

The spin pumping describes the conversion of electrons from the group of the spin-polarized electrons into the group of spin-unpolarized electrons. The conversion rate of the spin-pumping is described as

where tpump is the spin pumping time, nTIA and nTIS are the numbers of spin-polarized and spin-unpolarized electrons, respectively.

Details about different mechanisms of spin pumping is here

Spin-relaxation rate:

The spin damping describes the conversion of electrons from the group of the spin-polarized electrons into the group of spin-unpolarized electrons. The conversion rate of spin-relaxation can be described as

where trelax is the spin relaxation time.

Details about different mechanisms of spin relaxation is here

Spin polarization:

The spin polarization sp of electron gas can be found from the condition that in an equilibrium there is a balance between the spin pumping and the spin relaxation, which is described by the condition:

How to increase the spin polarization?

(1) Increase of spin pumping rate

(2) Decrease of spin relaxation rate

 

Why does the spin polarization become larger in a magnetic field

Alignment of spins of conduction electrons along a magnetic field is the reason of increase of the spin polarization of the electron gas

Fig.3 Precession of electron spin and the spin precession damping in a magnetic field. The red arrow represents electron spin. The grey arrow shows the direction of magnetic field. The data was calculated by solving Landau-Lifshiz equation

Spin precession and precession damping in a magnetic field. During the precession the spin aligns itself along the direction of the magnetic field. Click on the image to enlarge it.
 

 

There are two reasons:

(1) major reason: increase of spin pumping in a magnetic field

It is due to alignment of spins of spin-unpolarized electrons along a magnetic field (see Fig. 3)

(2) minor reason: suppressing of spin relaxation by a magnetic field

 

Spin-pumping rate induced by a magnetic field:

In a magnetic field, the spins of spin-unpolarized electrons aligns along the magnetic field due to the precession damping (See here). However, scatterings quickly re aligns spins of electrons into two groups of spin- polarized (all spins are one direction) and spin-unpolarized electrons (spins are equally distributed in all directions). (details See here) As a result, there are more spin-polarized electrons. The spin-pumping (See Eq.19 below)

where tH,pump is the spin pumping time in a magnetic field. The spin pumping time in a magnetic field can be calculated as

spin polarization of electron gas in a magnetic field

The spin polarization sp of electron gas can be found from the condition that in an equilibrium there is a balance between the spin pumping and the spin relaxation, which is described from Eqs (1.2),(1.5) (1.6) by the condition:

Substitution Eqs.(1.2) (1.6) Eq(1.7) into Eq.(1.1) gives the spin polarization sp of electron gas in a magnetic field (see more details here) as

where sp0 is the spin polarization in absence of an external magnetic field (a material parameter) , which is calculated as

Hpump is the pumping magnetic field (a material parameter), which is calculated as

 


Reduction of spin relaxation in a magnetic field

 

Is it possible that the spin relaxation is reduced in an external magnetic field? Does it influence the spin polarization?

Absolutely. An external magnetic field reduces the spin relaxation. This reduction should be included into an evaluation of the spin polarization.

An external magnetic field reduces all mechanism of the spin relaxation:

reduction of mechanism 1 : the spin-dependent scatterings: Even spin may be rotated after a spin-dependent scattering, in a magnetic field it quickly rotates back to be along the magnetic field. Therefore, the spin- dependent scattering does not lead to the increase of the spin relaxation.

reduction of mechanism 2 : incoherent spin precession in a spatially inhomogeneous magnetic field: A large magnetic field levels out and fully compensates any possible inhomogeneities of internal magnetic field in a metal. A magnetic field reduces or even this type of the spin relaxation.

 

 


Increase of magnetization in a magnetic field

Can direction or magnitude of the magnetization change in an external magnetic field.? How does it influence the spin polarization measurement?

A. It does influence very much. (1). Magnetization inclination. Keeping the magnetization in the same direction is very important for these measurements. The sample geometry and the scan range of magnetic field should chosen to avoid any (even slight) magnetization inclination or domain movement. (2) Change of magnetization magnitude. In a magnetic field the spin polarization sp increases. The increase of sp may cause the increase of the magnetization as well. The amount of increase of the magnetization is difficult to measure. The correct measurement of such increase is still a challenging task.

 

 

 

 

 

 


Spin polarization & VCMA effect

Dependence of spin polarization on gate voltage

 
  Fig. 14 Spin polarization of FeB nanomagnet as a function of the gate voltage

click on image to enlarge it

The VCMA effect describes the fact that in a capacitor, in which one of the electrodes is made of a thin ferromagnetic metal, the magnetic properties of the ferromagnetic metal are changed, when a voltage is applied to the capacitor. For example, under an applied voltage the magnetization direction of the ferromagnetic metal may be changed or even reversed. This magnetization-switching mechanism can be used as a data recording method for low-power magnetic random access memory and all metal transistor. Until now the physical origin of the VCMA effect has not been clarified. However, several possible physical mechanisms have been discussed. (See here for more details)

 

Figure 14 shows the gate-voltage dependence of the spin polarization in the FeB sample. The measured voltage-dependent change of the spin polarization is -0.245 %/V. The change is substantial and it can be reliably measured. For example, the observed voltage-dependence of 1st derivative of αAHE(Fig.3(c)) is about 5 %. For the FeCoB sample the voltage-dependence of sp was smaller ~ -0.0068%/V. For both samples, the sp depends linearly on the gate voltage and the polarity of the dependence is the same as the polarity of the voltage-dependence of anisotropy field Hanis, coercive field Hc , Hall angle αAHE, magnetization switching time tswitch , and retention time treten.

