The design of vehicle’s surroundings monitoring tools is a problem of great significance in area of modern vehicles development and their control automation. One of the essential safety characteristics of the system is its ability to detect objects and monitor surroundings in adverse conditions – in presence of fog, snow or rain, in night lighting, at traffic overload or while parking in a difficult environment. Leading manufacturers have proposed various solutions to this problem. Optical surveillance cameras work effectively only in good light conditions. Night vision systems operating in the IR range depend significantly on weather conditions: active ones on the presence of rain, fog or snow, passive ones on the temperature of objects in comparison to the environment. Sound vision systems (parking sensors) do not work well in conditions of heavy rain, noise, or a detection range of more than 3 m. For adequate and timely decision-making, some more detailed information on the detected objects is also required. In conditions of poor visibility, microwave vision devices are ones to prefer – centimeter-wave microwave radars are suitable for distant object detection even in bad weather conditions. There has recently [1-2] been shown the feasibility of holographic image registration in the microwave range using a Wi-Fi emitter. In this work, we study the potential of the method of holographic detection and visualization of objects using highly sensitive spin diodes. Those diodes possess advantages at low microwave irradiation power [3-6, 20-22] and can show a significantly higher sensitivity than the sensitivity of a Schottky diode at a similar level of equivalent input noise [16]. Estimates show that for a specified sensitivity, taking into account the radiation safety standards, there exists at least 25 m safe obstacle detection range. As it is well known, one of the key requirements for automotive recognition systems is an ability to stop the car upon the obstacle detection, i.e. registration of obstacles at a safe distance. While manually driving a car at a speed of 48 km/h, safely making a decision requires a distance to the object of about 23 m on a dry surface [7]. In bad weather conditions, it increases to 80 m. However, in the case of a fully automated car in good technical condition, if driving at a speed of 30 km/h, the braking capabilities provide a stop on dry surfaces at a distance of 6 meters or less, and a decrease in the collision speed to prevent injuries even at shorter distances. Thus, for a fully automated car in congested urban traffic conditions, in a traffic zone with a speed limit of 30 km/h, or while parking, the required level of security even at poor visibility and noise would be achieved when recognizing obstacles at a range of up to 5-10 m with data processing time less than 0.1 sec.
The holographic method of microwave visualization of images based on spin diodes has been previously under discussion regarding mainly near-field registration. The use of spin diodes in a device that does microwave visualization of objects for an autonomous driving system, as far as we know, was not under discussion before. Prior to moving on to methods of object recognition in microwave imaging with spin diodes, we discuss the requirements on the parameters of a recognition system in an autonomous driving environment.
The
holographic method of optical registration of images is well known [8] and is
widely used for visualization of various objects [9,10]. The possibility of
expanding the range of holographic visualization of images to the terahertz
range using the holographic principle of recording the wave front using a
high-resolution video camera pointed out, for example, in [11]. The earlier proposed
holographic vision systems assumed broadband multi-position irradiation with an
excess density of receiving elements. Unlike the photodetector matrix of an
optical video recorder, a microwave holographic system based on spin diodes not
only allows you to obtain information about both the amplitude and phase of the
received wave front, but also makes it possible to use the spin diode as a
microwave mixer, which simplifies the microwave recording system [4-6]. The
frequency at which the sensitivity of the spin diode is at an acceptable level
(see requirements for diodes below) can reach 10 GHz or higher, which
corresponds to a wavelength of
λ
= 3 cm or less.
