Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field
Formation of non-stationary nonequilibrium distribution functions (DF) of electrons and phonons is investigated under the action of a strong pulse electric field on metal. For concreteness parameters were taken for nickel having reference temperature of 20 K. It is shown: isotropization of electron...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1121582017-01-18T03:04:04Z Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field Karas, V.I. Potapenko, I.F. Vlasenko, A.M. Нелинейные процессы в плазменных средах Formation of non-stationary nonequilibrium distribution functions (DF) of electrons and phonons is investigated under the action of a strong pulse electric field on metal. For concreteness parameters were taken for nickel having reference temperature of 20 K. It is shown: isotropization of electron DF occurs as a result of impacts with lattice imperfections; •electron DF does not become thermodynamically equilibrium as electron-electron impacts in the given situation give essentially smaller contribution than electron-phonon collisions and collisions with a "another's" subsystem do not result to thermalization; •DF of electrons and phonons have high energy "tails" as in electronphonon impacts the momentum is transferred with sufficiently small transfer of energy; a lot of phonons with Debye energy are being born, i.e. phonon DF is being enriched by Debye phonons. Досліджено формування нестаціонарних нерівноважних функцій розподілу (ФР) електронів та фононів під дією сильного імпульсного електричного поля на метал. Для конкретності були взяті параметри для нікелю, що має спочатку температуру 20 K. Показано, що: •ізотропізація електронної ФР обумовлена зіткненнями з дефектами решітки; •електронна ФР не стає термодинамічно рівноважною, тому що електрон-електронні зіткнення в такій ситуації дають суттєво менший внесок, ніж електрон-фононні зіткнення, а зіткнення з «іншою» підсистемою не приводять до термалізації; •ФР електронів та фононів мають високоенергетичні «хвости», тому що в електрон-фононних зіткненнях передається імпульс з досить малою передачею енергії; •народжується багато фононів з дебаєвською енергією, тобто фононна ФР збагачена дебаєвськими фононами. Исследовано формирование нестационарных неравновесных функций распределения (ФР) электронов и фононов под действием сильного импульсного электрического поля на металл. Для конкретности были взяты параметры для никеля, имеющего начальную температуру 20 K. Показано, что: •изотропизация электронной ФР обусловлена столкновениями с дефектами решетки; •электронная ФР не становится термодинамически равновесной, так как электрон-электронные столкновения в данной ситуации дают существенно меньший вклад, чем электрон-фононные соударения, а столкновения с «другой» подсистемой не приводят к термализации; •ФР электронов и фононов имеют высокоэнергетичные «хвосты», так как в электрон- фононных столкновениях передается импульс с достаточно малой передачей энергии; •рождается много фононов с дебаевской энергией, т.е. фононная ФР обогащена дебаевскими фононами. 2013 Article Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field / V.I. Karas΄, I.F. Potapenko, A.M. Vlasenko // Вопросы атомной науки и техники. — 2013. — № 4. — С. 272-278. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 47.11.-j,47.27.-i,47.27.ek,52.25.Dg http://dspace.nbuv.gov.ua/handle/123456789/112158 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах |
| spellingShingle |
Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Karas, V.I. Potapenko, I.F. Vlasenko, A.M. Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field Вопросы атомной науки и техники |
| description |
Formation of non-stationary nonequilibrium distribution functions (DF) of electrons and phonons is investigated under the action of a strong pulse electric field on metal. For concreteness parameters were taken for nickel having reference temperature of 20 K. It is shown: isotropization of electron DF occurs as a result of impacts with lattice imperfections; •electron DF does not become thermodynamically equilibrium as electron-electron impacts in the given situation give essentially smaller contribution than electron-phonon collisions and collisions with a "another's" subsystem do not result to thermalization; •DF of electrons and phonons have high energy "tails" as in electronphonon impacts the momentum is transferred with sufficiently small transfer of energy; a lot of phonons with Debye energy are being born, i.e. phonon DF is being enriched by Debye phonons. |
| format |
Article |
| author |
Karas, V.I. Potapenko, I.F. Vlasenko, A.M. |
| author_facet |
Karas, V.I. Potapenko, I.F. Vlasenko, A.M. |
| author_sort |
Karas, V.I. |
| title |
Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field |
| title_short |
Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field |
| title_full |
Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field |
| title_fullStr |
Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field |
| title_full_unstemmed |
Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field |
| title_sort |
kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2013 |
| topic_facet |
Нелинейные процессы в плазменных средах |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112158 |
| citation_txt |
Kinetics of nonequilibrium electron-phonon system for semiconductors and metals in a strong electric field / V.I. Karas΄, I.F. Potapenko, A.M. Vlasenko // Вопросы атомной науки и техники. — 2013. — № 4. — С. 272-278. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T03:28:41Z |
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2025-07-08T03:28:41Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2013. №4(86) 272
KINETICS OF NONEQUILIBRIUM ELECTRON-PHONON SYSTEM FOR
SEMICONDUCTORS AND METALS IN A STRONG ELECTRIC FIELD
V.I. Karas΄1,3, I.F. Potapenko2, A.M. Vlasenko3
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2M.V. Keldysh Institute of Applied Mathematics of the Russian Academy of Science,
Moscow, Russian;
3V.N. Karazin Kharkov National University, Kharkov, Ukraine
E-mail: karas@kipt.kharkov.ua; firena@yandex.ru; tn32@yandex.ru
Formation of non-stationary nonequilibrium distribution functions (DF) of electrons and phonons is investigated
under the action of a strong pulse electric field on metal. For concreteness parameters were taken for nickel having
reference temperature of 20 K. It is shown: isotropization of electron DF occurs as a result of impacts with lattice
imperfections; •electron DF does not become thermodynamically equilibrium as electron-electron impacts in the
given situation give essentially smaller contribution than electron-phonon collisions and collisions with a "another's"
subsystem do not result to thermalization; •DF of electrons and phonons have high energy "tails" as in electron-
phonon impacts the momentum is transferred with sufficiently small transfer of energy; a lot of phonons with Debye
energy are being born, i.e. phonon DF is being enriched by Debye phonons.
