Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields
It has been shown that the dependence between the parameters of materials of electronic equipment and external fields is determined by the distribution function of the corresponding random variable. The obtained results have been applied to the analysis of a number of physical phenomena.
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2015
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| Цитувати: | Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields / G.V. Milenin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 4. — С. 456-459. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1212752017-06-14T03:07:03Z Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields Milenin, G.V. It has been shown that the dependence between the parameters of materials of electronic equipment and external fields is determined by the distribution function of the corresponding random variable. The obtained results have been applied to the analysis of a number of physical phenomena. 2015 Article Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields / G.V. Milenin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 4. — С. 456-459. — Бібліогр.: 10 назв. — англ. 1560-8034 DOI: 10.15407/spqeo18.04.456 PACS 61.72.-y, 75.78.-n http://dspace.nbuv.gov.ua/handle/123456789/121275 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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It has been shown that the dependence between the parameters of materials of electronic equipment and external fields is determined by the distribution function of the corresponding random variable. The obtained results have been applied to the analysis of a number of physical phenomena. |
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Milenin, G.V. |
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Milenin, G.V. Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields Semiconductor Physics Quantum Electronics & Optoelectronics |
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Milenin, G.V. |
| author_sort |
Milenin, G.V. |
| title |
Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields |
| title_short |
Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields |
| title_full |
Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields |
| title_fullStr |
Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields |
| title_full_unstemmed |
Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields |
| title_sort |
probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/121275 |
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Probabilistic approach to the analysis of regularities in behavior of material parameters of electronic equipment under action of external fields / G.V. Milenin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 4. — С. 456-459. — Бібліогр.: 10 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T19:30:57Z |
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1837108360146583552 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 4. P. 456-459.
doi: 10.15407/spqeo18.04.456
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
456
PACS 61.72.-y, 75.78.-n
Probabilistic approach to the analysis of regularities in behavior
of material parameters of electronic equipment
under action of external fields
G.V. Milenin
V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine,
41, prospect Nauky, 03028 Kyiv, Ukraine
Abstract. It has been shown that the dependence between the parameters of materials of
electronic equipment and external fields is determined by the distribution function of the
corresponding random variable. The obtained results have been applied to the analysis of
a number of physical phenomena.
Keywords: random variable, distribution function, defect generation, spin-dependent
reaction, magnetization of ferromagnetics.
Manuscript received 08.04.15; revised version received 04.08.15; accepted for
publication 28.10.15; published online 03.12.15.
1. Introduction
As it was noted in [1], the effects of electric, magnetic,
electromagnetic and radiation fields on materials of
semiconductor products lead to changes in states of
ensembles of particles and defects. Behavior of these
ensembles is caused by flow of random events in
physical and chemical processes. This concept allowed
to give mathematical description of proceeding with
time a number of physical and chemical processes from
the probability positions. Therefore, it is of interest to
develop a probabilistic approach to the analysis of the
dependence of parameters of electronic equipment
material on the values of the parameters characterizing
the external fields (hereinafter, in the broadest sense, we
will call these parameters as the force ones).
2. Statistic regularities of behavior inherent to
material parameters under action of external fields
Let the probability that a particle, defect or structural
formation in material does not change its state when the
parameter of the external field changes from 0 to f will
be P(f). Furthermore, changes in a state, position,
generation or annihilation of particle, defect and
structural formation under external influences will be
treated as a random event (or simply – event). Then the
probability of the lack of events when changing the force
parameter to dff can be presented in the form
)( dffP . The latter probability can be considered as
the probability of a complex event consisting in the
absence of the event when changing the force parameter
from 0 to f and in the range df. Let us assume that these
events are independent. When these events are
independent, the sought-for probability is equal to the
product of the probabilities of the component events:
dfPfPdffP . (1)
Let dff is the probability of event in the range
df . Then the probability of the opposite event is equal
to dff1 .
Consequently,
dfffPdffP 1 , (2)
and f will be called as the sensitivity of events to the
parameters of external fields (hereinafter – the
sensitivity of events). The mathematical definition of the
latter will be given below.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 4. P. 456-459.
doi: 10.15407/spqeo18.04.456
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
457
Considering the mathematical definition of a
derivative, we obtain
df
df
fdP
fPdffP . (3)
Substituting (3) into (2), we get
dffPffdP . (4)
Integrating (4), we find
f
dff
efP 0 . (5)
Then for the probability of event to the value of
parameter of the external field f, we get
f
dff
efF 01 . (6)
In other words, fF is the distribution function of
the random variable – the force parameter before the
event.