Fig. 1. Microwave holographic
registration scheme based on a spin diode:
1 - microwave generator, 2 - horn
antenna, 3 - antenna array, 4 - channel selector, 5 - recording unit with a
spin diode, 6 - registered object
According
to Abbe's classical theory of optical resolution for monochromatic irradiation,
see e.g. [12], the detection range is determined by the diffraction limit
, where
Y is the width of the object,
is the radius of the outer Fresnel zone, equal to the radius of the
reception window of the recorder. Therefore, for example, for an object with a
width of Y = 30 cm with a holographic recorder diameter of 2 m at a wavelength
of 3 cm, we get L = 13 m. The requirement for the density of the sensors along
the diameter of the recorder is defined by KotelnikovТs theorem. According to
this theorem, accurate signal reproduction is given when sampling a signal with
a resolution of at least a half-period of the maximum spatial frequency of the
amplitude and phase modulation of the wave front. The discreteness of the
reception of the wave front can be associated with the discreteness of the
zones of the zone plate. The minimum width of the Fresnel zone at the edge of
the zone plate sets the maximum frequency of the wave front from the point
image. Therefore, in accordance with the Kotelnikov theorem, this width
determines the maximum step distance of the signal sampling. The width of the outer
Fresnel zone is given by the approximate formula
|
|
|
(1)
|
where
is the radius of the n-th (outer)
Fresnel zone.
This shows that
the resolution for the registration of the wave front could be much lower than
the step in the usual registration of the hologram, which should be of the
order of half the wavelength. For example, at a distance of 12 m, a radius of 1
m, and a wavelength of 3 cm, the width of the zone is
= 18
cm. In
the
latter
case
, 14
sensors
are
sufficient
for
a
diameter
of
2
m.
The received
coherent wave front within the window of the holographic recorder is used to
obtain an image in a given window and for further numerical processing. In this
regard, the task of recognizing objects includes separating the signal from
interfering noise and restoring the shape of the initial distribution of
scattering density. The restoration of the initial distribution may use the inverted
matrixes of the convolution kernel of the direct image (the inverse problem of
image restoration). In the case of a noisy signal, the resolution and detection
range will depend on the effectiveness of the signal extraction algorithm.
We consider the
following form of the image restoration problem. Given are the positions of one
monochromatic radiation source and several receiving sensors. The formula for
calculating the scattered signals (the values getting both their phase and
amplitude) -
P(p)
given by:
|
|
|
(2)
|
There
is
the value of the scattering density at the point
of
the image recovery region;
is
the radiation intensity in the direction
;
p
is the position of the sensor.
is the signal frequency;
R
is the length of the broken line
that connects the source position to the point
, and
then to the position of sensor;
is
the wave vector and
= 6
cm is the radiation wavelength.
Our algorithm to
reconstruct the scattering density field from the holographic scattering
pattern is based on solving the inverse problem in Fresnel – Kirchhoff integral
representation with consideration for the amplitude-phase modulation of the
wave front by the scattering object and it was first described in [13, 14]. To improve
the stability of the solution for the inverse scattering problem, the Tikhonov
regularization algorithm was applied [15] in the Fourier space for the
harmonics of both the microwave field and the kernel of the inverse scattering
problem, as well as the threshold-based signal reconstruction algorithm. The
algorithm is robust against computational errors and the presence of random noise;
it is efficient in the situation of a small number of sensors and an object
irradiation field restricted to a single emitter.
The restoration
made for the scattering density breaks into two successive separate stages, different
both in substantial motivation and in solution technology. At the first stage, we
restore the assumed suitable form of the scattering density field, described by
the sum of low-order Fourier harmonics, ensuring the smoothness of this
solution. Suppression of high harmonics made by a suitable choice of the
Tikhonov regularizing matrix. The second stage, on the contrary, is aimed at
finding the shape of the restored density in the form of a piecewise constant
function with predefined fixed local density values (i.e., already known
typically encountered values of the scattering density of known materials). Clustering,
by assigning clusters with threshold cutoff, the local density values restored
at the first stage around predefined density values achieves that. Such a two-stage
regularization scheme (based on the Tikhonov matrix at the first stage and
local clustering of density values at the second) allows one to achieve high
resistance to signal noise even at a low number of receiving channels (antenna
array sensors). At the second stage, the form of the problem solution defines a
scene built of bounded objects of a given density. This is exactly the representation
required by the scene visualization task. However, both the initial
regularization of the solution and the noise suppression are much more
effective in the Fourier space.
