Stabilization of thermal breakdown development in semiconductor films
The analysis of fixed points of the evolution equation for temperature in thermal fluctuation area in semiconductor film is done. It is shown that there exists a stable fixed point being more than the threshold of the breakdown regime development. Проведено аналiз нерухомих точок еволюцiйного рiвн...
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2004
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| Zitieren: | Stabilization of thermal breakdown development in semiconductor films / N.V. Andreyeva, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2004. — № 5. — С. 126-128. — Бібліогр.: 2 назв. — англ. |
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Andreyeva, N.V. Virchenko, Yu.P. 2015-04-18T20:21:02Z 2015-04-18T20:21:02Z 2004 Stabilization of thermal breakdown development in semiconductor films / N.V. Andreyeva, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2004. — № 5. — С. 126-128. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 75.50.L; 75.30.С https://nasplib.isofts.kiev.ua/handle/123456789/80553 The analysis of fixed points of the evolution equation for temperature in thermal fluctuation area in semiconductor film is done. It is shown that there exists a stable fixed point being more than the threshold of the breakdown regime development. Проведено аналiз нерухомих точок еволюцiйного рiвняння для температури в областi теплової флуктуацiї у напiвпровiдниковiй плiвцi. Доведено, що існує нерухома стійка точка, яка є більшою за величиною порогу виникнення режиму пробою. Проведен анализ неподвижных точек эволюционного уравнения для температуры в области тепловой флуктуации на полупроводниковой плёнке. Показано, что существует устойчивая неподвижная точка, большая по величине порога возникновения режима пробоя. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Взаимодействие релятивистских частиц с кристаллами и веществом Stabilization of thermal breakdown development in semiconductor films Стабiлiзацiя розвитку теплового пробою у напiвпровiдникових плiвках Стабилизация развития теплового пробоя в полупроводниковых плёнках Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Stabilization of thermal breakdown development in semiconductor films |
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Stabilization of thermal breakdown development in semiconductor films Andreyeva, N.V. Virchenko, Yu.P. Взаимодействие релятивистских частиц с кристаллами и веществом |
| title_short |
Stabilization of thermal breakdown development in semiconductor films |
| title_full |
Stabilization of thermal breakdown development in semiconductor films |
| title_fullStr |
Stabilization of thermal breakdown development in semiconductor films |
| title_full_unstemmed |
Stabilization of thermal breakdown development in semiconductor films |
| title_sort |
stabilization of thermal breakdown development in semiconductor films |
| author |
Andreyeva, N.V. Virchenko, Yu.P. |
| author_facet |
Andreyeva, N.V. Virchenko, Yu.P. |
| topic |
Взаимодействие релятивистских частиц с кристаллами и веществом |
| topic_facet |
Взаимодействие релятивистских частиц с кристаллами и веществом |
| publishDate |
2004 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Стабiлiзацiя розвитку теплового пробою у напiвпровiдникових плiвках Стабилизация развития теплового пробоя в полупроводниковых плёнках |
| description |
The analysis of fixed points of the evolution equation for temperature in thermal fluctuation area in
semiconductor film is done. It is shown that there exists a stable fixed point being more than the threshold of the
breakdown regime development.
Проведено аналiз нерухомих точок еволюцiйного рiвняння для температури в областi теплової флуктуацiї
у напiвпровiдниковiй плiвцi. Доведено, що існує нерухома стійка точка, яка є більшою за величиною порогу
виникнення режиму пробою.
Проведен анализ неподвижных точек эволюционного уравнения для температуры в области тепловой
флуктуации на полупроводниковой плёнке. Показано, что существует устойчивая неподвижная точка,
большая по величине порога возникновения режима пробоя.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/80553 |
| citation_txt |
Stabilization of thermal breakdown development in semiconductor films / N.V. Andreyeva, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2004. — № 5. — С. 126-128. — Бібліогр.: 2 назв. — англ. |
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| first_indexed |
2025-11-24T18:04:41Z |
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2025-11-24T18:04:41Z |
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1850491347158433792 |
| fulltext |
STABILIZATION OF THERMAL BREAKDOWN DEVELOPMENT
IN SEMICONDUCTOR FILMS
N.V. Andreyeva1, Yu.P. Virchenko2
1Belgorod State University, Belgorod, Russia
e-mail: virch@bsu.edu.ru
2Single Crystal Institute, NASU, Ukraine
e-mail: virch@isc.kharkov.com
The analysis of fixed points of the evolution equation for temperature in thermal fluctuation area in
semiconductor film is done. It is shown that there exists a stable fixed point being more than the threshold of the
breakdown regime development.
