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Original papers Nanostructured, nanoscale materials and nanodevices
УДК 537.622.6 DOI: 10.17277/jamt.2021.03.pp.167-178
The effect of Dzyaloshinskii-Moriya interaction on direct and backward transition between magnetic states of Pt/Co/Ir/Co/Pr synthetic ferrimagnet
Artem D. Talantseva, Ekaterina I. Kunitsynaa, Roman B. Morgunova,b^
a Institute of Problems of Chemical Physics, 1, Academician Semenov avenue, Chernogolovka 142432, Russian Federation; b Tambov State Technical University, 106, Sovetskaya St., Tambov 392000, Russian Federation
И morgunov2005@yandex.ru
Abstract: In this paper, we present the study of domain structure accompanying interstate transitions in Pt/Co/Ir/Co/Pr synthetic ferrimagnet (SF) of 1.1 nm thick and 0.6 - 1.0 nm thin ferromagnetic Co layers. Variation in the thickness of the thin layer causes noticeable changes in the domain structure and mechanism of magnetization reversal revealed by MOKE (Magneto-Optical Kerr Effect) technique. Magnetization reversal includes coherent rotation of magnetization of the ferromagnetic layers, generation of magnetic nuclei, spreading of domain walls (DW), and development of areas similar with strip domains, dependently on thickness of the thin layer. Inequivalence of the direct and backward transitions between magnetic states of SF with parallel and antiparallel magnetizations was observed in sample with thin layer thicknesses 0.8 nm and 1.0 nm. Asymmetry of the transition between these states is expressed in difference fluctuation fields and shapes of reversal magnetization nucleus contributing to the correspondent forward and backward transitions. We proposed simple model based on asymmetry of Dzyaloshinskii-Moriya interaction. This model explains competition between nucleation and domain wall propagation due to increase/decrease of the DW energy dependently on direction of the spin rotation into the DW in respect to external field.
Keywords: synthetic ferrimagnets; perpendicular anisotropy; magnetic domains; Dzyaloshinskii-Moriya interaction; MOKE.
For citation: Talantsev AD, Kunitsyna EI, Morgunov RB. Effect of Dzyaloshinskii-Moriya interaction on direct and backward transition between magnetic states of Pt/Co/Ir/Co/Pr synthetic ferrimagnet. Journal of Advanced Materials and Technologies. 2021;6(3):167-178. DOI: 10.17277/jamt.2021.03.pp.167-178
Влияние взаимодействия Дзялошинского-Мория на прямой и обратный переход между магнитными состояниями синтетического ферримагнетика Pt/Co/Ir/Co/Pr
А.Д. Таланцев", Е.И. Куницына", Р.Б. Моргунов3^
a Институт проблем химической физики, пр-т Академика Семенова, 1, Черноголовка 142432, Российская Федерация; b Тамбовский государственный технический университет, ул. Советская, 106, Тамбов 392000, Российская Федерация
И morgunov2005@yandex.ru
Аннотация: Представлено исследование динамики переходов между стабильными состояниями в синтетических ферримагнетиках (СФ) Pt/Co/Ir/Co/Pr с толстыми (1,1 нм) и тонкими (0,6-1,0 нм) слоями ферромагнитного Co. Методом MOKE (магнитооптический эффект Керра) установлено, что изменение толщины тонкого слоя вызывает заметные изменения механизма перемагничивания. В зависимости от толщины тонкого слоя перемагничивание может происходить за счет когерентного вращения намагниченности ферромагнитных слоев, генерации зародышей намагниченности, расширения доменных стенок (ДС) и роста областей, подобных полосовым доменам. Неэквивалентность прямых и обратных переходов между магнитными состояниями синтетического ферримагнетика с параллельными и антипараллельными намагниченностями наблюдалась в образце с толщиной
тонкого слоя 0,8 и 1,0 нм. Асимметрия перехода между этими состояниями выражается в разных флуктуационных полях и формах зародышей намагниченности, вносящих вклад в соответствующие прямые и обратные переходы. Мы предложили простую модель, основанную на асимметрии взаимодействия Дзялошинского-Мория. Данная модель объясняет конкуренцию между зародышеобразованием и распространением доменной стенки за счет увеличения/уменьшения энергии ДС в зависимости от направления вращения спина в ДС по отношению к внешнему полю.