The density reconstruction
and the regularization with the high harmonics suppression at the first stage
of the algorithm is as follows. The integral equation (2) for
in the space of Fourier harmonics of given orders is a linear
equation, the kernel of it given by a fixed finite matrix
. Accordingly,
its solution might proceed by applying the inverse matrix
. However, the naive form of the matrix
does not exist, since
is,
generally speaking, a rectangular matrix, and, practically, the size of the
vector
(the number of harmonics) is significantly larger than the size of
the vector
(the number of sensors). That is, the system is underdetermined. With
that, the solution with Tikhonov regularization proceeds by substituting the
(non-existent) matrix
with the matrix
, where
and
are
Hermitian conjugates to
and
. The
matrix
is a Tikhonov regularizing matrix (a matrix for a quadratic form in
the space of solutions) – the choice of which is ours and must depend on the essential
formulation of the regularization problem. In our case (suppression of high
harmonics), the suitable form for
is a
diagonal matrix, the weights in its diagonal being proportional to the product
of the orders of the corresponding harmonics for each of the spatial
coordinates. This way, optimal suppression is ensured for high order harmonics
– the experiments showed that using, say, the product of harmonics orders in
degrees higher than the first gives too much suppression of high harmonics with
a deterioration of reproduction quality for the scattering object shape and its
visualization. Alternatively, an excessively weak suppression of high harmonics
leads to the same.
The task of
clustering local density values around given known values is a one-dimensional
clustering problem. It is solvable by many standard methods; one of the
simplest is the threshold cutoff at the point of the lowest local density of
values within the interval – we used it successfully here to get a very
adequate reconstruction of the external shape of the object both to visualize
it and to apply for the task of recognizing obstacles.
To assess the
effectiveness of the objects visualization algorithm we developed, we carried series
of computer simulation numerical experiments for model problems detecting a
number of three-dimensional, two-dimensional and one-dimensional objects, for a
more detailed study of the algorithm behavior for various parameters and of the
theoretical capacities of the method.
Below are the examples
of visualization of three-dimensional obstacles in the frontal plane of the XY
projection - the contours of the reconstructed image as shown in comparison
with the contour of the original object. The virtual model of the experiment involved
the following parameters: distance to the object, rotation of the object,
position of the object along the sceneТs width, the direction of the radiation emitter,
radiation scattering pattern, and configuration of the array of sensors. Series
of experiments made it possible to establish the boundaries of applicability of
the system.
For the job, we
chose two distinct objects, that is, three-dimensional densities
, which
we then projected onto the XY plane. The first object had a simpler form than
the second one. We chose the cube as the first object (Fig. 2a). As the second one
(Fig. 2b) we chose a human figure.
Fig. 2: Densities used in the simulation (a - density in the
shape of a cube, b - density in the shape of a male human figure).
To
apply the proposed algorithm, we had to calculate the signals gotten from the
receiving elements. The emitter wavelength:
λ
= 6 cm, the
sensor array 13x13, the distance between the sensors 20 cm. After the numerical
calculation of the integrals (formula 2), we obtained the phase and amplitude
values on each of the sensors. The figures below demonstrate an example of the
obtained values of phases and amplitudes. Fig. 4 shows data for an object in
the form of a cube located at a distance of 360 cm from the holographic stand
(Fig. 3a). Fig. 5 shows similar data for an object in the form of a man (Fig. 3b)
located at the same distance from the stand.
Fig. 3: Projection of density in the shape of a cube (a) and
of a man (b) on a plane.
Fig. 4: Example of received signals for the density of a
cube (a - amplitude, b - phase).
Fig. 5: Example of received signals for the density in the human
shape (a - amplitude, b - phase).
Then,
upon getting the signals at the receiving sensors, the proposed restoration
algorithm is applied. Figure 6a and Figure 6b show examples of reconstructed
two-dimensional densities.