PACS: 47.11.-j,47.27.-i,47.27.ek,52.25.Dg
INTRODUCTION
In the sixtieth of the twentieth century the phenome-
non of sharp reduction of resistance of metals to plastic
deformation was revealed in case of excitation of their
conductivity electron subsystems by either irradiation or
by transmission electric current of high density
j=108…109 А/m2. It could not be explained by trivial
thermal influence (in macroscopic display) of current
therefore there was an assumption of existence of elec-
tron-dislocation interaction influencing on mechanical
properties of crystals [1].
The phenomenon was offered to be named electro-
plastic effect (EPE). EPE mechanism was associated
with the increase of disposition mobility in the neigh-
borhood of sources and therefore with intensification of
work of sources. In the purest state EPE was investi-
gated on metal monocrystals of Zn, Cd, Sn, Pb [1]. If
during deformation of samples of these materials the
current pulses of magnitude 102…103 А/mm2 and dura-
tion of 10-4 are being passed through them or the sam-
ples are being irradiated by accelerated electrons if the
direction of sliding the softening of the samples is re-
vealed which is expressed by spasmodic recessions of
deforming stress. Pressure oscillations are connected
with jumps of plastic deformation of objects. It is estab-
lished that synchronously with passage of pulses of a
current and deforming stress decrease the packs of slip
bands appear and also that deforming stress drop is in-
commensurably less in the section of quasi-elastic de-
formation than beyond a the yield point. These oscilla-
tions are abnormally high in the area of yield stress of
the material. Deforming stress drops on the diagrams
decrease when testing in a mode of stress relaxation.
Application of a pulse current has allowed to provide
high density current at simultaneous exception of domi-
nating influence of macroscopic heating by Joule heat.
In case of the pulse durations of 100…200 μs, current
amplitudes of 50…1000 A and frequency of pulse re-
currences of 0,1·104 Hz a constant component of a cur-
rent only a little exceeded 1 A and could not provide
heating of a sample more than on 1…5°С. The intensity
of jumps of deformation at action of pulses of the cur-
rent in single crystals with slipping anisotropy has
strongly pronounced orientation dependence. So in Zn
the maximum of effect and the minimum of stress at
which it starts to become apparent when the deformable
samples are oriented for easy basal slip. These facts
obviously denote the dislocation nature of EPE.
EPE has threshold character, i.e. it appears at a par-
ticular value of pulse current density which depends on
a grade of the deformable crystal and temperature-speed
condition. So for zinc at Т=77 K it is equal to
400…500 А/mm2.
It has been established that EPE is rather sensitive to
external factors. First, the effect is amplified by action
of surface-active media. For example specific crystallo-
graphic shift of amalgamated single crystals of zinc at
temperature 300 K and under conditions of current puls-
es influence with parameters: j=600…1000 А/mm2,
frequency of recurrence of pulses of 0.1…0.5 Hz and
duration of a pulse tp=10-4 s increases by 50…60%.
As a result of alloying the drop magnitude under ac-
tive stress can grow by tens percent (up to 100%). In
limits of relatively small replacement impurity content
the magnitude of the effect grows linearly with concen-
tration as it is shown in experiments with Zn, alloyed
Cd from 10-3 up to 10-1 аt. % (the content of other impu-
rities did not exceed 2⋅10-3 аt. %). At the same time
threshold value of stress from which the effect begins
also increases when alloying. However this fact can be
connected to the common increase of the critical shear-
ing stress in alloyed crystals.
The growth of speed of test first results in weak in-
crease of the magnitude of the effect and then in falling
of it. The increase of pulse frequency lowers the com-
mon level deforming stress but also reduces the stress
drop amplitude ΔΠ. The increase of duration of current
pulses at constant amplitude linearly increases depth of
stress drop. Last phenomenon is fixed both in stress
relaxation and creep experiments. Under conditions of
zinc monocrystals creep the reduction of the threshold
value of duration of pulses at which EPE becomes ap-
parent is revealed. As it was already mentioned EPE is
registered beginning from some threshold amplitude of
ISSN 1562-6016. ВАНТ. 2013. №4(86) 273
pulses of current and then the magnitude of EPE grows
linearly with the pulse amplitude. The given phenome-
non is typical both for active stressing and for creep. It
is necessary to note that electro-plastic effect of current
can be activated also by change of the direction of cur-
rent in the following pulses while the amplitude remains
constant (polarity of action of current).
