Tunnel Magnetic Contacts with Perpendicular Anisotropy of Magnetic
Electrodes as Promising Elements for Recording
Abstract: - This paper describes the mechanism of the appearance of the magnetic capacitance in tunnel
magnetic contacts with magnetic electrodes that have perpendicular anisotropy, presents the results of
measurements of the value of tunnel magnetic resistance and tunnel magnetic capacity in Tb22-Co5Fe73/
Pr6O11/Tb19-Co5Fe76 tunnel contacts. The work also provides a structural diagram of the construction of an
information carrier based on tunnel magnetocapacitance and describes the principle of recording information
in such a structure. This paper describes the mechanism of appearance of magnetic capacity in tunnel magnetic
contacts with magnetic electrodes that have perpendicular anisotropy, presents the results of measurements
of the value of tunnel magnetic resistance and tunnel magnetic capacity in Tb22-Co5Fe73/Pr6O11/Tb19-
Co5Fe76 tunnel contacts, where the value of tunnel magnetic resistance is almost 120%, and the value of the
tunnel magnetic capacity is more than 110%. The work also provides a structural diagram of the construction of
an information carrier based on tunnel magnetocapacitance and describes the principle of recording information
in such a structure.
Key-Words: - tunnel magnetic contacts, resistance, capacitance, perpendicular anisotropy. recording
information
Received: June 28, 2022. Revised: October 19, 2023. Accepted: November 20, 2023. Published: December 31, 2023.
1 Introduction
Although studies of the characteristics of magnetic
tunnel junctions (MTJs) have been studied for a
long time, the prospect of such structures as basic
elements of spintronics began to be discussed after
a large change in resistance under the influence of a
magnetic field was obtained in them [1, 2]. This
effect was called tunnel magnetoresistance (TMR),
and its value in the best samples reached a value of
up to 500%. The main efforts of scientific research
were directed to the development of methods of
controlling the conductivity of MTJ, to establishing
regularities of the process of switching their
conductivity and optimizing the design and
composition of MTJ. Very few works were devoted
to the influence of temperature on the technical
characteristics of tunnel magnetic contacts and the
determination of their aging parameters, although
these processes have a strong influence on the
possibility of practical use of MTJ in memory
elements and other spintronic devices.
In addition to the effect of tunnel
magnetoresistance in magnetic tunnel contacts,
there is an effect of a change in capacitance upon
remagnetization of one of the magnetic electrodes
[3-8]. This effect is called tunnel magnetic
capacitance (TMC) and the reason for this change
in capacitance is the appearance of additional
capacitance that occurs in the MTJ with antiparallel
magnetization of the magnetic electrodes.
Additional capacity arises due to spin-dependent
diffusion of polarized electrons and is called spin
capacity. The spin-dependent diffusion of polarized
electrons leads to the spatial separation of major
and minor polarized electrons and changes in the
characteristics of the dielectric constant in the area
of the magnetic metal/insulator interface. The effect
of tunnel magnetic capacitance is intensively
studied, although today there are already talks
about a good prospect of its practical use.
The results of experimental studies show that
the value of the tunnel magnetic capacitance, as
well as the value of the tunnel magnetic resistance,
depend not only on the value of the spin
polarization of the magnetic electrodes, but also
strongly depend on the material and structure of the
magnetic metal/insulator interface. Record high
values of TMC and TMR were obtained in MTJ, in
of which magnesium oxide is used as an insulator.
In Fe/MgO/Fe tunnel contacts, the value of TMC
reaches values of more than 400%, and the value of
TMR can be even greater than TMR500%. Such
high values of TMR and TMC can be obtained only
with a very good agreement between the crystal
lattice of the barrier nanolayer and the crystal
lattice of the magnetic electrode, which is achieved
when using epitaxial methods of obtaining such
МYKOLA KRUPA
Institute of Magnetism National Academy of Science of Ukraine, 03143 Kiev, Vernadsky
bul., 36 UKRAINE
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nanolayers. However, even when the initial ideal
alignment of these lattices is achieved, significant
temperature stresses will occur in the interface
region, which can greatly reduce the values of
tunnel magnetic depends and tunnel magnetic
resistance. The reason for such thermal stresses is
the difference in the coefficients of thermal
expansion of the grids of the magnetic metal
electrode and the oxide dielectric. All this greatly
complicates the technology of manufacturing
tunnel magnetic electrodes of the Fe/MgO/Fe type
and narrows the operating temperature range of
spintronics elements based on them.