As it is seen from (5) and (6), the sensitivity of
events is determined as follows:
df
fdF
fP
f
1
, (7)
where dffdF – density of distribution of the force
parameter before the event (the probability density of the
event).
By analogy with [1], where the term of event
intensity is used, we can write an expression for the
sensitivity of events:
ffnn
fn
f
0
, (8)
where n0 is the total number of events, n(f) – number of
events to the force parameter value f, ffn –
number of events when changing the parameter of the
external field by f (the number of events per unit f).
Let us define the quantities in the formula (8) by
using an example of some physical phenomena.
1. Generation of defects in semiconductor materials
under the influence, for example, of laser
irradiation. A random event – generation of a
defect: (f) – the sensitivity of generations of
defects to the intensity of laser irradiation f = I; n0
– total number of generated defects; n(f) – number
of defects generated to the value of the laser
irradiation intensity f = I; ffn – number of
the defects generated per unit of the laser
irradiation intensity f = I.
2. The spin-dependent reactions. A random event is a
change in the electron state caused by a change in
spin orientation: (f) – sensitivity of changes in
electron states to the magnetic induction f = B; n0 –
total number of electrons changing their state; n(f)
– number of electrons that have changed their state
to the magnetic induction value f = B; ffn –
number of electrons changed their state by the unit
of magnetic induction f = B.
3. Magnetization of ferromagnetics. A random event
is the domain wall displacement: f – sensitivity
of displacements of the domain walls to the
magnetic field strength f = H; n0 – total number of
the displacing domain walls; n(f) – number of
domain walls, displaced to the value of the
magnetic field strength f = H; ffn – number
of domain walls that have displaced per the unit of
magnetic field strength f = H.
Since, by definition 0nfnfF , accounting
(6), we get
f
dff
enfn 0
0 1)( . (9)
From (9) it follows that at f , fn
asymptotically approaches the value n0.
For f so small that 1
0
dff
f
, expanding the
exponent in series and limiting only by the first term of
the expansion, we obtain
dffnfn
f
0
0 . (10)
In its turn, the changes in the parameter values of
electronic equipment materials control the changes in
n(f) and are proportional to n(f). Considering that in the
initial state the material parameter had a value yin, then
during its growth with the f increase, taking (9) into
account, we get
f
dff
in eyyfy 0
0 1 , (11)
where y0 is the asymptotically attainable value
inyfy when f .
If y(f) decreases with increasing f, we obtain
f
dff
in eyyfy 0
0 1 , (12)
where y0 is the asymptotically attainable value
fyyin when f .
For f so small that 1
0
dff
f
, according to (10)
we have
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 4. P. 456-459.
doi: 10.15407/spqeo18.04.456
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
458
dffyyfy
f
in
0
0 , (13)
dffyyfy
f
in
0
0 . (14)
Therefore, the change in the material parameters is
determined by the form of the f function that depends
on the type of the used distribution F(f). To analyze the
force dependences of the change of material parameters,
as in the case of the mathematical description of the
flowing physical and chemical processes with time [1], we
use the distribution of Weibull–Gnedenko, the function of
which takes the form [2]:
m
f
f
efF
1 , (15)
where f and m are the distribution parameters.
The expression for f , in accordance with (7)
and (15), is as follows:
1
m
m
f
f
m
f . (16)
In future, we will call m as the form factor of
distribution of a random variable – force parameter
before the event, and we mean that f (that is the scale
parameter) is the constant of the force parameter. When
ff , the probability of a random event is 11 e ,
that is, about 63% of the events. When m = 1, the
sensitivity of the events is a constant, and when 1m it
changes with increasing the parameter of the external
field by a power law.
We note three different types of F(f) behavior over
the whole range f. When 0 < m < 1, the curve F(f) is
always convex and tends asymptotically to unity. At
m > 1, the curve in the initial section of change in f is
concave, and then after passing through an inflection
point it becomes convex, tending to the asymptote.
When m = 1, the Weibull-Gnedenko distribution
transforms into the exponential law that is characterized
by a linear behavior of F(f) in the initial section of
change in f.
The Weibull–Gnedenko distribution has the
following important feature [3]. At m > 1, when
3 < m < 4, the Weibull–Gnedenko distribution reduces to
the normal one. The function of the Weibull–Gnedenko
distribution is symmetrical at the point F(f) = 0.5 at
m = 3.26, that is 1
2ln1
, and the inflection point
changes with m remarkably slower. When 0 < m < 0.7,
the Weibull–Gnedenko distribution is reduced to the
normal logarithmic one. Finally, as it was already noted,
when m = 1 the Weibull–Gnedenko distribution
transforms into the exponential one.