Prominent feature of EPE in single crystals is ab-
sence of temperature dependence in a wide interval
from 77 up to 300 K. It is established at research of
stress relaxation in single crystals of zinc where the
jump deforming stress was measured stimulated by the
first pulse of a current after the loading device stop.
The analysis of all marked laws shows that EPE
cannot be reduced simply to thermal heating and should
be anyhow connected with a dislocation subsystem of
deformable object. More detailed research of EPE in
modes of creep and stress relaxation has allowed to es-
tablish additional features of the effect [1]. It is estab-
lished that the current influences during the first seconds
after the loading device stop at relaxation and for elec-
trically stimulating influences at the bottom relaxation
holes change of direction of current is necessary at the
same amplitude. After current switching the delay be-
fore the beginning of additional plastic deformation is
observed. It testifies that the factor of a current is equiv-
alent to occurrence of effective additional stress ΔΠ.
ΔΠ grows with the increase of Π and decreases in due
course from the beginning of relaxation. Hence, it
should be connected with the presence of mobile dispo-
sitions in the sample. The further experiments on EPE
research in modes of creep, internal friction and stress
relaxation pressure have shown that EPE is caused by
the growth of effective short-range stress. At last bipolar
pulses of current following without interval slow down
plastic deformation and they accelerate it as it was al-
ready marked earlier if between them there is an interval
of about 3⋅10-3 s.
The basic laws of EPE which have been found out
while deforming single crystals are observed also in ex-
periments with polycrystalline materials. So the occur-
rence of spasmodic deformation is established when
stretching of samples of polycrystalline zinc, cadmium,
lead, indium and tin at temperature 77 K and under in-
fluence of single pulses of current of value ~103 А/mm2.
However the magnitude of the effect in this case was
5 times lower than in case of single crystals under corre-
sponding conditions and did not exceed 6…8% [1]. In
[1] the increase of creep speed of wire polycrystalline
samples of W, Mo, Zn, and also of alloys Мо-Re, W-
Re, steel etc. due to action of a constant electric current
with density up to 5⋅103 А/mm2 is described. It is estab-
lished that logarithms of speed of creep grow linearly
with the square of current density and it has allowed the
authors to refuse the hypothesis of only thermal influ-
ence of current because the sample was being cooled
during the experiment.
At application of current pulses while stressing
(stretching) metal crystals the process of deformation
turns from naturally sporadic and non-uniform into or-
dered discrete. The maximum of the effect is observed
in the close to yield stress of samples. In case of sam-
ples alloyed by a small amount of impurity the effect
grows. There is an optimum interval of speeds of de-
formation in which the electric current in the maximal
degree lowers the resistance of metal to plastic deforma-
tion. In a wide interval of temperatures (from 80 up to
300 К) the magnitude of effect practically does not de-
pend on temperature [2].
The similar phenomena are observed at irradiation of
metal by packages of accelerated electrons pulses.
Combination of influence of current and irradiation
leads to intensification of effect of durability decrease of
metal. Under electronic influence (current and irradia-
tion) the probability of brittle fracture of samples at ini-
tial stages of deformation decreases and the crystals
being deformed under simultaneous influence of current
and irradiation are characterized by decrease of the crit-
ical cleavage stress, reduction of the factor of hardening
and increase of the speed of creep [2].
It is shown, that with increase in electron energy be-
yond a atom knocking-out threshold (in case of zinc
7,0≈trE МeV) the effect of radiation hardening is im-
posed on radiation plasticizing of metal due to creation
of additional stoppers for dispositions as dot defects and
their ensembles. At increase of electron density in one
pulse the effect of radiation plasticizing at first grows
and then decreases. Recession of the effect is explained
by influence of probable partial degeneration of elec-
tronic gas in metal on movement and interaction of dis-
positions [2].
It is shown that the activation volume after irradia-
tion of metal by electrons essentially does not change
and also the increase of creep speed is explained by re-
duction of time (increase of frequency) of the process of
thermally activated overcoming of obstacles by disposi-
tions [2].
1. MATHEMATICAL MODEL
For quantitative description of dynamics of electron-
phonon system of a metal film in work [3] an important
simplifying assumption about Fermi form of isotropic
parts of electron distribution function with time-
dependent electron temperature has been used. Though
authors [3] also mention that introduction of electron
temperature being equivalent to frequently used as-
sumption about instant thermalization of electron sub-
system not always can be strictly proved. So in area of
very low temperatures
*TTe < (temperature FDTT ε/* 2≈ ,
TD − Debye temperature) where electron-electron colli-
sions dominate above the electron-phonon collisions the
electron distribution function becomes thermodynami-
cally equilibrium (Fermi-Dirac distribution function)
during characteristic time periods of electron-electron
interaction eeτ . In usual rather pure metals Т* ~ 1 K and
in deliberately polluted films where electron-electron
interaction is being amplified because of effects of weak
localization Т* can be about 10 K. At temperatures of
*TTe > but De TT < electron thermalization in rather
thick films occurs not as a consequence of direct elec-
tron-electron interaction but because of indirect interac-
tion in form of phonon exchange. It has been shown
earlier by one of the authors [3] that the electron distri-
bution function which is close in the form to Fermi dis-
tribution function is formed also in rather thin films
ISSN 1562-6016. ВАНТ. 2013. №4(86) 274
(nonequilibrium phonons leave it and go to the substrate
without being reabsorbed by electrons) as a result of
only the process of phonon emission by "hot" electrons.