Fig. 6: Examples of reconstructed
densities (a - density in the shape of a cube, b - density in the shape of a man).
We used these three
indicators to assess the quality of visualization: precision (accuracy), recall
(completeness), F1-measure. Recall definition was the ratio of the area of the
correctly identified part of the initial density to the area of the initial
density; precision definition was the ratio of the area of the correctly identified
part to the area of the entire reconstructed contour. The F1 measure defined as
the harmonic mean of recall and precision:
|
|
For both objects
presented above, we obtained the dependencies of the F1 measure on the distance
from the emitter to the object and the horizontal position of the center of
mass of the object (see Fig. 7 and Fig. 8). Figure 7 shows the dependence of
the F1 measure (indicated by color, on the scale given on the right) for the density
in the shape of a cube, and Figure 8 shows the dependence for the density in
the shape of a man. It is evident there that an acceptable recognition quality
is maintained up to 20 m - which is consistent with the stated goals for
detecting road obstacles in the near distance range.
Fig. 7: F1-measure dependence for the density in the shape
of a cube (the
x-axis defines
the vertical position of the object; the
z-axis defines the distance from the
sensors to the object).
Fig. 8: F1-measure dependence for the density in the shape
of a man (the
x-axis defines
the vertical position of the object; the
z-axis defines the distance from the
sensors to the object).
Another
parameter for evaluating the quality of the algorithm was noise tolerance. The
noise added as follows – independent normally distributed random variables
added to the signal of each sensor calculated by the formula (2). The noise
level measured in fraction of the mean square amplitude of the signal over the
entire field of the sensors. Using this, we obtained the dependence of the immunity
of the algorithm to the noise, on the object distance. Below is a plot of the
maximum noise, at which there is still an opportunity to detect the object, in
dependence of the distance. Here we used no formal detection criteria,
assessing the detectability "by eye".
Fig. 9: Dependence of the maximum permissible noise on the
distance to the object.
From this data,
the following conclusions can be drawn: an optimal distance with respect to
noise tolerance is up to 6 m; noise tolerance is quite high: a practically
acceptable resistance level is maintained up to 20 m.
The simulation
results show that the resolution efficiency (i.e., the minimal resolvable strip
gap width) for one-dimensional objects in the form of two strip reflectors
depends on the distance as
, where
(n = 2) is the ratio of the gap
width to the size of the Fresnel zone. Table 1 below shows the probability of recognition
error for a given distance to the object and the number of receiver sensors.
The probability of error defined as the percentage of cases with resolution of
the objects not achieved, out of all checked combinations of object parameters
(reflector width, gap width) satisfying the above resolution conditions (limits
for
Y
).
Table 1.
The
dependence of the probability of recognition error on the distance (L)
to the object and on the number of sensors (NS)
in
the line of receivers at a reception window width of 2 m.
|
L\NS
|
4
|
8
|
16
|
32
|
64
|
|
0.3 m
|
---
|
18.1%
|
1.7%
|
0.2%
|
0.0%
|
|
0.45 m
|
13.3%
|
27.4%
|
5.3%
|
2.6%
|
2.2%
|
|
0.6 m
|
31.4%
|
4.8%
|
5.1%
|
4.8%
|
3.4%
|
|
1.5 m
|
0.0%
|
9.7%
|
1.9%
|
1.3%
|
2.9%
|
|
3 m
|
24.0%
|
17.6%
|
4.2%
|
4.3%
|
3.0%
|
|
4.5 m
|
8.3%
|
16.7%
|
5.6%
|
6.3%
|
6.3%
|
|
6 m
|
---
|
30.0%
|
11.6%
|
9.5%
|
7.4%
|
|
15 m
|
---
|
20.0%
|
20.0%
|
53.3%
|
53.3%
|
It follows from
the table that the most reliable recognition conditions for a wavelength of 3
cm and a reception window of 2 m, for an object 40 cm wide, are: the distance
to the object not more than 6 m, and the number of receiving antenna sensors along
the width of the reception window of at least 16.