It is clear that resistance and capacitance effects
must always arise in tunnel magnetic contacts when
one of the magnetic electrodes is remagnetized.
Usually, the magnitude of such effects depends on
the characteristics of the interface and the
electronic structure of the contacting materials,
magnetic electrode/dielectric barrier nanolayer, but
the effects themselves are due to the
magnetoelectric effect, which occurs at the
interface between the dielectric and the spin-
polarized metal, and which describes the response
of the electrical polarization of the electronic
system to the applied magnetic field.
In this work, we want to present the results of
our research on the effects of changing resistance
and capacitance in magnetic tunnel contacts, in
which the magnetic electrodes have perpendicular
anisotropy, to show that high TMC and TMR
values can be obtained in such tunnel contacts. The
paper proposes a mechanism that explains the
appearance of tunnel magnetic capacitance in
magnetic tunnel contacts with electrodes that have
perpendicular anisotropy and provides a scheme for
constructing a spin information carrier based on the
TMC effect. We want to show that tunnel magnetic
contacts with perpendicular magnetization of
electrodes can have a good prospect of practical use
in the development of spintronics elements.
2. Research methodology and
obtained results
A feature of tunnel magnetic contacts with
perpendicular magnetization of the electrodes is a
strong change in the configuration and direction of
the magnetic field in the barrier layer during the
transition from the variant with parallel
magnetization to the antiparallel magnetization of
the magnetic electrodes. With parallel
magnetization of the electrodes in the tunnel
contact, there is an almost uniform magnetic field
in the barrier layer. When the electrodes are
antiparallel magnetized, a very strong magnetic
field gradient is formed near each electrode in the
barrier layer. Moreover, the strongest changes in
the intensity of the magnetic field occur in the
direction parallel to the direction of magnetization
of the magnetic electrodes dH/dx>>dH/dy and
dH/dx>>dH/dz (Fig. 1). From the results of works
[9, 10] it can be shown that the intensity of the
magnetic field
()
xi
Hx
in the direction of
magnetization
x
of the magnetic electrodes rapidly
decreases from
0
H
to zero as the coordinate
i
x
approaches the centre of the gap between the
magnetic electrodes.
Fig.1. Magnetic field distribution scheme in tunnel
magnetic contacts with perpendicular anisotropy of
magnetic electrodes (I) and magnetization reversal
curves of magnetic electrodes in Tb22-
Co5Fe73/Pr6O11/Tb19-Co5Fe76 tunnel magnetic contacts
(II): Tb19-Co5Fe76– from above and Tb22-Co5Fe73–from
below
In the region of the interface between the metal
tunnel contact and the barrier nanolayer (dielectric
or wide-band semiconductor), the conduction
electrons are redistributed in the tunnel magnetic
contact, which is formed under the action of the
electric field of the contact potential difference
between the metal contact and the barrier
nanolayer. In most cases, this potential difference
which depends on the output of electrons from the
metal conductor and the material of the barrier
layer, is negative
c
W
<0, so conduction electrons
move from the magnetic electrode to the barrier
nanolayer. This leads to the appearance of an
inverse nanolayer thick in it
i
d
2
0
( ) / ( )
i i i c e
d A W e n
, (1)
Where
e
n
is the concentration of conduction
electrons in the magnetic electrodes,
e
is the
electron charge,
i
is the dielectric constant of the
barrier layer,
0
is the absolute dielectric constant,
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1
is the coefficient that characterizes the
transition of conduction electrons from the
magnetic electrode to the barrier layer,
i
A
is the
proportionality coefficient.
With a parallel orientation of the magnetization
of the electrodes, the concentration of major
()
i
ns
and minor
()
i
ns
spin polarized electrons
in the inverse nanolayer
i
d
will be close to their
initial concentration in the magnetic electrodes of
the tunnel contact. With the antiparallel orientation
of the magnetization of the electrodes, the
magnetomotive force
(() B)Fs
acts on the
major and minor polarized electrons in the interface
region. The magnetomotive force of interaction
with major polarized electrons is almost equal in
magnitude to the force of interaction with minor
polarized electrons, but these forces have the
opposite direction. This magnetomotive force
causes in the inverse nanolayer the separation of
major and minor polarized electrons and the uneven
distribution of electrons along the direction
x
.