In both cases a characteristic time of electron thermali-
zation is the time electron-phonon collisions epτ .
Let's notice also that in optically thick films the uni-
formity of electron temperature on thickness of a film is
provided by the fast leaving of electrons the area of the
skin-layer and high electron heat conductivity in com-
parison with the phonon one [3]. Owing to additional
diffusion reduction of "hot" electron density the ther-
malization speed of the electron subsystem essentially
grows and consequently in optically thick films the ap-
proach of instant thermalization gives good agreement
of the theory with experiment. In work [3] the case of
small "heatings" was considered, we when considering
EPE deal with very significant "heatings", therefore it is
necessary for us to carry out consecutive kinetic consid-
eration of both electron and phonon subsystems what
makes the basic content of the given article.
At the kinetic description the electron behaviour
submits to Boltzmann equation for electron distribution
function ),,( tprf
rr with corresponding integrals of col-
lisions
edepee III
p
f
dt
pd
r
f
t
f
++=
∂
∂
⋅+
∂
∂
+
∂
∂
r
r
r
r
υ , (1)
[ ]{ }),(,),( trBtrEe
dt
pd rrrrrr
υ+= ,
where eeI is the integral of collisions of electrons with
electrons, epI is the integral of collisions of electrons
with phonons, edI is the integral of collisions electrons
with impurities and lattice defects υ
r
is the velocity, r
r
is the radius-vector, pr is the momentum, t is the time,
E
r
is the electric field intensity, B
r
is the magnetic in-
duction. Further we shall consider the magnetic field
absent.
Let's write down the integrals of collisions:
( ) ( )
×−−−+
×−−+= ∫
))(1)(()()){()()()((
,/,
32321
321321321
pfpfpfpppp
ppppppppwpdpdpdI eeee
rrrrrrr
rrrrrrrrrrr
εεεεδ
δ
))},(1))((1)(()())(1( 3211 pfpfpfpfpf
rrrrr
−−−−×
( ) ( )
[ ]
[ ]
( )},)()()(
}1)())(1)(()())(1(
)({)}())(1)((1)())(
1)(({)()()({
qpqp
qNqpfpfqNpf
qpfqNqpfpfqNpf
qpfqpqpqwqdIep
r
h
rrr
rrrrrr
rrrrrrrr
rrr
h
rrrrr
Ω+−+×
×+−−−−×
×−++−−+
−+Ω−−+= ∫
εεδ
εεδ
( ) ( ){ },)()()()({ pfpfppppwpdI eded
rrrrrrr
−′−′−′′= ∫ εεδ
mpp 2/)( 2=
r
ε , υ
rr
fj = is the density of electric cur-
rent, the symbol of averaging
is understood as mul-
tiplication by 3)2/(2 hπ and integration over pdv , h −
Planck's constant, m is the electron mass, q
r is the pho-
non momentum.
The phonon distribution function also submits to the
kinetic equation with integrals of collisions:
pdpppeq III
r
qN
t
qN
++=
∂
∂
+
∂
∂
r
r
r
r
)()( υ , (2)
where peI is the integral of collisions of phonons with
electrons, ppI is the integral of collisions of phonons
with phonons, pdI is the integral of collisions phonons
with impurities and defects of lattice, qq
r
h
r
∂Ω∂= /υ is
the phonon speed.
( ) ( )
[ ] )},())(1)((1)())(1)(({
)()()(
qNqpfpfqNpfqpf
qpqpqwpdI pe
rrrrrrrr
r
h
rrrrr
+−−+−+×
×Ω−−+= ∫ εεδ
( )[ ],)()( qNqNqI Tpppp
rrr
−−= ν
( )[ ],)()( qNqNqI pdpd
rrr
−−= ν
where ( )[ ] 11/exp)( −−Ω= TqNT h
r
is the thermodynami-
cally equilibrium phonon distribution function (Bose-
Einstein's distribution function), ∫= dOqNqN )(
4
1)(
r
π
is the phonon distribution function averaged by angles.
Taking into account that impacts of electrons with im-
purities, phonons and defects result in isotropization of
electron distribution function we shall search it as
p
ppfpftpf
rrr
))(())((),( 1 εε += . (3)
Taking into account that in collisions of electrons
with phonons the transfer of energy is very small we
shall simplify the integrals of collisions of electrons
with phonons, namely we shall expand the isotropic a
part of electron distribution function into series by small
transfer of energy down to square-order summand:
( )
+Ω
∂
∂
±=Ω±≡± hh
rrr
ε
εεε )())(())()(()( pfpfqpfqpf
( ) ( )22
2
( )
.