 

 

VCMA and TMR effects. It would be interesting to correlate this gate tuning of spin polarization with the bias-dependent TMR experiments which have been traditionally explained with barrier height modulation.

A. I guess there are several contributions to the bias-dependence of TMR. The voltage dependence may have some contribution. However, it should be a contribution, which is polarity dependent. The voltage-control change of the spin polarization always changes its sign, when the voltage polarity is reversed. I have checked many samples already. The spin polarization always linearly increases under a negative gate voltage and it always linearly decrease under a positive gate voltage.

 

 

 

 

 

 


Spin polarization & SOT effect

Dependence of spin polarization on polarity of bias current (SOT effect)

(a) Sample: FeB. Spin polarization as function of current density. At a higher current, the spin polarization decreases due to the sample heating. However, the decrease is different for opposite polarities of the current (a) Sample: FeB. Change of the spin polarization under reversal of current direction (c) Sample: FeCoB. Change of the spin polarization under reversal of current direction

Fig.14 click on image to enlarge it

 

 

 

 

The effect of Spin-Orbit torque (SOT effect) describes the fact that magnetic properties of ferromagnetic nanowire may depend on the magnitude and polarity of an electrical current flowing through the nanowire. For example, under a sufficiently large current the magnetization of the nanowire may be reversed. The direction of the magnetization reversal depends on the polarity of the current. The effect may be used as a recording mechanism for 3-terminal MRAM. The origin of the SOT effect is the spin Hall effect, which describes the fact that an electrical current may create a spin accumulation.

Figure 14 (a) shows the measured spin polarization sp of the FeB sample as a function of the current density. The sp decreases for both polarities of the current. The used current density is relatively large and the decrease of sp is assumed to be due to the heating of the nanomagnet. The increase of the nanowire resistance confirms the increase of the nanowire temperature. In order to exclude the influence of heating, the spin polarization was measured at the same current, but for two opposite current directions. Figure 14(center) shows the change of the spin polarization as the polarity is reversed.  The change of the sp linearly depends on the current. The measured slope is +0.00524 %/(mA/mm2) for the FeB sample and -0.018 %/(mA/mm2) for the FeCoB sample.

Why are the slopes of Figs. 14 (b) and 14(c) different?

A. The polarity of the spin polarization generated by the Spin Hall effect are different at opposite sides of nanomagnet (see here). In the case of a symmetrical nanomagnet, the generated spin polarization is the same at opposite sides of the nanomagnet, the total generated spin polarization is zero and there is no SOT effect.  Our studied samples are asymmetric. The ferromagnetic metal is contacting the MgO at one side and the Ta at another side. As a result, the total spin polarization generated by the Spin Hall effect is non-zero. However, in this case the contributions from each interface are nearly equal. It can explain the observed substantial change of the slope for different nanomagnets on the same wafer and the different slope polarities for the FeB and FeCoB samples. The enlargement of structure asymmetry and optimizing interfaces may increase the current- induced change of sp.

 

 


 

Oscillations of 2nd derivative

Oscillations of measured 2nd derivative vs H

     
There are clear oscillations of measured 2nd derivative of αAHE vs H of an unknown origin.

 

 

The physical origin of the oscillations is not yet understood.

 

The oscillations exists for all measured samples.

The period and amplitude of the oscillations are different from a sample to sample

 

 

 

 

 

 

 

 

 

 

 


Spin polarization vs film thickness & interface type

nanowire with two pairs of Hall probes

Measurement setup

spin polarization

hysteresis loop

Backside Hall probe is connected to a nanomagnet. FeB is thicker in this region and top of FeB is covered by MgO. The Front side Hall probe is connected at side of a nanomagnet. FeB is thinner in this region and at its top covered by SiO2. The distance between two Hall pairs is 11 μm. Measured spin polarization vs applied perpendicular magnetic field; Fit by Eq.(1.8) gives (region of nanomagnet):sp0=81.2% ; Hpump=0.425 kG ; (region at side): sp0=63.5 % ; Hpump=0.88 kG ;  

Volt 54B ud40 Ta(2.5)/FeB(1.1)/ MgO(6)/ nanowire width 1000 nm nanomagnet length 500 nm. Sample is soft. Hc~0 Oe, Hani~1.7 kG,

click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measurement of spin polarization. Comparison of different thicknesses of the same nanowire

nanomagnet (thicker FeB covered by MgO)
Hall angle αAHE 1st derivative 2nd derivative
sp0=81.2% ; Hpump=0.425 kG ;
side (thinner FeB covered by SiO2)
Hall angle αAHE 1st derivative 2nd derivative
sp0=63.5 % ; Hpump=0.88 kG ;
Volt 54B ud40 Ta(2.5)/FeB(1.1)/ MgO(6)/ nanowire width 1000 nm nanomagnet length 500 nm. Sample is soft. Hc~0 Oe, Hani~1.7 kG,

triangles show the measured data. The solid line shows the fitting by Eq.(1.12)

Click on image to enlarge it.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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