PACS: 75.50.L; 75.30.С
1. INTRODUCTION
In this communication, it is analysed the stabilisation
condition for the evolution regime which can appear
when the direct electrical current getting through the
thin semiconductor film. Such a regime generates the so-
called thermal breakdown. This phenomena consists of
the localisation of the extracted Joel heat in those areas
on the film where sufficiently large thermal fluctuation
are concentrated, it gives the strong heating of these
areas. Due to increasing dependence of the electrical
conductivity of the material on the temperature. Such
positive feedback may give the local increasing of the
temperature up to the melting temperature of the
material or to its eutectic point in the case of an intrinsic
semiconductor. This physical process is called {\it the
thermal breakdown}, it is developed during several
microseconds and gives the functional destruction of the
material.
2. THE EVOLUTION EQUATION AND THE
WAGNER APPROXIMATION
It has been obtained in Ref. [1] the evolution integral
and differential equation of the temperature field on the
film, which describes the breakdown development and
also one-dimensional solutions of this equation have
been analysed. It has been used such an approximation
when the effect of the voltage transfer during the
breakdown regime is neglected. It may; really, neglect
this effect at the origin stages of the breakdown regime
and if the circuit resistance being external to the film is
not sufficiently large. In this work, we shall analyse the
evolution equation taking into account the influence of
the voltage transfer. We shall show that if the external
resistance is sufficiently large then it is possible the
stabilisation of the evolution regime. As a result, the
melting temperature (the eutectic one) is not attained.
We start from the thermal conductivity equation in
the form
,)())(( Τ+∇∇=
⋅
QTTT κ (1)
where
.
)),()(1(
)()( 2''1
2
∫−+
ΤΕ=Τ
rdtrdS
Q σσ
σ
It describes the temperature distribution ),( ' trΤ on the
film when there are non-linear dependencies of the
temperature conductivity coefficient κ(T) and the
electrical conductivity )(Τσ . The integral denominator
in the second summand takes into account the effect of
the change of the voltage applying to the film during the
evolution process. In Eq. (1) E is the voltage on the film
in the equilibrium state, σ is the average electrical
conductivity characterising the external resistance, S is
the film square, d and is its thickness. Further, we shall
consider in Ref. [2] that κ(T)=κ is constant and
)2/)(1()( 2
0 mTTT −+= νσσ ,
mT is the temperature of the electrical conductivity
minimum.
We realise the investigation of the solution stability
of Eq. (1) using the approximation, which we name the
Wagner one. It was used in the thermal breakdown
theory in dielectrics. We consider that the heat
localisation may be modelled by introducing of the
thermal channels having a critical diamete D. These
channels are passing through the film. The temperature
in these channels is approximately constant when the
spatial point changing in them. It is changed essentially
in each of them only in thin boundary layer having the
thickness l. We put that the temperature is equal T(t)
126 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2004, № 5.
Series: Nuclear Physics Investigations (44), p. 126-128.
in channels outside these boundary layers. Outside
channels, it is put to constant temperature T0 of the
thermal surrounding.
The breakdown originates at mTT =0 . At these
conditions, the equation for relative temperature
fluctuation
0),(),(/2 '' TtrTtr −=Θ ν
is obtained on the basis of Eq. (1), if there is one
cylindrical channel with diameter D on the film. This
value is not equal to zero only in the channel. The
equation has the form
.
)))(1(1(
)1(2/
22
22
0
tΘ++
Θ+Ε+Θ∆=Θ
⋅
η
σνκ (2)
Here, we introduce non-dimensional parameter
)4/)(/( 2
0 SDπσση =
characterising the speed of the voltage transfer and we
denote 0)()( TtTt −=Θ . Averaging Eq. (2) over the
channel domain having the volume V and using the
transformation of integral on the volume to the integral
on the surface (it is considered that the heat flow aside
of the channel is absent), we obtain
=Θ∇=∆ Θ ∫∫ −− )(11 sdVdV υ
.4
lDlV
dD Θ−=Θ−= π
Here, we take into account that the heat flow through
cylinder side surface having the square Ddπ is directed
along the radial temperature gradient. This gradient we
change approximately by the finite difference lt /)(Θ .
At last, we simplify the averaging equation changing the
temporal scale tt ⇒Ε 22/ν and introducing the
decrement )/4(/2 2
00 Ε= σκνα lD ,
.
)))(1(1(
)(1)()( 22
2
t
ttt
Θ++
Θ++Θ−=Θ
⋅
η
α (3)
Our further analysis is reduced to the investigation of
the solution stability of this equation.
3. ANALYSIS OF THE SOLUTION STABILITY
If Θ in Eq. (3) is small then one can consider that
1)()( +Θ−=Θ
⋅
tt α
and, therefore, solutions are stable. If )(tΘ is
sufficiently large but the value η is very small such as
we may neglect by the value ))(1( 2 tΘ+η in
denominator then we obtain the equation
)()()( 2 ttt Θ+Θ−=Θ
⋅
α possessing the peaking regime
which describes the thermal breakdown development,
i.e. its solution
1).)