2
f pε
ε
∂ Ω
+ ⋅
∂
h (4)
Also we shall substitute this decomposition into the
integrals of collisions that will allow to simplify them
essentially so the integral of collisions of electrons with
phonons will become
{ }
( ) [ ] )5(},)()())(1)(()(
2
1)()({2)(
8
0
2
8
0
2
∫
∫
Ω−+Ω×
×⎥⎦
⎤
⎢⎣
⎡ +
∂
∂
∂
∂
=
ε
ε
εε
εεε
πε
m
m
ep
qqdqqwffq
qNqqdqwfmfI
hh
{ } [ ]
( ) }.)(
2
)()](1)([
))(1)(({)(2)(
2
22
8/
2
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂Ω
+
∂
∂
Ω−+×
×−= ∫
∞
ε
ε
ε
εε
εεεπ
fffqN
ffd
q
qwmqNI
mq
pe
h
h
(6)
Taking into account a concrete kind of dependence
of probability of transition )(qw and also frequencies of
impacts of phonons with phonons )(qppν
qwqw 0)( = ,
( ) s
w A
ρπ
ε
hh 3
2
1
0 22
= and 2
0)( qq pppp νν = ,
where
cDc
pp MTa
sT
4
3
0 =ν , sqq =Ω )(h , A1ε is a constant
of deformation potential, T is the temperature of lat-
tice, DT is the Debye temperature, ca is the lattice
spacing, cM is total weight of two atoms, s is the sound
ISSN 1562-6016. ВАНТ. 2013. №4(86) 275
speed, ρ is the substance density.
As a result of a concretization we shall obtain
{ }
[ ] )7(,}))(1)((
2
1)({2)(
8
0
3
0
8
0
42
0
2
∫
∫
−
+⎥⎦
⎤
⎢⎣
⎡ +
∂
∂
∂
∂
=
ε
ε
εε
εεε
πε
m
m
ep
dqqswff
qNdqqswfmfI
{ } [ ]
( ) )8(},)(
2
)()](
1)([))(1)(({2)(
2
22
8/
0
2
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
∂
∂
−
−++−= ∫
∞
ε
ε
ε
εε
εεεπ
fsqfsqf
qNffdwmqNI
mq
pe
[ ],)()(0 qNqNqI Tpppp
rr
−−= ν (9)
p
pf
p
pfI eded
rrrr
)()( 11 ενε −=
⎭
⎬
⎫
⎩
⎨
⎧
, (10)
,)()( 11 p
pf
p
pfIep
rrrr
ενε −=
⎭
⎬
⎫
⎩
⎨
⎧
(11)
where ⎥⎦
⎤
⎢⎣
⎡ += ∫ 2
1)()(
8
0
3
3
0 qNdqq
m
w mε
ε
πεν , edν is the
frequency of collisions of electrons with impurities and
lattice defects which in the case considered (low tem-
peratures) determine the isotropization of the electron
distribution function
( )
=
∂
∂
−
∂
∂
p
pfEe
p
p
t
f
rrrr
ε
ευ1
p
pfed
rr
)(1 εν− . (12)
Considering the anisotropic additive to electron dis-
tribution function 1f
r
as stationary and neglecting spa-
tial dispersion we obtain final system of two equations
for isotropic distribution functions of electrons )(εf
and acoustic phonons )(qN which is to be solved:
[ ] ,}))(1)((
2
1)({2
1
3
2
8
0
3
0
8
0
42
0
2
2/3
2/1
2
2
ee
m
m
ed
Idqqswff
qNdqqswfm
fE
m
e
t
f
+−+
+⎥⎦
⎤
⎢⎣
⎡ +
∂
∂
∂
∂
=
=⎥⎦
⎤
⎢⎣
⎡
∂
∂
∂
∂
⋅−
∂
∂
∫
∫
ε
ε
εε
εεε
π
ε
ε
εεν
+−++=
∂
∂
∫
∞
))(1)(({2)(
8/
2
0
2
εεεπ ffdmwII
t
qN
mq
pdpp
[ ] ( ) })(
2
)()(1)( 2
22
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
∂
∂
−++
ε
ε
ε
εε fsqfsqfqN . (13)
Both electron )(εf and phonon )(qN distribution
functions are dimensionless values which satisfy such
normalization conditions
ndfm
=⎟
⎠
⎞
⎜
⎝
⎛
∫
∞
εεε
π
)(2
2
1
0
2/1
2/3
22 h
, (14)
where n is the electron density in the valence band (for
metals it is also the conduction band as it is filled only
in part).
∞<⎟
⎠
⎞
⎜
⎝
⎛
∫ dqqNq
Dq
)(1
2
1
0
2
32 hπ
. (15)
Where Dq is Debye phonon momentum which is
determined from the equality
DDB sqTk = ,
where 161038.1 −⋅=Bk erg/К is the Boltzmann constant,
s is the sound speed, DT is the Debye temperature.