In addition to
the conditions of optical resolution, the sensitivity and the level of the
equivalent input noise of the detecting device also affect the detection range.
Let us consider these conditions for a spin diode, which can demonstrate the
sensitivity exceeding the sensitivity of a Schottky diode at a similar level of
equivalent input noise [16]. The decrease in power coming to the receiving
antenna, at the angle of the horn antenna
, the
size of the reflecting mirror
Y
, the distance to the mirror object
L
and the width of the receiving detector
,
one
may approximately estimate by the formula
|
|
|
(3)
|
The
safe dose of volumetric exposure for humans is, according to the published
data,
[16]. Therefore, with a weight of
G
=
50 kg and an irradiated frontal surface of a person of approximately
, we
obtain the intensity of safe exposure
. This threshold intensity is created by
a horn antenna at a distance of
R
= 1 m with a source of power
. On the other hand, the signal
detection range is determined by the relation
,
where
N
is the signal-to-noise ratio,
is
the input threshold sensitivity of the spin diode, which is proportional to the
noise level of the diode voltage
and inversely proportional to its sensitivity
ε
.
Assuming that the wavelength determines the useful area of the microwave source
and receiver, i.e.
, where
the parameter
takes into account the
utility coefficient of the receiving antenna, the efficiency of the
transmitting antenna and the matching of the receiving path, from the formula
(3) we find
|
|
|
(4)
|
For the noise equivalent
power of a spin diode
[16], for the maximum sensitivity
, achieved with a bias current, the
noise threshold power with a bandwidth
∆f = 10 kHz
is
. For estimates, we set
η
=
0.1,
N
= 4,
R
= 1 m,
λ
= 3 cm, then for the adopted
noise level of the spin diode we obtain the detection range
L
= 250 m.
For a Schottky diode, with a noise equivalent power
, we get roughly the same estimate for
the threshold noise power.
In the absence
of a bias current, the sensitivity
ε
of the spin
diode decreases by a factor of 100 or more. In the latter case, the range of
safe detection (with the accepted range of safe exposure
R
= 1 m),
according to estimates based on formula (4), will be about
L
= 25 m.
With a detection range of up to 10 m, the input power of the generator should
be at least
. The
irradiation safety requirements definitely met here.
The spin diode
is a magnetic tunnel junction with a high magnetoresistance. As a result of †the
non-linearity of the current-voltage characteristic, the constant voltage
occurs at the junction with a microwave change in the tunnel current. The
effect of rectifying a microwave signal by a spin diode has several mechanisms
based on current-induced ferromagnetic resonance, thermal transfer of the spin
(spin Seebeck effect), and modulation of perpendicular anisotropy induced by
the voltage at the junction [4, 16, 17]. Other physical mechanisms of spin
systems response discussed in [18, 19]. Spin diodes with a noncollinear spin configuration
of adjacent magnetic layers of the tunnel junction under resonant excitation of
spin vibrations due to current transfer of rotational moment in the presence of
a bias current near and above the boundary of micromagnetic instability have
the highest microwave sensitivity to microwave irradiation [4, 17, 20-22]. We
calculated the resonance dependence of the microwave sensitivity of the spin
diode based on the Landau – Lifshitz equations modified by Slonczewski, for
several values of the bias current [20]. The calculated parameters are close to
the parameters of the spin diode of [17], namely: the nanocolumn tunnel
structure had a diameter of
D
= 140 nm, the thickness of the free layer
d
= 2 nm, the magnetization
and
the magnetoresistance
,
, spin-polarization factor
, magnetic attenuation parameter
.