Ключевые слова: синтетические ферримагнетики; перпендикулярная анизотропия; магнитные домены; взаимодействие Дзялошинского-Мория; MOKE.
Для цитирования: Talantsev AD, Kunitsyna EI, Morgunov RB. Effect of Dzyaloshinskii-Moriya interaction on direct and backward transition between magnetic states of Pt/Co/Ir/Co/Pr synthetic ferrimagnet. Journal of Advanced Materials and Technologies. 2021;6(3):167-178. DOI: 10.17277/jamt.2021.03.pp.167-178
1. Introduction
Planar ferromagnetic heterostructures based on ultrathin Co layers and containing Co/Ir, Co/Pt interfaces demonstrate many unusual properties in respect of domain wall dynamics [1-19], formation of the skyrmions [20-28], and spin-orbital torque [19, 29-32]. Most of the new phenomena can be explained by contribution of the Dzyaloshinskii-Moriya interaction (DMI) in the Co/Ir and Co/Pt interfaces [1-3, 10, 13-15, 20, 24, 29, 33-41]. Synthetic ferrimagnets (SF) of Co/Ir/Co and Co/Pt/Co types are quite promising heterostructures for ultrafast magnetic memory devices [16, 26, 42]. Although magnetoresistance of the Co multilayer structures is usually small (~ 1 %), their main assignment is development of alternative technologies for information storage and processing based on spin orbit torque [42-44]. Magnetization dynamics in SF is governed by thicknesses of their layers. The variation of spacer thickness controls the sign and value of interlayer exchange coupling between ferromagnetic layers of SF [45-48]. The variation of ferromagnetic layer thickness in SF with perpendicular magnetic anisotropy (PMA) allows one to understand the role of crystalline magnetic anisotropy and its contribution to the multilayer sample properties together with interface magnetic anisotropy [12, 33, 35, 36, 40, 49]. Dependence of Co bilayer properties as a function of thickness of the Co layers, tCo , demonstrates the occurrence of the perpendicular anisotropy and enhancement of DMI below tCo = 2 nm [6, 8]. Since the analysis of net magnetic moment is often not accompanied by domain structure visualization, details of the local magnetization reversal tend to be overlooked. This can result in wrong judgment on dependence of DMI on Co layer thickness. For that reason, in this paper analysis of integral magnetic moment on tCo are accompanied by detailed analysis of the magnetic nuclei.
Four stable magnetic states of SF are determined by mutual alignment of magnetizations in the layers and controlled by, both, the magnetic interlayer exchange interaction and the Zeeman energy. These energies were extracted from temperature dependences of magnetic hysteresis parameters [50, 51]. The four types of magnetic nuclei corresponding to four horizontal levels of magnetization are presented in Figure 1, in contrast with a single monolayer. Two stable "parallel" states two
"antiparallel" (AP+ and AP ) states correspond to pure domainless states typical for nanosized SF. A variety of multidomain states in large ~ 1 mm SF can be considered as intermediate states.
In this paper, we report on alteration of a physical origin of magnetization reversal in series of the Pt/Co/It/Co/Pt samples with progressively decreasing thicknesses of one of the Co layers. We show progressive changes in the dominating mechanism of magnetization reversal from the coherent rotation of magnetization to the multidomain magnetic structure and strip domain propagation. Competition of different channels of the magnetization reversal provides domination of one of them depending on sweeping rate of the external magnetic field We analyzed the switching field HC for different interstate transitions in the series of the samples with different
tCo as a function of the magnetic field sweeping rate.
2. Materials and Methods
Multilayered structures SiO2/Pt(3.2 nm)/Co (1.1 nm)/Ir(1.3 nm)/Co( tCo)/Pt(3.2 nm), tCo = = 0.60 nm, 0.70 nm, 0.80 nm and 1.0 nm of 4 x 4 mm sizes were grown by magnetron sputtering at T = 300 K. The methods of sample preparation and their preliminary chemical, structural and magnetic attestation were described in [50]. The thickness of 1.3 nm of the Ir spacer was selected to provide antiferromagnetic interlayer exchange coupling
comparable with the Zeeman energy. This circumstance gives an ample opportunity to switch magnetic stable states of the SF using external magnetic field.