Particularly further all quantities are taken for nickel
(for which it is possible to compare calculations with the
experimental results [4] s =5 105 cm/s, DT = =375 K,
whence the maximal phonon momentum will be
1019 g cm/s, the electron density in the valence band
n =2.5·1022 cm-3, the density of nickel 9.8=ρ g cm-3,
the lattice spacing (distance between the neighbouring
atoms) 8105.2 −⋅=a cm.
In the state of thermodynamic equilibrium the elec-
tron distribution function )(εf is the Fermi-Dirac func-
tion
1
1exp)(
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
=
eB
F
Tk
f εεε , (16)
where Fε =5 10-12 erg, eT is the temperature of electron
component (in experiments it was equal to 20 and 80 К)
and initially was equal to lattice (phonon) temperature
(before applying of the electric field E =0.08 CGSE
(24 V/cm)). By the residual resistance of nickel
curρ =1.25·10-6 Ohm cm, we find the frequency of elec-
tron collisions with impurity and lattice defects
edν =7.5·1012 s-1. For convenience of the further re-
search and numerical modelling of the system of equa-
tions for electron and phonon distribution functions (13)
we shall introduce dimensionless variables: momenta of
electrons and phonons divided by ( ) 2/12 eBTmk , thus
energies of electrons ε and phonons sqph =ε will be
divided by eBTk , time we shall measure in characteris-
tic times of electron-phonon impacts
( ) 7
2
1
3
3
0 102 −==
A
ep sm επ
ρπτ hh s. Then the system of equations
becomes:
[ ] )17(},~~2))~(1)(~(
2
1)~(
~~
~{~~~
~
~~
4~
~
~4
0
3
~4
0
4
2/1
2/3
2/1
∫
∫
−+⎥⎦
⎤
⎢⎣
⎡ +
×
∂
∂
∂
∂
=⎥⎦
⎤
⎢⎣
⎡
∂
∂
∂
∂
⋅Δ−
∂
∂
εα
εα
εεαεεε
εε
εεε
β
ε
ε
εε
ε
phphph
phph
dffN
dff
t
f
[ ]
}~
)~(~
~
)~(~2)]~(1
)~([))~(1)(~({~
~
)~(
2
2
22/1
16
~2
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
+
∂
∂
−+
++−=
∂
∂
∫
∞
ε
εε
ε
εεαε
εεεεγ
ε
α
ε
fff
Nffd
t
N
phph
ph
ph
ph
(18)
where
0
~
ep
tt
τ
= ,
eBeB Tmk
p
Tk 2
~
2
==
εε ,
eBTmk
pp
2
~ = ,
eBTmk
qq
2
~ = ,
eBTk
ms
2
2
=α ,
eBed
ep
Tkm
Ee
ν
τ
ε
6
~ 0
22
=Δ ,
2/51 )(8 αβ ⋅=− ,
eB
ph Tk
sq
=ε~ , αγ 21 =− .
ISSN 1562-6016. ВАНТ. 2013. №4(86) 276
Let's copy system of the equations (17)-(18) with in-
tegration over momenta
[ ]
2
2 2
2 2
0
2
3
0
( ) 1 ( ) 1 ( ){
1 1( ) ( )(1 ( )) }, (19)
2
p
p
f p f p f pp dqq
t p p p p pp p
N q f p f p dqq
ε
α
⎡ ⎤∂ ∂ ∂ ∂ ∂
− Δ = ×⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦
⎡ ⎤× + + −⎢ ⎥⎣ ⎦
∫
∫
%
%
% % %
% % % %
% % % % % %% %
% % % % %
[ ]
( )20},~~
)~(
~~2
~
~~
)~(~1)]~(
1)~([))~(1)(~(1{~~~
)~(
2
2/~
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
∂
∂
+
∂
∂
−
−++−=
∂
∂
∫
∞
pp
pf
pp
q
pp
pfqpf
qNpfpfppd
t
qN
q
α
α
where 85~20 ≤< p , 3.0~ =Δε , 1800~0 ≤< ε ,
18~0 ≤< phε , 5.42~~0 =≤< Dqq , 045.0=α .
Integration of the resulted system of equations was
carried out with the help of completely conservative
difference schemes. Conservatism of the scheme is the
obligatory requirement as it provides absence of accu-
mulation of mistakes in calculations on long time inter-
vals (see, for example, [5]). It is essential to satisfy sev-
eral conversation laws. In this case the satisfaction of
energy and particles conversation laws if needed.
2. RESULTS OF NUMERICAL MODELING
AND THEIR DISCUSSION
As a result of the numerical calculations carried out
non-stationary distribution functions of electrons )~( pf
and phonons )~(qN of momenta have been found. On fig.
1 the dependence of distribution function of electrons on
the dimensionless momentum in various time moments
is shown. The leftmost curve corresponds to a thermo-
dynamically equilibrium distribution function (16)
which is taken as initial at the solution of equation sys-
tem (19), (20).