Major polarized electrons are concentrated in the
yz-plane at the border of the inverse nanolayer with
the magnetic electrode, and minor polarized
electrons are concentrated in the parallel yz-plane
on the opposite border of this inverse nanolayer
i
d
.
Since the number of majorly polarized electrons in
the inverse nanolayer significantly exceeds the
number of minor polarized electrons, an increased
concentration of electrons and a negative electric
charge
s
Q
appear at the boundary of the inverse
nanolayer with the magnetic electrode relative to
the opposite boundary of this inverse nanolayer.
This electric charge can be called a spin
nonequilibrium charge. That is, an additional
capacity appears in the interface nanolayer, which
can be called spin capacity. Such two additional
capacities, which arise near each of the magnetic
electrodes of the tunnel contact with their
antiparallel magnetization, reduce the total capacity
of the tunnel contact, and the additional negative
electric charge increases its resistance.
The electric field of the nonequilibrium spin
charge
s
Q
opposes the magnetomotive force, which
limits the maximum value of the charge value. The
estimate of maximum value of the quantity
0
Q
can
be obtained from the condition that the energy of
the electrostatic interaction
e
W
of an electron
e
with a non-equilibrium magnetically induced spin
charge will be equal to the energy
W
of the
magnetic interaction of its magnetic moment with
the gradient dВ/dx of the magnetic field in the
inverse interface layer of the barrier layer
e
WW
.
00
0
/4
2/
e e i i
i e c i o
W AQ e d
W A H d d
, (2)
where
e
A
and
A
are the coefficients of
proportionality.
When an electric voltage is applied to the tunnel
contact, the thickness of the inverse nanolayer
i
d
increases near one magnetic electrode and
decreases near the opposite electrode. This
difference is due to the fact that the electric
potential difference
i
U
is added to the contact
potential difference for the first electrode, and it is
subtracted for the second electrode.
2
10
2
20
( ) / ( )
( ) / ( )
i i i c i e
i i i c i e
d A W U e n
d A W U e n
(3)
Therefore, the conductivity and capacity of the
tunnel contact with parallel magnetized electrodes
will be determined with great accuracy by the
characteristics of the passage of electrons through
the barrier dielectric layer. At low values of the
applied electric voltage
V
, when the electron
energy
e
E eV
is much lower than the energy
height of the tunnel barrier
0
U
e
E eV
<
0
U
the
transparency coefficient of the tunnel contact with
parallel magnetized electrodes
D
can be written
as
0 0 0
2
exp[ 2 ( )]
ee
D D d m U E
h
, (4)
In tunnel contacts with antiparallel magnetized
electrodes, the conductivity will depend on the
tunnel characteristics of electrons through the
dielectric barrier layer and on the tunnel
characteristics of these electrons through additional
energy barriers that arise in such contacts near each
magnetic electrode. With a certain approximation,
three different energy barriers can be introduced in
this case. The main barrier
0
U
determines the
conductivity and the amount of resistance
0
r
when
electron tunneling through the barrier dielectric
layer, and as well two additional barriers. The first
of them is the Coulomb barrier
e
U
, which is
created due to the appearance of a non-equilibrium
magnetically induced spin charge
0
Q
. The second
barrier is a pseudo-barrier
s
U
, which describes the
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passage of spin polarized electrons and introduces
additional resistance
s
r
into the overall
conductivity of tunnel contacts with antiparallel
magnetized electrodes. It is clear that the effective
thickness and energy height of the spin-dependent
and Coulomb barrier are significantly smaller than
the analogous parameters of the dielectric barrier
layer
0
U
>>
s
U
e
U
and
0
d
>>
s
d
e
d
. At low
values of the applied electric voltage
V
, when the
electron energy is less than the energy height of the
Coulomb barrier
e
E eV
<
e
U
the transparency
coefficient
D
of the tunnel contact with
antiparallel magnetized electrodes can be written as
00
22
2 ( ) 2 ( )
0
22 ( )
e e e e e e
s e s e
d m U E d m U E
hh
d m U E
h
D D e e
e
(5)
Here
h
is the Planck constant,
e
m
is the mass of
the electron,
0
d
is the thickness of the dielectric
barrier layer,
e
d
is the effective thickness of the
Coulomb barrier,
s
d
is the effective thickness of
the spin-dependent barrier,
0
D
is the coefficient
that depends on the material of the electrodes and
the barrier layer.