Local M-H hysteresis loops and magnetic domain images were recorded by a Durham Magneto-optics NanoMOKE3 microscope based on the Kerr effect. The microscope was equipped with an electromagnet with ± 2000 Oe field range and 0.1 Oe resolution of magnetic field. MOKE measurements were performed in polar geometry (P-MOKE). The local M-H loops were collected from the area of a focused laser spot of 6 ^.m diameter. The domain images were recorded in a scanning mode from 350 x 350 ^m square areas with magnification value x 270. Velocity of the domain walls (DW) was determined from a comparison of domain boundary coordinates in a series of the MOKE images recorded with 0.6 s time interval. Prior to recording, the samples were exposed in + 2000 Oe magnetic field, which exceeded the saturation field in all the samples. After that, the magnetic field was switched down to a varying negative value, and the domain images were recorded. Field stabilization time was ~ 0.1 - 0.5 s, which was less than the time interval between the frames. Time interval between frames was more, than 10 times longer than scanning time. The depth of light penetration 30-40 nm allows us to observe magnetic nuclei in top and bottom Co layers, both. We can clear distinguish four types of areas of different brightnesses corresponding to the P+, P-, AP+ and AP- states.
Brillouin light scattering (BLS) in the Damon-Eshbach geometry at room temperature was used to measure DMI coupling energy by the asymmetric frequency shift appeared from the annihilation or generation of spin waves (SWs). The laser wavelength was 532 nm; the diameter of the laser spot was 50 ^m. The asymmetric frequency shift of the BLS spectrum was determined by measuring for in-plane fields HIP = +8 kOe and HIP = - 8 kOe. As the Co films were very thin, counterpropagating surface spin waves on the top and bottom surfaces of each layer as well as spin waves in the top and bottom Co layers were simultaneously observed in the BLS experiments, averaging frequency shifts of the all layers.
3. Results and Discussion
3.1. Shape of magnetic nuclei and origin of magnetization reversal
Magnetic hysteresis loops corresponding to variation of integral Kerr rotation are presented in Figs. 1a-d, left panel, for the four samples with
tCo = 0.6 nm, 0.7 nm, 0.8 nm and 1.0 nm. These loops
have four single domain states, Depending on thin Co layer thickness and initial state of the sample, the transition between the states proceeds by different manners.
In the sample with tco= 0.6 nm, the P+ ^ AP+
and AP+ ^ P+ processes are implemented by coherent rotation of magnetization in the entire area of the film (lines 1, 2 in Fig. 1a). The MOKE images for these transitions demonstrate homogeneous changes in "background" from dark to white color occurring under sweeping magnetic field. On the contrary, the AP- ^ AP+ transition is provided by generation and expansion of magnetization nuclei. The AP- ^ AP+ transition is accompanied by replacing black to white areas with no intermediate colors (line 3 in Fig. 1a). There are two magnetization phases AP- and AP+ separated by 180° domain walls.
The rest of the samples with tCo = 0.7 nm - 1.0 nm show the formation of nuclei of several types. In the tCo = 0.7 nm sample, the P+ and AP+ domains formed
in the P+ ^ AP+ and AP+ ^ P+ transitions are of similar shapes (lines 1, 2 in Fig. 1b). However, the shape of the AP+ areas formed in the AP- ^ AP+ transition (line 3 in Fig. 1b) differs from the shape of the AP+ areas formed in the P+ ^ AP+ transition (lines 1, 2 in Fig. 1b). Thus, the shape of the nuclei is not sensitive to mutual orientation of magnetization in a final state of transition (P+ or AP+), but the shape depends on the layer (top or bottom), where transition occurs. This fact is expectable because the individual maps of nucleation centers (defects) distribution are different in top and bottom layers. Reproducible decoration of the horizontal scratch in lines 1, 2 in Fig. 1b indicates the nuclei in the same top layer, while the absence of scratches in line 3 in Fig. 1b indicates nuclei of the bottom layer.
On the contrary, in the samples with tCo= 0.8 and 1.0 nm, the P+ ^ AP+ and AP+ ^ P+ transitions accompanied with domains of different shapes (compare lines 1, 2 in Figs. 1c, d). The AP+ ^ P+ transition is provided by nucleation of many small round areas of reversed magnetization, decorating scratches, while backward process P+ ^ AP+ is accompanied by stripe domains as well as for AP- ^ AP+ transition.
