Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим
У роботi представлено результати аналiзу, якi пояснюють загальну тенденцiю в особливостях процесу поширення тепла в напiвпровiдникових структурах на основi Si з модифiкованими властивостями приповерхневого шару при опромiненнi їх коротким лазерним iмпульсом. Показано, що наявнiсть структурної неодно...
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| Дата: | 2010 |
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| Мова: | Українська |
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Відділення фізики і астрономії НАН України
2010
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим / Р.М. Бурбело, М.В. Ісаєв, А.Г. Кузьмич // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 318-322. — Бібліогр.: 7 назв. — укр. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860249889893515264 |
|---|---|
| author | Бурбело, Р.М. Ісаєв, М.В. Кузьмич, А.Г. |
| author_facet | Бурбело, Р.М. Ісаєв, М.В. Кузьмич, А.Г. |
| citation_txt | Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим / Р.М. Бурбело, М.В. Ісаєв, А.Г. Кузьмич // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 318-322. — Бібліогр.: 7 назв. — укр. |
| collection | DSpace DC |
| description | У роботi представлено результати аналiзу, якi пояснюють загальну тенденцiю в особливостях процесу поширення тепла в напiвпровiдникових структурах на основi Si з модифiкованими властивостями приповерхневого шару при опромiненнi їх коротким лазерним iмпульсом. Показано, що наявнiсть структурної неоднорiдностi (модифiкованого шару) та врахування впливу нелiнiйної залежностi коефiцiєнта температуропровiдностi приводить до суттєвої трансформацiї областi (її зменшення), локалiзацiї теплової енергiї та збiльшення температури в приповерхневому шарi матерiалу.
В работе представлены результаты анализа, которые объясняют общую тенденцию процессов распространения тепла в полупроводниковых структурах на основе Si с модифицированными свойствами приповерхностного слоя при облучении их коротким лазерным импульсом. Показано, что наличие структурной неоднородности (модифицированного слоя) и учет нелинейной зависимости коэффициента температуропроводности приводит к существенной трансформации области (ее уменьшению) локализации тепловой энергии и увеличению температуры в приповерхностном слое образца.
We present the results explaining the general tendency in peculiarities of the process of heat distribution in semiconductor structures with modified properties of the surface layer under a pulse laser irradiation. It is shown that the presence of a structural inhomogeneity (modified layer) and the influence of a nonlinear dependence of the thermal diffusivity coefficient result in both a substantial transformation of the area of localization (its decrease) of thermal energy and an increase of the surface temperature.
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| first_indexed | 2025-12-07T18:41:50Z |
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| fulltext |
EVOLUTION OF TEMPERATURE DISTRIBUTION
EVOLUTION OF TEMPERATURE DISTRIBUTION
IN IMPLANTED Si-BASED STRUCTURES: PULSE
MODE OF LASER IRRADIATION
R. BURBELO, M. ISAIEV, A. KUZMICH
Taras Shevchenko National University of Kyiv, Faculty of Physics
(64, Volodymyrs’ka Str., Kyiv 01601, Ukraine; e-mail: rmb@ univ. kiev. ua )
PACS 65.40.-b
c©2010
We present the results explaining the general tendency in pecu-
liarities of the process of heat distribution in semiconductor struc-
tures with modified properties of the surface layer under a pulse
laser irradiation. It is shown that the presence of a structural
inhomogeneity (modified layer) and the influence of a nonlinear
dependence of the thermal diffusivity coefficient result in both a
substantial transformation of the area of localization (its decrease)
of thermal energy and an increase of the surface temperature.
1. Introduction
Photoacoustic (PA) diagnostics has lately been a rapidly
developing field of investigation of material properties.
This method is grounded on the photothermal trans-
formation (PTT) under no stationary heating of the
medium by electromagnetic radiation (e.g., by laser ra-
diation).
The recent growth of the interest in the PA diagnostics
is mainly related to the development of new PA meth-
ods and a successful application in materials science and
particularly in micro- and optoelectronics [1]. It should
be mentioned that the number of realized applications
of the PA effect is relatively large, while the physical
nature of the phenomenon is not well understood be-
cause of its complexity. It is mainly due to the fact
that the overall description of the effect involves a prop-
agation of fields of at least three types (light, thermal,
and elastic ones), the energy exchange between them,
and even the consideration of the electron-hole subsys-
tem in semiconductor materials. It is clear that there
are significant unanswered questions in the description
of the effect even for homogeneous continuous media.