)0(
1(1()( −
Θ
−−=Θ tet ααα
α>Θ )0( goes to infinity during the finite time ∞t ,
which is identified with the breakdown time. Let us
study now the possibility of stabilisation in the case
when ))(1( 2 tΘ+η is sufficiently large in comparison
with the unit. In this case we neglect the unit at the
denominator of second summand in Eq. (3) and after
that we take away the extra parameter in the obtained
equation introducing the new temperature
Θ⇒Θ2/1η , the decrement ααη ⇒2/1 and the
time tt ⇒2/1η . As a result we obtain the equation with
one parameter
)).((
))(1(
)()()( 22
2
tf
t
ttt Θ≡
Θ+
Θ+Θ−=Θ
⋅
α (4)
Let us find some fixed points of this equation, i.e.
solutions 0)( =θf . Such solutions consist with 0=θ
and solution of the equation 22 )1( θαθ += . Let us
introduce the notation 21 θ+=X . Then this equation
has the form
.01)( 4 =+−= XXXP α
There is the unique minimum of the polinom )(XP in
the point 3/12 )4(*
−= αX . It satisfies the equation
0)(' =XP . There are not other fixed points different
from 0=θ at 0)( * ≥XP (the breakdown regime is
not realised). If
04/)4(31)( 3/12
* ≤−= −αXP ,
i.e. 4/)4/3( 32 ≤α , then there are two solutions
−+± > XXX , of the equation 0)( =XP . They are
both positive since 1)0( =P . At this case 1>+X is
fulfilled for sure, since 1* >X . Consequently, 1>−X
, since 0)1( 2 >= αP . Therefore, two fixed points ±θ
correspond to solutions ±X . It is easy checked that these
fixed points tend as 3/1~,~ −
+− αθαθ
asymptotically at small α .
Let us analyse the stability of those found fixed
points. For this, it is necessary to set the sign of the
derivative θθ ddf /)( in each of these points. It is
obvious in the point 0=θ that
αθθ −=0)/)(( ddf .
127
This is in accordance with the above conclusion about
its stability. In points ±θ we obtain
)1)(31()( 22'
±±± +−= θθαθf
on the basis 0=±θ . Since the following conditions
,3/4)4( 3/12
* >=> −
+ αXX
1)1(3)1(33 *
2 =−>−= ++ XXθ
take place. Consequently, the point +θ is stable and, the
point −θ should be unstable on the basis of some
topological arguments. Just this point corresponds to the
threshold of the fluctuation value from which the
breakdown is developed.
4. CONCLUSIONS
Thus, the stable fixed point +θ places above the
threshold point −θ . The "breakdown" solutions )(tΘ
are attained to this point and if its value corresponds to
the temperature +T being less the melting one (or the
eutectic one) then the breakdown is not realised. In this
case only some areas having very large temperature (the
mesoplasma channels) may be occur.
REFERENCES
1. Yu.P. Virchenko, A.A. Vodyanitskii. Heat
localisation and formation of heat breakdown structure in
semiconductor materials // Functional Materials. 2001,
v. 8, p. 428-434.
2. N.V. Andreyeva, Yu.P. Virchenko. Analysis
of the mathematical model of semiconductor
material thermal breakdown // Functional
Materials. 2003, v. 10,
p. 591-598.
СТАБИЛИЗАЦИЯ РАЗВИТИЯ ТЕПЛОВОГО ПРОБОЯ
В ПОЛУПРОВОДНИКОВЫХ ПЛЁНКАХ
Н.В. Андреева, Ю.П. Вирченко
Проведен анализ неподвижных точек эволюционного уравнения для температуры в области тепловой
флуктуации на полупроводниковой плёнке. Показано, что существует устойчивая неподвижная точка,
большая по величине порога возникновения режима пробоя.
СТАБIЛIЗАЦIЯ РОЗВИТКУ ТЕПЛОВОГО ПРОБОЮ
У НАПIВПРОВIДНИКОВИХ ПЛIВКАХ
Н.В. Андреєва, Ю.П. Вiрченко
Проведено аналiз нерухомих точок еволюцiйного рiвняння для температури в областi теплової флуктуацiї
у напiвпровiдниковiй плiвцi. Доведено, що існує нерухома стійка точка, яка є більшою за величиною порогу
виникнення режиму пробою.
128
1Belgorod State University, Belgorod, Russia
PACS: 75.50.L; 75.30.С
2. THE EVOLUTION EQUATION AND THE WAGNER APPROXIMATION
4. CONCLUSIONS
REFERENCES
Н.В. Андреева, Ю.П. Вирченко
Н.В. Андреєва, Ю.П. Вiрченко
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