Thus, the shape of domains in "thick" samples becomes independent of the reversing layer, but the shape is sensitive to mutual orientation of magnetizations at the final state of transition.
1.0 0.5
w"5 ^ 0.0
-0,5
-1.0
- = 0.6 ii in
Co
II <1 v> rr.'l;.'
i i» i »>
P*-» AP<
AP + PH
-2 -10 12 AP" —» AP * H (kOe)
B Q U u M~llr 1
(a)
1.0 0.5
gt>
0.0 -0.5 -1.0
iGo = 0.7nm
-2 -10 12 H (kOe)
P*-> AP +
AP+ —► P*
AP" —► APH
r m
23 s is a! f-
A2 ll fi
(b)
1.0
0.5 ^ 0.0 -0.5 -1.0
t„ =0.8 n in
Co
JT
P + AP +
AP
-2-1012 AP " —»■ AP ' H (kOe)
JUJ
(c)
-2-1012 II (kOe)
P + Ap +
AP + —»P '
AP " — AP
« 1
u
{d)
Fig. 1. Left panel: M-H loops for the tCo = 0.6 - 1.0 nm samples recorded at 0.1 kOe/s magnetic field sweeping rate. Right panel: MOKE images captured for Pt/Co(x)/Ir/Co(1.0 nm)/Pt heterostructure (x = 0.6, 0.7, 0.8 and 1.0 nm),
for P+ ^ AP+, AP+ ^ P+ and AP+ ^ AP transitions
Variation of the top Co layer thickness in the 0.6 -1.0 nm range enables to tune the magnetic nuclei shapes. There are two intervals of Co layer thickness: (1) presence of the thin layer the domain shape depends on the layer, which magnetization is changed in external field (independently on direction of the transition), (2) in the case of thicker top layer, mutual orientation of magnetizations of the layers in their final states (parallel or antiparallel) fully predetermines nuclei shapes (independently on that, which layer magnetization is reversed).
3.2. Thermal activation of the magnetic nuclei
The regularities of nuclei formation depend on thermodynamic parameters of nucleation such as fluctuation field Hf, activation volume VA, and activation energy EA. In this section, we will determine these parameters and analyze their sensitivity to magnetic field sweeping rate.
The dependences of switching fields HC on magnetic field sweeping rate are not linear for all Co thicknesses and all transitions (Fig. 2), except two rate independent transitions in the sample tCo = = 0.6 nm (Fig. 2a, b).
Fluctuation field Hf was determined from the slope of the dependence of switching field HC on sweeping rate dH/dt plotted in a semi-logarithmic scale. Linear approximations of these dependences at low (below 0.01 kOe/s) and high (over 1 kOe/s) sweeping rates determine the range of fluctuation fields for each transition. Fluctuation field Hf ranges are represented for different samples in Fig. 3. The mean values of fluctuation fields for different transitions in all samples are summarized in Fig. 4.
The width of an accessible fluctuation field range and the middle of fluctuation field range grow with increasing tCo, and suddenly decrease at
tCo = 1.0 nm for the AP — AP+ and AP+ — P+ transitions, both. The fluctuation field of the P+ — AP+ transition varies non-monotonically with tCo. It is surprising, that the Hf ranges for the
AP —> P+ and P+ — AP transitions do not coincide.
The activation volume was estimated from fluctuation field using the fact of proportionality of the layers saturation magnetizations MS1 and MS2 to correspondent thicknesses t1 and t2 :
Va = kT (t2 + ti )/ 2Hf M sh (1)
for the P —>
Va = kT (t2 +11 )/2H f M s (t2 -11 ) (2)
for the AP
AP transition. The difference
between expressions (1) and (2) is that the P+ ^ AP+ process occurs by upper layer magnetization reversal, while the AP- ^ AP+ process proceeds by magnetization reversal of both layers possessing integral magnetization proportional to MS2 - MS1 «t2 -11.
The activation volume of the AP- ^ AP+ transition (green bars in Fig. 4b) is almost the same for 0.6, 0.7 and 0.8 nm samples. Therefore, the thickness of free layer does not affect the mechanism of the AP- ^ AP+ transition.