The maximum
microwave sensitivity occurs at the resonance frequency that depends on the
effective magnitude of the magnetic field acting on the spins of the free
layer. Maximum sensitivity falls when frequency increases. Table 3 shows the
literature data on the microwave sensitivity of spin diodes, obtained from
theoretical calculations and experimental measurements. It is apparent that the
sensitivity of the spin diode can be comparable with the sensitivity of a
Schottky semiconductor diode in the range up to 10 GHz.
Table 2.
Microwave sensitivity of the spin diode and its controlling parameters –
frequency, external magnetic field, bias current.
|
, mV/mW
|
f, GHz
|
H, Oersted
|
J, A/cm2
|
Experiment/
Theory
|
Link
|
|
12000
|
1.53
|
1100
|
3.8x10-6
|
experiment
|
[16]
|
|
970
|
1.2
|
0
|
0
|
experiment
|
[17]
|
|
75000
|
1.1
|
0
|
2.7x10-6
|
experiment
|
[17]
|
|
30
|
13.5
|
0
|
0
|
experiment
|
[4]
|
|
20000
|
9.2
|
500
|
7x10-6
|
theory
|
[20]
|
|
40
|
10
|
3200
|
0
|
theory
|
[22]
|
From formulas (2) and (3), it follows that the detection range
associated with microwave resolution increases with decreasing wavelength, and the
power limited one decreases due to a decrease in the useful area of the
emitting and receiving antenna element. In this regard, there is an optimal
wavelength where the holographic detection range while using Schottky diodes as
microwave receivers reaches maximum
|
|
|
(5)
|
Spin diodes, as
follows from Table 2, are close in microwave sensitivity to the sensitivity of
Schottky diodes and close to optimal in the frequency range. Unlike a semiconductor
measurement-base that includes an additional unit with a microwave vector
spectrum analyzer for conducting fast and stable phase measurements, spin
diodes can directly be used as nanoscale nonlinear elements for mixing signals,
in combination with an RF synchronous amplifier for reading a rectified signal [5,23].
This contributes to the structural simplification and miniaturization of the
holographic device.
The analysis of
numerical estimates, numerical simulation results, and literature data shows
that spin diodes can be promising for creating a holographic microwave-imaging
device designed for autonomous driving of a vehicle in restricted traffic
conditions with poor visibility and high noise. Their high microwave
sensitivity and frequency range are close to the optimal parameters of
microwave holographic imaging for these conditions. An acceptable recognition
quality maintained up to 20 m, which is in good agreement with the stated goals
of detecting road obstacles at the near distance. The most safe reception
conditions for a wavelength of 3 cm and a reception window of 2 m met with the
number of receiving antenna sensors along the width of the reception window of
at least 16. The optimal distance with respect to noise tolerance is up to 6 m;
noise resistance is quite high: a practically acceptable resistance level maintained
up to 20 m.
The Russian Science Foundation (project No.
19-12-00432) supported this work.
1.
Holl P.M., Reinhard F., Holography of wi-fi radiation // Phys.
Rev. Lett. 118, 183901 (2017).
2.
Chetty K., Smith G.E., Woodbridge K., Through-the-wall
sensing of personnel using passive bistatic wifi radar at standoff distances
//IEEE Trans. on Geoscience and Remote Sensing 50(4), 1218-1226 (2012).
3.
Fu L. et al., Microwave holography using
a magnetic tunnel junction based spintronic microwave sensor // J. Appl. Phys. 117(21),
213902 (2015).
4.
Gui Y.S. et al., New
horizons for microwave applications using spin caloritronics // Solid State
Communications 198, 45-51 (2014).
5.
Yao B.M. et al., Rapid microwave phase
detection based on a solid state spintronic device //
Appl. Phys. Lett. 104, 062408 (2014).
6.
Cao Z.X. et al., Spintronic microwave imaging //
Appl. Phys. A 111(2), 329–337 (2013).
7. Speed,
speed limits and stopping distances // Brake: the road safety charity.
URL
:
http://www.brake.org.uk/fundraise/15-facts/1255-speed
(retrieval
date: May 05, 2020).