The resistance of the tunnel contact with
antiparallel magnetized electrodes can be
represented as the sum of the tunnel resistances for
each barrier
0es
R r r r
. Formulas (4) and
(5) show that the conductivity of tunnel contacts
with antiparallel magnetized electrodes will
increase and its resistance will decrease with an
increase in the applied voltage much more strongly
compared to tunnel contacts with parallel
magnetized electrodes.
The dependence of the capacitance of tunnel
contacts with antiparallel magnetized electrodes on
the applied voltage
V
can be more complex. With a
certain approximation, it can be assumed that the
total capacitance
C
of tunnel contacts with
antiparallel magnetization consists of two
successive capacitances: the capacitance of the
contact with the dielectric barrier layer
0
C
and the
additional spin capacitance
i
C
, which arises in the
inverse nanolayer due to the separation of major
and minor polarized electrons. At a low value of
the applied electric voltage
V
, when the electron
energy is less than the energy height of the
Coulomb barrier
ee
EU
, the effective value of
the additional spin capacitance may even
increase with increasing voltage, which will
lead to a decrease in the total capacitance of
tunnel contacts with antiparallel magnetized
electrode. As the applied voltage increases, the
value of the spin capacitance will decrease and
the total capacitance of tunnel contacts with
antiparallel magnetized electrodes will approach
the capacitance of these tunnel contacts with
parallel magnetized electrodes. The frequency
dependence of the capacitance of tunnel
contacts with antiparallel magnetized
electrodes can also be complex, which is related
to the resonance frequency of the spin capacitance.
3. Experimental results
In experimental studies, we used
amorphous ferrimagnetic TbCoFe films for the
manufacture of magnetic tunnel contacts.
Magnetic and magneto-optical characteristics of
amorphous rare-earth-transition-metal films are
well studied as materials for magneto-optical
recording of information [11]. Similar studies can
be found in [12], [14]. Analysis of literature data
and the results of our previous studies [13]
showed that TbCoFe ferrimagnetic films are
a good material for the manufacture of tunnel
magnetic contacts with perpendicular
magnetization of electrodes. This is ensured by
the large energy of the perpendicular magnetic
anisotropy of TbFe films, large values of the
coercive force near the compensation point
(Tb22Fe78), and the dependence of the coercive
force on the concentration of the components in the
film. Replacing iron atoms with cobalt atoms up to
10% does not change the characteristics of
the films, but significantly reduces their aging rate.
Therefore, for the manufacture of tunnel
contacts with perpendicular anisotropy of magnetic
electrodes, we used films produced by magnetron
sputtering of Tb22Co5Fe73 and Tb19Co5Fe76 alloy
targets. Remagnetization curves of Tb22Co5Fe73 and
Tb19Co5Fe76 films are presented in Fig. 1. The
dielectric barrier layer was made from
praseodymium oxide Pr6O11, which is a low-
temperature paramagnet.Tunnel magnetic contacts
Tb22-Co5Fe73/Pr6O11/Tb19-Co5Fe76 were fabricated
by photolithography in the multilayer film structure
Au/Tb22-Co5Fe73/Pr6O11/Tb19-Co5Fe76/Au. Such a
multilayer film structure was produced on a
substrate of fused quartz S=14x14 mm by
magnetron sputtering of the corresponding targets.
The thickness of film Tb22-Co5Fe73/Pr6O11 and
Tb19-Co5Fe76 was dm≈ 40 nm. The area of each
magnetic electrode was approximately equal to S
50μ2. We investigated tunnel contacts with two
different thicknesses of Pr6O11 (d1=1-1,2 nm or
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d2=1,5-1,8 nm).The distance between individual
tunnel contacts was at least 5 mm.
We measured tunnel magnetocapacitance and
magnetoresistance in high-resistance contacts
Tb22-Co5Fe73/Pr6O11/Tb19-Co5Fe76 using a
measuring bridge and the four-probe method.
Registration and processing of measurement signals
was carried out by a personal computer. The
accuracy of capacitance measurement was at the
level of 3 picofarads in the frequency range of 0-
300 Hz, and the accuracy of resistance
measurement did not exceed 1 microohm in the
frequency range of 0-30 kHz. The measurement
results are presented in Figure 2.