The high value of activation volume for 1.0 nm sample can be caused by the difference between the actual value of Co layer thickness and that determined from the deposition rate. As the difference between t1 and t2 is in the denominator of Eq. (2), and in the 1.0 nm sample the thicknesses of the thin and thick layers are of almost same values (1.0 and 1.1 nm), the error of 10 % in both thicknesses will result in variations of VA up to 6 times.
P+ ^ AP+
o-o-o-a
tCo= 0.6 nm HNS
tCo= 0.7 nm
tCo= 1.0 nm
0.01 0.1 1 10 dH/dt (kOe/s)
(a)
1.2
1.0
0.8
0.6
AP+ P+
- tC0 = 0.6 nm
- tCo = 0.7 nm
- tCo = 0.8 nm
- tC0 = 1.0 nm
0.01 0.1 1 10 dH/dt (kOe/s)
(b)
o
AP- ^ AP+
0.6 tCo= 1.0 nm
^Co = 0.8 nm
0.4
0.7 nm
0.2 . tCo = 0.6 nm
0.01 0.1 1 10 dH/dt (kOe/s)
(c)
Fig. 2. Dependences of switching fields on magnetic field sweeping rate for P+ ^ AP+ (a), AP+ ^ P+(b) and AP- ^ AP+(c) transitions
—>
0.6 nm
0.7 nm
0.8 nm
• AP- AP+
— P+ ^ AP+ -AP+ P+
■AP- ^ AP+
— P+ ^ AP+
■ AP+ ^ P+
AP- ^ AP+ 1
in " P+ ^ AP+ 1.0 nm -AP+ ^ P+
■AP- ^ AP+
0 10 20 30 40 50
H (Oe)
Fig. 3. Ranges of fluctuation fields for samples with tCo = 0.6, 0.7, 0.8 and 1.0 nm for the P+ ^ AP+ ,
AP+ ^ P+ and AP- ^ AP+ transitions. Lines corresponding to the same types of transitions are shown by the same color
O
40 -
20
^ 3
s 3 2
k 1 0
1 1 P+ ^ AP+ —
. AP+ P+
• AP- ^ AP+
0.6
0.7
0.8
1.0
tCo (nm)
(b)
,_„ 80
E
40
o
0
P+ ^ AP+ AP+ ^ P+ AP- ^ AP+
: .n. iïh ,ri:n.
0.6 0.7 0.8
0.7 0.8 1.0 tCo (nm)
(c)
Fig. 4. Dependences of fluctuation field (a), activation
volume (b) and lateral size of domain nuclei (c) on thickness tco of Co thin layer for the P+ — AP+, AP+ — P+ and AP- — AP+ transitions
In contrast to the AP — AP transition, the activation volume for the P — transitions (orange and blue bars in Fig. 4b) are sensitive to the thickness of Co free layer. From the comparison of 0.7 and 0.8 nm samples (for 0.6 sample the fluctuation field of these transitions is
zero and the activation volume is not determined), one can conclude that the increase in thickness of Co free layer results in reduction of activation volumes of the P —> AP+ and AP+
From the determined values of activation volume, the lateral size of domain nuclei can be estimated. As layer thicknesses are
Co
t1~ 1 nm = 10 cm and activation volumes are
17 3
of an order of 10
cm
-10
the domain nucleation area
is of an order of 10 cm , and lateral size of nuclei is L0~ 10-5 cm = 100 nm. Lateral size of nuclei of different types can be calculated accurately by formula L) = V Va/ nt
transitions, and by formula
transition.
AP+ ^ P+
L0 = VVJn(t1 +12) for the AP — AP+ The size of nuclei for the P+ — AP+, AP+ — P+ and AP- — AP+ transitions are given in Fig. 4c.
The lateral nuclei size is always out of resolution of Kerr microscope, being ~10 times lower than light wavelength. Thus, we can not judge about initial stages of nucleation. For that reason, very fast generation of small non resolvable nuclei in the sample with tCo = 0. 6 nm can give images similar to images in lines 1 and 2 in Fig. 1. Zero value of fluctuation field in this sample disproves this assumption because thermally activated nuclei generation should possess finite activation parameters and volume compatible with values in Fig. 4c. Thus, we cannot explain the absence of magnetization nuclei in images in lines 1 and 2 of Fig. 1 by low resolution of Kerr microscope. Reasonable origin of the magnetization reversal in the P+ — AP+, AP+ — P+ transitions is coherent rotation of magnetization under external magnetic field.