Fig. 1. Dependence of electron distribution function f in
various moments of time (t=0; 0.1; 0.2; 0.3; 0.4) on the
dimensionless momentum, curves are located from the
left – to the right accordingly
One can see from Fig. 1 (curves with increase of
time interval displace from the left − to the right) that in
time the electron distribution function differs from
thermodynamically equilibrium more significantly get-
ting high-energy "tail".
Thus it is established that energy received from ex-
ternal electric field as a result of electron-phonon colli-
sions partially (a small part because of quasielastic elec-
tron-phonon collisions) is transferred from the electron
subsystem to the phonon subsystem and most part it is
being spent not on establishment of thermodynamically
equilibrium electron distribution function (as it is fre-
quently supposed (see [1 - 4])) but on formation of in-
tensive high-energy "tails" of the electron distribution
function.
Fig. 2. Dependence of the phonon distribution function
in various dimensionless time moments ( t =0; 0.1; 0.2;
0.3; 0.4; 0.5) on the dimensionless momentum for the
electric field E=24 V/cm. Curves are located from
below - upwards accordingly
Such cardinal change of the electron distribution func-
tion results in formation of phonon distribution function
that is very strongly enriched by phonons with the en-
ergy close to the Debye energy (see, Fig. 2 and Fig. 3)
in contrast to work [4] in which the phonon distribution
function in this area of momentum practically corre-
sponds to Bose-Einstein function but with the tempera-
ture corresponding to temperature of the electron sub-
system.
Fig. 3. Dependence of phonon distribution functions in
various dimensionless time moments ( t =0; 10; 15) on
the dimensionless momentum for small intensity
of electric field E=1 V/cm. Curves are located from
below - upwards accordingly
As if has been shown by our numerical modelling
the "temperature" (to speak more precisely average elec-
tron energy because the obtained electron distribution
function strongly differs from thermodynamically equi-
librium) electron distribution function changes insig-
nificantly, i.e. thermalization of the energy received
from electric field does not occur but high-energy "tails"
which lead to such cardinal to change of phonon distri-
bution functions are formed.
Further we shall carry out comparison of depend-
ences of the product of phonon distribution functions on
the dimensionless momentum cubed for thermodynami-
cally equilibrium situation (Bose-Einstein's function (at
the moment of time t=0) Fig. 4) and phonon distribution
functions in various time moments after the beginning
of the electric field action (Fig. 5)
ISSN 1562-6016. ВАНТ. 2013. №4(86) 277
Fig. 4. Dependence of the product the thermodynamic-
cally equilibrium phonon distribution function
(Bose-Einstein's functions (at the dimensionless time
moment t =0)) on cube of the dimensionless momentum
vs the dimensionless momentum
Apparently from Fig. 5 the phonon distribution func-
tions in due course have more and more powerful high
energy "tails" as at electron-phonon impacts the mo-
mentum is transferred by enough small transfer of en-
ergy; much phonons with Debye energy is born i.e. pho-
non DF is enriched by the Debye phonons.
Fig. 5. Dependence of the product of the phonon
distribution function (in the dimensionless time moments
t =0.1; 0.2; 0.3; 0.4) on cube of the momentum
vs the dimensionless momentum. Curves are located
from below - upwards accordingly
As
( )
17
3
2
1
3
1 10
2
−− == csm A
ep
ρπ
επτ
hh
, cep
710−=τ is elec-
tron-phonon interaction time on which the system of the
connected equations for electron and phonon distribu-
tion functions are normalized.
The neglect during the calculation by electron-
electron impacts will be fair on times on which the en-
ergy received by the electron subsystem from electric
field would not exceed the initial energy. Therefore we
shall find the time τ for which the energy received by
unit of volume from electric field with intensity of
24=E V/cm, will heat up sample with specific resis-
tance currρ =1.25·10-6 Ohm cm on eTT =Δ =20 K, i.e.
will double its reference temperature
e
curr
TcE
Δ= ρτ
ρ
2
,
( )
1.0
10124
201025,19.81025
72
63
2 =
⋅⋅
⋅⋅⋅⋅
=
Δ
=
−
−−
ep
curr
ep E
Tc
τ
ρρ
τ
τ .
The increase of energy received by unit of volume of
a sample from electric field for this time will make
e
curr
TcE
Δ= ρτ
ρ
2
=4.5 J/cm3.
Full energy which will be contained in unit of vol-
ume of the sample by the moment τ, appears equal
=⋅⋅= − 409.810382)40( 3
eTKc ρ 13.6 J/cm3.
Energy which is contained in unit of volume of the
sample in its phonon subsystem we find from expres-
sion
( )
( ) ( ) ( ) =⋅= ∫∫
DD qq
eB qNqdqqNqdqTmks
~
0
32
~
0
32
3
~~~106~~~2
2
4
hπ
π
= 71018⋅ erg/cm3=18 J/cm3.
The integral is found from the schedule rice 5 for the
moment of time 0.1.
In papers [6, 7] a detailed analysis of influence of
such abnormal behaviour is carried out by study of elec-
tron-phonon systems in a strong electric field on behav-
iour of samples under mechanical load that allows to
explain abnormal electro-plastic properties of metals
and semiconductors observable in experimental re-
searches.