Fig. 2. Change in the capacitance and resistance of
tunnel contacts Tb22-Co5Fe73/Pr6O11/Tb19-Co5Fe76
depending on the direction and magnitude of the
magnetic field: the thicknesses of Pr6O11 nanolayer
d1=1-1,2nm, curve 1 describes the process when the
field H changes from 0 to –400 kA/m, curve 2 describes
the process when the field H changes from –400 kA/m
to+400 kA/m.
The results of the measurements showed that the
capacity of tunnel contacts with a greater thickness
of the barrier nanolayer Pr6O11 d2=1,5-1,8 nm
changes during remagnetization more strongly than
in tunnel contacts with a smaller thickness d1=1-
1,2nm of the barrier nanolayer. The resistance of
tunnel contacts with a smaller thickness of the
barrier nanolayer Pr6O11 d1=1 -1,2 nm changes
more strongly when the magnetic electrodes are
remagnetized than in tunnel contacts with a
thickness d2=1,5-1,8 nm of the barrier nanolayer.
The value of TMR and the value of TMC is defined
[14] as TMR=(Rmax– Rmin)/Rmin and
TMC=(CP−CAP)/CAP. Here Rmin, Rmax and CP, CAP
is the resistance and capacitance in the parallel and
antiparallel magnetization states for both magnetig
tunnel contacts. The value of TMC in the best MTJ
samples reached values of TMC=110% for MTJ
contacts of the second type (Pr6O11 d1=1 -1,8 nm)
and TMC=75% for MTJ contacts of the first type
(Pr6O11 d1=1 -1,2 nm). The value of TMR in the
best MTJ samples reached TMR=120% for
contacts of the first type (Pr6O11 d1=1 -1,2 nm) and
TMR=70% for contacts of the second type (Pr6O11
d1=1 -1,8 nm).
4. Magnetocapacitance and
information recording
It is clear that for the practical use of tunnel
magnetocapacitance and magnetoresistance in
tunnel contacts with perpendicular magnetization of
electrodes, it is necessary to conduct detailed
experimental and technological developments.
However, we would like to propose in this work the
principle of recording information and the scheme
of building an information carrier based on tunnel
magnetocapacitance and magnetoresistance in
tunnel contacts (Fig. 3).
Fig. 3. Scheme of the information carrier based on
the tunnel magnetocapacitance in tunnel contacts with
perpendicular magnetization of the electrodes: 0 –
substrate of the information carrier, 1 and 4 – magnetic
electrodes with a fixed direction of magnetization, 2 –
dielectric barrier nanolayer, 3 – magnetic electrode with
a small coercive force
The magnetic spin information carrier consists
of a substrate 0 (material: glass, quartz, silicon,
etc.), on which a highly coercive magnetic layer 1
is applied, the material of which has a high spin
polarization of electrons and perpendicular
anisotropy. Structurally, layer 1 is made in the
form of a system of m separated flat electrodes with
a thickness of several tens nanometers and a width
of about one micron. A continuous thin dielectric
barrier nanolayer 2 with a thickness of 1-3
nanometers made of a dielectric non-magnetic
material is applied to the magnetic layer 1. A
magnetic layer 3 is applied to the barrier nanolayer,
which also has a high electron spin polarization and
perpendicular anisotropy, but a small coercive force
H3 compared to the coercive force H1 of the
magnetic layer 1 H3 0,1H1. The magnetic layer 3
is also made in the form of a system of separated
flat electrodes with a thickness of several tens of
nanometers and a width of about one micron. These
flat electrodes are oriented perpendicular to the m
flat electrodes of magnetic layer 1. A similar
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dielectric barrier nanolayer 2 is deposited on
magnetic layer 3, and a magnetic layer 4 is
deposited on it, the material of which also has high
electron spin polarization and perpendicular
anisotropy, but its coercive force is much greater
even than the coercive force of the magnetic layer 1
H4 > H1. The design of the electrodes of the
magnetic layer 4 is the same as in the magnetic
layer 1. Then, layer 5 can be successively deposited
on layer 5 with nanolayer 2, layer 3, nanolayer 2,
layer 1, etc.