Replacing f in the formulas (9) and (10) to (16)
and integrating, we obtain
m
f
f
enfn 10 , (17)
m
f
f
nfn
0 . (18)
Accordingly, substituting (16) into (11) and (12),
we will get the following force dependences of changes
in the material parameters:
m
f
f
in eyyfy 10 , (19)
m
f
f
in eyyfy 10 . (20)
At the values of force parameters so small that
1 m
ff , as a result of expansion of the exponent
in series and limitation by the first term of the
expansion, these dependences can be represented as:
m
in
f
f
yyfy
0 , (21)
m
in
f
f
yyfy
0 . (22)
Thus, in the case of using the Weibull–Gnedenko
distribution, the change in material parameters for small
f is described by the power functions. On the other hand,
taking into account that in practice for small f, the
changes in the respective parameters are well
approximated by power laws, it once again confirms the
correctness of using the Weibull–Gnedenko distribution
for mathematical analysis of random events in the
materials of electronic equipment under the influence of
external fields.
In the presence of the sensitivity threshold f0 to the
action of external fields, the expressions (17)-(22) can be
rearranged via replacing f by 0ff .
3. Some applications of the obtained results
The relation (19) well approximates the dependence of
the relative conductivity of InSb on the laser irradiation
intensity, which is presented in [4] (Fig. 1, curve 1). This
dependence is rectified in the Weibull-Gnedenko
coordinates:
1
0
1lnln
II
IIII
z in and
th
IIx ln
(Fig. 1, curve 2), in this case m 3.2, that is the sensiti-
vity of generations of defects is a power function of the
laser irradiation intensity.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 4. P. 456-459.
doi: 10.15407/spqeo18.04.456
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
459
The formula (19) with the form factor m = 1
describes given in [5, 6] regularities of the behavior of
the microhardness of crystals NaCl with paramagnetic
impurities Ca and the photocurrent in fullerene and
tetracene crystals depending on magnetic induction
caused by spin-dependent reactions. In these cases, there
is a constant sensitivity of changes in the states of
electrons in the defects to the magnetic induction.
The obtained results can be recommended to use
for describing the magnetization curve of ferromagnetics
with the polydomain structure. The magnetization of
such ferromagnetics is caused by the process of
displacements of domain boundaries (domain walls) and
growth of volumes of those domains, in which the
magnetic moment vector forms the smallest angle with
the magnetic field direction [7]. With increasing
magnetic field strength, the state of material reaches the
technical saturation. If we continue to strengthen the
field, then there observed is the paraprocess comprising
weak linear increase of magnetization [7].
Thus, the dependence B on H can be expressed as
B(H) = B1(H) + B2(H): where B1(H) describes the process
related with motion of the domain walls, which obeys the
laws of probability and is described by the expression (19)
with f = H, y(f) = B(H), yin = 0 (previously demagnetized
material). In its turn B2(H) = 0H is the paraprocess
component [8], here μ0 is the magnetic constant, μ is the
magnetic permeability for the paraprocess.
In particular, the curve of magnetization of gray
cast iron is presented in the work [9]. The analysis
shows that it is approximated by the expression (19)
with m 1.7. It indicates that the sensitivity to
displacements of the domain walls is a power function of
the magnetic field.
Similarly, one can analyze the dependences of
polarizability of ferroelectrics with a polydomain
structure on the electric field. Polarizability of
ferroelectrics P(E) = P1(E) + P2(E) consists of the
nonlinear orientational polarizability P1(E) caused by
the process of repolarization of domains possessing a
statistical character [7, 10].
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
-5
-4
-3
-2
-1
0
1
2
0 2 4 6 8
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
X
Z
1
2
I/
I
I/I
th
Fig. 1. Relative change of the InSb sample conductance as
a function of the laser irradiation intensity.
It can be represented in the form (19) with f = E,
y(f) = P(E), yin = 0 as well as the contribution of the
mechanisms of elastic and relaxation polarization P2(E) =
0E [10] (0 is the dielectric constant of vacuum, is the
dielectric susceptibility by induced field of elastic and
relaxation polarization) causing a weak linear increase in
the segment of domain polarization saturation.
4. Conclusions
Behavior of material parameters of electronic equipment
under the influence of external fields obeys the laws of
statistics. Analytical expressions establishing the relation
between the parameters of materials and external fields
are the distribution functions of the corresponding
random variables. The use of the Weibull–Gnedenko
distribution for this purpose has been substantiated. A
number of applications of the obtained results has been
demonstrated.
References
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physical and chemical processes flowing in
materials of semiconductor products under external
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Physics, Quantum Electronics & Optoelectronics,
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