8.
Gabor D., A new microscopic principle // Nature 161, 777–778 (1948).
9.
Yamaguchi I., Zhang
T., Phase-shifting digital holography // Optics letters 22(16), 1268-1270
(1997)
10.
Kreis T., Handbook of holographic interferometry: optical and digital methods – John Wiley & Sons, 2006.
11.
Êàëåíêîâ Ã.Ñ., Êàëåíêîâ Ñ.Ã., Øòàíüêî À.Å., Ãèïåðñïåêòðàëüíàÿ ãîëîãðàôè÷åñêàÿ ôóðüå-ìèêðîñêîïèÿ // Êâàíòîâàÿ ýëåêòðîíèêà. – 2015. – Ò. 45. – ¹. 4. – Ñ. 333-338. (Kalenkov G.S., Kalenkov, S.G., Shtan'ko A.E., Hyperspectral Holographic Fourier-Microscopy // Quantum Electronics 45(4), 333–338 (2014) – in Russian).
12.
Born M., Wolf E. Principles of Optics – Cambridge
Univ. Pr., 1999.
13.
Leshchiner D. et al., Resolution Limits in
Near-Distance Microwave Holographic Imaging for Safer and More Autonomous
Vehicles // J. of Traffic and Transportation Engineering 5, 316-327 (2017).
14. Leshchiner D. et al., Image reconstruction
algorithms for the microwave holographic vision system with reliable gap
detection at theoretical limits // EPJ Web Conf. 185 (2018).
15.
Tikhonov A.N. et al., Numerical methods for the
solution of ill-posed problems // Mathematics and Its Applications 328 –
Springer Science & Business Media, 1995.
16.
Miwa S. et al., Highly sensitive nanoscale
spin-torque diode // Nature Materials 13(1), 50-56 (2014).
17.
Fang B. et al., Giant spin-torque diode
sensitivity in the absence of bias magnetic field // Nature Communications 7, 11259
(2016).
18. Plokhov D.I. et al., Current-driven quantum
dynamics of toroidal moment in rare-earth nanoclusters // EPJ Web Conf. 185
(2018).
19. Petrov P.N. et al., Inverse spin Hall effect in heterostructures “nanostructured ferromagnet/topological insulator” // EPJ Web Conf. 185 (2018).
20.
Êóëàãèí Í.Å. è äð., Íåëèíåéíûé òîêîâûé ðåçîíàíñ â ñïèíîâîì äèîäå ñ ïëîñêîñòíûì íàìàãíè÷èâàíèåì // Ôèçèêà íèçêèõ òåìïåðàòóð. – 2017. – Ò. 43. – ¹. 6. – Ñ. 889–897. (N.E. Kulagin et al., Nonlinear Current Resonance in the Spin-Torque Diode with a Planar Magnetization // Low Temperature Physics 43, 708 (2017) – in Russian).
21. Demin G., Popkov A., Spin-torque
quantization and microwave sensitivity of a nano-sized spin diode // EPJ Web Conf.
185 (2018).
22.
Ïîïêîâ À.Ô. è äð., Ïîëåâûå îñîáåííîñòè ìèêðîâîëíîâîé ÷óâñòâèòåëüíîñòè ñïèíîâîãî äèîäà ïðè íàëè÷èè òîêà ñìåùåíèÿ // Èçâåñòèÿ âûñøèõ ó÷åáíûõ çàâåäåíèé. Ýëåêòðîíèêà. – 2017. – Ò. 22. – ¹. 2. – Ñ. 109-119. (Popkov, A.F et al., Field Features of Spin-Torque Diode Microwave Sensitivity Presence of Bias Current // Proceedings of Universities. Electronics 22, 109-119 (2017) – in Russian).
23.
Hemour S. et al., Towards low-power high-efficiency RF and microwave energy harvesting // IEEE transactions on microwave theory and techniques 62(4), 965-976 (2014).