The unresolvable size of the nuclei obviously plays a role in explanation of the stripe like areas limited by scratches (lines 1 and 3, Fig. 1c, d). The stripe domains have not been observed yet in structures with perpendicular anisotropy due to their energetically unfavourable structure and size. For that reasons seeming of the strip domain in our experiments is probably an artefact caused by low resolution of Kerr microscope. Most probably the space limited by scratches was rapidly filled out by small non resolvable nuclei seeming like continuous area of the same continuous direction of magnetization. Thus, one should distinguish magnetization areas filled by nuclei of the magnetic phase and expanded magnetic nuclei also observable when velocity of the domain walls is high enough to enlarge nuclei size until their visible length of ~ 1 ^m.
0
3.3. Contribution of Dzyaloshinskii-Moriya interaction to magnetization reversal
The Dzyaloshinskii-Moriya antisymmetric exchange interaction leads to noncollinear and chiral ordering of spins. The DMI plays an important role in spin orbitronics [2, 3]. The interfacial DMI arises in thin ferromagnetic films at the interface with heavy metals, where the spin-orbit interaction (SOC) of two adjacent spins of the ferromagnetic film Si and Sj
propagates through the exchange interaction with a heavy metal atom located in an adjacent nonmagnetic layer. This type of DMI implying the absence of the inversion symmetry in interface region is described in the framework of the three-site Levy-Fert model [1].
The interfacial DMI was experimentally studied in wide series of Co based samples, where Co/Ir and Co/Pr interfaces are presented. Asymmetry of bubble expansion in the in-plane magnetic field allowed to estimate equivalent DMI field in Pt/Co/Pt, Pt/Co/Ir/Pt [7], Pt/Co/Ir [12], Pt/Co/Pt [8], Pt/Co/Ir [18], Pt/Co/Pt [19] lying in the 0.5 - 2.5 erg/cm range dependently on interface quality, Co thickness, type of the heavy metal and spacer properties.
In the case, when both the top and bottom ferromagnetic layers of the SFs are coupled with the interlayer by DMI, different directions of the domain wall magnetization should be considered. Since symmetry of the DMI and Heisenberg exchange energy J are different, the change of the spin rotation direction in the Neel wall (Fig. 1) causes change of the sign of DMI energy density D, and does not affect
J value. The P+ — AP+
and AP+ — P+ transitions,
both, have same spin rotation direction in the domain wall of upper Co layer. Another reason to change the sign of DMI is to change the direction of the domain wall propagation. In this case both J and D change their sign. The difference in the sign of DMI in the upper Co layer, when P+ ^ AP+ transition is changed to AP+ ^ P+ transition leads to difference between energy of the domain wall and to corresponding change in the domain wall velocity.
One of the most reliable techniques to measure coupling energy D is Brillouin light scattering (BLS) used in our experiments.
A typical BLS spectrum recorded in direct (red line) and opposite (blue line) directions of in-plane applied 8 kOe magnetic fields at T = 300 K and wave vector kx = 11 ^m-1 is shown in Fig. 5.
The Damon-Eshbach geometry was used to determine D value, i.e. magnetization of sample lied in sample plane perpendicularly to the wave
I (a. u)
0.9
0.6 0.3
10 10
15 20 f (GHz)
Fig. 5. Normalized BLS spectra recorded in direct (red line) and opposite (blue line) directions of applied magnetic fields 8 kOe at T = 300 K and wave vector kx = 11 ^m 1
vector kx. We applied in plane field HIP equal to magnetic anisotropy field H A= 10 kOe determined by SQUID magnetometer to satisfy the above mentioned conditions. One can see presence of the two peaks with maximums in positive and negative magnetic fields. These two peaks correspond to the Stokes and anti-Stokes BLS components. Change of magnetic field HIP to opposite direction causes shift of the Stokes and anti-Stokes maximums, replacing their frequency positions f, but frequency difference Af between them remains constant (Fig. 5). The origin of the alteration of the Stokes and anti-Stokes maximums is interfacing DMI, which coupling energy D is directly proportional to Af value:
A f nM s
D = -
2jkx
(3)
Y is gyromagnetic ratio, kx is the x component of the wave vector, depending on incidence angle, MS is saturation magnetization. Extraction of frequencies of the Stokes and anti-Stokes lines from Fig. 5 and calculation of DMI by formula (3) results in D = 1.1 erg/cm2, quite similar with the value obtained by other authors in similar samples.