CONCLUSIONS
Formation of non-stationary non equilibrium distri-
bution functions of electrons and phonons is investi-
gated at action on metal of a strong pulse electric field.
For concreteness parameters are taken for nickel having
reference temperature of 20 K. It is shown: isotropiza-
tion of electron DF occurs as a result of impacts with
imperfections of lattice; •electron DF does not become
thermodynamically equilibrium as electron-electron
impacts in the given situation give essentially smaller
contribution than electron-phonon collisions and colli-
sions with "another's" subsystem do not result in ther-
malization; •Distribution functions of electrons and
phonons have high energy "tails" as in electron-phonon
impacts the momentum is transferred at enough small
transfer of energy; much phonons with Debye energy
are born i.e. phonon DF is enriched by Debye phonons.
Such behaviour of electron-phonon system in strong
electric field allows to explain abnormal electro-plastic
properties of metals and semiconductors observable in
experimental researches (in more detail see [6, 7]).
REFERENCES
1. V.E. Gromov, V.J. Tsellermayer, V.I. Bazaykin.
Electrostimulation drawing: the analysis of process
and a microstructure. M.: «Nedra», 1996, 166 p.
2. V.I. Spitsyn, O.A. Troitsky. Research of electronic
influence on plastic deformation of metal // Metal
Physics. 1974, v. 51, p. 18-45.
3. A.I. Bezuglyj, V.A. Shklovsky. Kinetics of low tem-
peratures electron-phonon relaxations in a metal film
after instant electron heating // JETP. 1997, v. 111,
iss. 6, p. 2106-2133.
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4. N. Perrin and H. Budd. Phonon Generation by Joule
Heating in Metal Films // Phys. Rev. Lett. 1972,
v. 28, iss. 26, p. 1701-1703.
5. V.I. Karas`, I.F. Potapenko. Quasistatsionary distri-
bution functions of particles for the equations of
Landau-Fokker-Planck type at presence of sources //
Journal of Calculation Mathematics and Mathe-
matical Physics. 2006, v. 46, № 2, p. 307-317.
6. V.I. Dubinko, V.I. Karas`, V.F. Klepikov,
P.N. Ostapchuk, I.F. Potapenko. Modelling of in-
crease of plasticity of materials under action of puls-
es of an electric current // Problems of Atomic Sci-
ence and Technology. Series «Physics of Radiating
Damages and Radiating Material Science». 2009,
v. 4-2, p. 158-164.
7. V.E. Zakharov, V.I. Karas`. Nonequilibrium Kol-
mogorov-type particle distribution and their applica-
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Article received 17.04.2013.
КИНЕТИКА НЕРАВНОВЕСНОЙ ЭЛЕКТРОН-ФОНОННОЙ СИСТЕМЫ
ДЛЯ ПОЛУПРОВОДНИКОВ И МЕТАЛЛОВ В СИЛЬНОМ ЭЛЕКТРИЧЕСКОМ ПОЛЕ
В.И. Карась, И.Ф. Потапенко, А.М. Власенко
Исследовано формирование нестационарных неравновесных функций распределения (ФР) электронов и
фононов под действием сильного импульсного электрического поля на металл. Для конкретности были взя-
ты параметры для никеля, имеющего начальную температуру 20 K. Показано, что: •изотропизация элек-
тронной ФР обусловлена столкновениями с дефектами решетки; •электронная ФР не становится термодина-
мически равновесной, так как электрон-электронные столкновения в данной ситуации дают существенно
меньший вклад, чем электрон-фононные соударения, а столкновения с «другой» подсистемой не приводят к
термализации; •ФР электронов и фононов имеют высокоэнергетичные «хвосты», так как в электрон-
фононных столкновениях передается импульс с достаточно малой передачей энергии; •рождается много
фононов с дебаевской энергией, т.е. фононная ФР обогащена дебаевскими фононами.
KІНЕТИКА НЕРІВНОВАЖНОЇ ЕЛЕКТРОН-ФОНОННОЇ СИСТЕМИ ДЛЯ НАПІВПРОВІДНИКІВ
І МЕТАЛІВ У СИЛЬНОМУ ЕЛЕКТРИЧНОМУ ПОЛІ
В.І. Карась, І.Ф. Потапенко, А.М. Власенко
Досліджено формування нестаціонарних нерівноважних функцій розподілу (ФР) електронів та фононів
під дією сильного імпульсного електричного поля на метал. Для конкретності були взяті параметри для ні-
келю, що має спочатку температуру 20 K. Показано, що: •ізотропізація електронної ФР обумовлена зіткнен-
нями з дефектами решітки; •електронна ФР не стає термодинамічно рівноважною, тому що електрон-
електронні зіткнення в такій ситуації дають суттєво менший внесок, ніж електрон-фононні зіткнення, а зітк-
нення з «іншою» підсистемою не приводять до термалізації; •ФР електронів та фононів мають високоенер-
гетичні «хвости», тому що в електрон-фононних зіткненнях передається імпульс з досить малою передачею
енергії; •народжується багато фононів з дебаєвською енергією, тобто фононна ФР збагачена дебаєвськими
фононами.
|