Recording of information on the described
tunnel spin carrier is carried out in the following
way. Before recording information, magnetization
is carried out in the constant magnetic field of
magnetic electrodes 1 and 4. The constant magnetic
field is applied to the medium, the intensity of
which is perpendicular to the plane of magnetic
electrodes 1 and 4 and the magnitude of the field
intensity H0 exceeds the coercive force H4 of the
magnetic layer 4 H0>H4. Then an oppositely
directed magnetic is applied to the carrier, the
intensity of which H01 exceeds the coercive force
H1 of the magnetic layer 1, but is significantly less
than the coercive force H4 of the magnetic layer 4
H4>>H01>H1.
When writing "1" to the ml memory cell, a
powerful recording pulse JW is applied to the m flat
electrode of magnetic layer 1 and the l flat
electrode of magnetic layer 3. Moreover, the
electric field voltage to the m electrode of layer 1
is negative in relation to the l electrode of layer 3.
When writing "0" in the ml memory cell, the same
powerful recording pulse JW is applied to the m flat
electrode of magnetic layer 4 and the l flat
electrode of magnetic layer 3. The negative electric
field voltage is also applied to the m electrode of
layer 4.
The amplitude of the write pulse JW is
determined by the amount of current that must be
passed through the tunnel contact to obtain a local
remagnetization of l flat electrode of magnetic layer
3 in the ml memory cell
0
4
ae
W
sB
H S he
J
, (6)
where
W
J
is the magnitude of the current through
the contact, Se and h is area and thickness of the
magnetic electrode 3 in the ml memory cell,
and
s is magnetic permeability and spin polarization
relaxation time in the material of the magnetic layer
3,
<1 is the coefficient characterizing the value of
spin polarization in magnetic materials of magnetic
layers 1 or 5, e is electron charge,
0 is absolute
magnetic permeability.
Estimates show that even with a write current of
J=0,1
A through the tunnel contact with the area
of the magnetic electrodes one square micron, h=40
nm,
s =10-9 с,
=500 H/m and
=0,5 the spin
current from electrodes 1 and 4 creates a magnetic
field Hs>106A/m in magnetite electrode 3, which,
without a doubt, will significantly exceed the
anisotropy field of the magnetic material of layer 3.
When reading information from any ml
memory cell, two identical reading pulses JR are
sent simultaneously to the m electrode of magnetic
layer 1 and the m electrode of magnetic layer 4.
The amplitude of the reading pulse JR is much
smaller than the amplitude of the writing pulse
JR0,1JW, and the polarity of such a pulse
coincides with the polarity of the recording pulses.
Then, with the help of the processing unit, the
phase difference between the two pulses that passed
through the ml tunnel contact 1-2-3 between
magnetic layers 1 and 3 and the pulse that passed
through the ml tunnel contact 4-2-3 between
magnetic layers is recorded 4 and 3.
The magnitude of the phase shift between the
reading pulses will depend on the difference in
capacitance between tunnel magnetic contacts
1-2-3 and 4-2-3 =f(C13-C43). The capacity of
these contacts will vary depending on the
mutual orientation of magnetization of magnetic
electrodes 1 and 3 or 4 and 3 in the ml memory
cell. If "1" is written in the ml memory cell, then
the capacity between contacts 1-2-3 will be
greater than the capacity of contacts 4-2-3 C13
C43. When "0" is written in the ml memory cell,
the capacity between contacts 1-2-3 will be less
than the capacity of contacts 4-2-3 C13
C43. The
method of measuring the phase difference between
signals is much more sensitive compared to the
method of measuring the difference of amplitudes
between these signals, which makes it possible
to obtain high sensitivity and reliability of
reading information from the described spin
media.
4 Conclusion
In the final part of our work, we would like to
emphasize that although the tunnel magnetic
capacitance effect is considered one of the most
promising basic effects for use in spintronics
elements and information recording, for the
practical application of this effect, detailed studies
of the main technical characteristics of the
magnetic spin capacitance in tunnel contacts must
be carried out. find the optimal construction
materials. The results of this work show that tunnel
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magnetic contacts with magnetic electrodes that
have perpendicular anisotropy are not only an
interesting object for research although the effect of
tunnel magnetic capacitance, but they may also
have a good perspective of practical use.
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DESIGN, CONSTRUCTION, MAINTENANCE
DOI: 10.37394/232022.2023.3.24
Мykola Krupa
E-ISSN: 2732-9984
259
Volume 3, 2023