In the presence of DMI in the sample, an effective field consists of the external constant field HIP and HDMI directed, both, in plane of the sample. The effective in-plane field of the sample affects the velocity of the domain walls. This circumstance can be used to estimate the strength and sign of DMI contributing to domain wall velocity [7, 8]. We use the equation for the motion of the domain wall in creep mode to find the relationship between the speed
of the domain wall v and the HDMI field in creep mode:
v = v0 exp
(h )-m),
(4)
v0 is constant velocity, H is out-of-plane magnetic field, Z is scaling coefficient, exponent ^ = 1/4 is parameter corresponding to creep regime.
According to [8], Z parameter depends on energy density of the domain wall and in-plane field HIP:
Z=Z 0 [DW (DMI )/<G0 ] ,
(5)
where Zo is scaling coefficient,
Cdw is energy
density of the domain wall, o0 is energy density of the Bloch-type domain wall.
According to [8], dependence of the density of domain wall energy cDW on DMI field HDMI can be expressed as:
An2M s2 H DMI
G
DW
= G ±-
8K D
(6)
|HIP + HDMI I
Expression (6) can be applied with a limitation |< 4Kd/%Ms . In the studied sample
with anisotropy field ~ 8 kOe, this limitation is fulfilled in zero in-plane fields. The sign "+" or "-" in expression (8) is determined by direction of DMI field in respect to the magnetic nuclei border. Energy density of the AP+/P+ border is obviously controlled by DMI, which sign (positive or negative) depends on mutual orientations of the top and bottom Co layers. Substitution of (6) to (5) and to (4) results in expression:
ln(v / v 0) = Z 0
1 ±
An2M s2 H DMI 8g 0 K d
\1/4
H
-1/4
(7)
Approximations of field dependences by formula (7) are shown in Figs. 6 and 7 for different transitions and different thicknesses of the upper layer.
One can find a different sign of the slope in vDW (H) coordinates (Fig. 6a) and very close slopes for the samples of different upper layer thicknesses (Fig. 6b). Figure 6b allows one to conclude very high accuracy of comparison of the slopes even in different samples.
If we modify Fig. 6a, plotting modulus of the DW velocity (Fig. 7), we can compare slopes of the vDW (H) dependences for opposite processes P —> AP+ and AP+ — P+. There is an obvious difference in the slopes of the vDW (H) dependences. This difference indicates different activation energies
Vdm, Mm/s
100
10
P+ ^ AP+
AP+ ^ P
520 540
-ff-
ln(v 100
10
, Mm/s
0.7 nm
620 640 660 H, Oe
(a)
1 nm
4.75
4.80
—ff— 5.70
(b)
5.75
5.80
H1/4, Oe1/4
Fig. 6. Field dependences of the velocity of motion of the
domain wall AP+ during the transition P+ — AP+ and AP — P (a); comparison of the field dependences for P+ — AP+ transition in samples of different thicknesses of upper layer (b). Solid lines are approximations by formula (7)
ln(v), 100
10
Mm/s
5.00
5.05
t85
5.90
H1/4, Oe1/4
Fig. 7. Field dependences of the modulus of the velocity of motion of the domain wall P+ during the transition AP+ — P+
1
1
1
for direct and backward transitions. According to formula (7) this difference can be explained by contribution of the DMI field, increasing and decreasing potential barriers in direct and backward processes, respectively.
4. Conclusions
1. The physical process driving magnetization reversal of the thin layer without changes in magnetization of the thick 1.1 nm layer (the P —> AP+, AP+ — P transitions) is controlled by thin layer thickness.
For tCo < 0.6 nm magnetization reversal is caused by coherent rotation of magnetization of the layer on the whole in the field interval of 1-20 Oe. The fluctuation field of this process was determined as zero indicating the absence of the magnetization nuclei.
For tCo = 0.7 nm the forward and reverse
P —> AP+, AP+ —> P transitions are equivalent and they proceed by generation of nuclei.
For tCo = 0.8-1.0 nm the forward P —
backward AP+ — P+ processes become inequivalent. Forward transition is provided by small number of nuclei rapidly expanding by domain wall motion. Backward transition AP+ — P+ is provided by generation of great number of nuclei with small contribution of their expansion.
2. Activation volume of the P —
AP+ — P+ transitions depends on the ratio of thicknesses of thin and thick layers. Similarity of shape of the magnetization areas filled by nuclei of the final state for the P+ — AP+ and AP+ — P+ transitions correlates with ratio of activation volumes of these transitions. When the activation volumes of the P+ — AP+ and AP+ — P+ transitions are close to each other ( tCo = 0.7 nm), magnetization reversal areas shapes are determined by those layer, which magnetization reverses. At higher thickness ( tCo= 0.8 - 1.0 nm) activation volumes of the
P —> AP+ and AP+ —> P transitions become different, and the shape of magnetization area is determined by final state of transition independently on that, which layer is involved in magnetization reversal.
3. Change of the DMI sign in forward P+ — AP+ and backward AP+ — P+ domain wall motion makes different energy densities of the domain wall in these DW motion modes. The correspondent increase in the DW velocity in
forward P+ ^ AP+ transition provides domination of the magnetization reversal by magnetic nuclei expansion. Decrease in the DW velocity in backward AP+ ^ P+ transition provides domination of nucleation type of the magnetization reversal.
5. Funding
This study was supported by Grant of President of Russian Federation for Scientific School 2644.2020.2 and the State Assignment of the Institute of Problems of Chemical Physics, Russian Academy of Sciences (Project No. AAAA-A19-119092390079-8).
6. Acknowledgements
We are grateful to Prof. A. Fert, Prof. S. Mangin and Prof. A.K. Zvezdin for fruitful discussions.
7. Conflict of interests
The authors declare no conflict of interest.
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Информация об авторах / Information about the authors
Таланцев Артем Дмитриевич, кандидат физико-математических наук, младший научный сотрудник, ФГБУН «Институт проблем химической физики РАН», Черноголовка, Московская обл., Российская Федерация; ORCID 0000-0002-2223-4145; e-mail: artgtx32@mail.ru
Куницына Екатерина Игоревна, кандидат физико-математических наук, научный сотрудник, ФГБУН «Институт проблем химической физики РАН», Черноголовка, Московская обл., Российская Федерация; ORCID 0000-0002-8621-0740; e-mail: kunya_kat@mail. ru
Моргунов Роман Борисович, доктор физико-математических наук, профессор, главный научный сотрудник, ФГБУН «Институт проблем химической физики РАН», Черноголовка, Московская обл.; ФГБОУ ВО «Тамбовский государственный технический университет», Тамбов, Российская Федерация; ORCID 0000-0002-4859-2733; e-mail: morgunov2005@yandex.ru
Artem D. Talantsev, Cand. Sc. (Physics and Mathematics), Junior Researcher, Institute of Problems of Chemical Physics of Russian Academy of Sciences, Chernogolovka, Moscow region, Russian Federation; ORCID 0000-0002-2223-4145; e-mail: artgtx32@mail.ru
Ekaterina I. Kunitsyna, Cand. Sc. (Physics and Mathematics), Researcher, Institute of Problems of Chemical Physics of Russian Academy of Sciences, Chernogolovka, Moscow region, Russian Federation; ORCID 0000-0002-8621-0740; e-mail: kunya_kat@ mail.ru
Roman B. Morgunov, D. Sc. (Physics and Mathematics), Professor, Chief Researcher, Institute of Problems of Chemical Physics of Russian Academy of Sciences, Chernogolovka, Moscow region; Tambov State Technical University, Tambov, Russian Federation; ORCID 0000-0002-4859-2733; e-mail: morgunov2005@yandex.ru
Received 25 June 2021; Accepted 09 August 2021; Published 30 September 2021
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Copyright: © Talantsev AD, Kunitsyna EI, Morgunov RB, 2021. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.Org/licenses/by/4.0/).