While developing a model of the PA effect for inhomo-
geneous media (e.g., layered structures), this problem
becomes even more complicated. There are a lot of sim-
ilar unresolved problems ranging from the description of
a light absorption mechanism in inhomogeneous media
to the influence of interfaces on the propagation of ther-
mal waves induced by nonstationary light absorption,
and so on.
Today, the problem of calculating the time evolution of
a spatial distribution of temperature fields in a material
is very actual for various areas of materials science (for
example, for the determination of thermal parameters;
the calculation of a PA signal, which allows one to find
elastic constants of a material; the laser processing, etc.).
In [2], the analytical solution of the heat equation was
analyzed, by taking the outflow of heat from the surface
into account. The results testify that the temperature
curve maximum moves in the depth of a sample. In
that work, only structural homogeneous materials were
analyzed; the results obtained are not suitable for cal-
culating the thermal fields in inhomogeneous structures.
This problem was partially solved in [3], where the ther-
mal structure of a specimen was modeled with separate
layers, so that the coefficient of thermal diffusivity was
constant within the limits of each layer. In other words,
a homogeneous heat equation can be written for each
layer. By applying the Laplace transformation and intro-
ducing the resistivity matrix that describes the boundary
thermal resistance between adjacent layers, the transfer
matrices were deduced. Thus, the description of the dif-
fusion of heat in inhomogeneous structures was given,
by introducing the effective thermal conductivity (gen-
eralized to the entire structure). But the nonlinear pro-
cesses of heat diffusion cannot be considered within such
a method.
The purpose of the given work is to analyze the for-
mation of temperature profiles, as a result of the ac-
tion of short laser impulses (∼ 10−8 s) on semiconductor
silicon-based structures (Si is one of the basic materials
of modern microelectronics), in which the near-surface
layer properties have been essentially modified according
to technological requirements. These changes (e.g., see
[2], where the influence of the implantation of monocrys-
talline silicon on the coefficient of thermal diffusivity of
a material with modified structure was investigated) can
be as large as several orders of magnitude for the quan-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 317
R. BURBELO, M. ISAIEV, A. KUZMICH
tities that define thermal properties of materials. Since
the action of high-energy laser radiation on media repre-
sents a great interest, we will also analyze the nonlinear
dependence of the coefficient of thermal diffusivity on
the temperature.
2. Mathematical Model
Let us consider the following nonlinear equation of ther-
modiffusion dependent on time:
∂T
∂t
=
∂
∂z
(
D(T, z)
∂T
∂z
)
+ f(z)g(t), (1)
where D is the coefficient of thermal diffusivity which
depends in the general case on a spatial coordinate (in
inhomogeneous samples) and on the temperature; f(z)
is the function which characterizes a spatial distribution
of heat sources in the sample. In a case which is con-
sidered with regard for the Bouguer–Lambert law (the
heat source is a laser radiation absorbed by a material),
we have
f(z) =
I(1−R)α exp(−αz)
cρ
.
The function g(t) describes the temporal distribution of
the incident light intensity. In the case of a single im-
pulse, g(t) = H(t)−H(t−τ), whereH(t) is the Heaviside
function.
In all calculations, we will take I = 10 MW/cm2, τ =
20 ns, R = 0.37, α = 5×104 cm−1, c = 0.8 J/(g·K),
ρ = 2.3 g/cm3 as constant and will trace a change of the
coefficient of thermal diffusivity only.
The following boundary conditions are more often re-
alized in practice:
– (∂T/∂z)|z=0 = 0 – absence of heat outflow from the
sample’s surface in an external environment;
– T |z=zmax = 0 – contact of the sample’s bottom surface
with the thermostat (zmax = 300 µm – thickness of a
sample);
The initial conditions are as follows:
– T |t=0 – the uniform distribution of the temperature
in the sample before the irradiation (we will accept that
the initial temperature is zero without any loss of gen-
erality: we will consider only a rising over the initial
temperature).
According to our purpose, we will consider the cases
of irradiation of a homogeneous sample and a sample
with modified properties of a subsurface layer (two-
layer structure) by short laser impulses. For definite-
ness, we will consider monocrystalline silicon (p-type,
D0 = 0.94 cm2/s, Np ∼ 1012 cm−3) and two-layer struc-
ture “implanted layer + crystal substrate Si” – Sip+p+
(Np ∼ 1020 cm−3, Dp+ = 0.25 cm2/s). We set the mod-
ified layer thickness dp+ = 0.6 µm and assume that,
in the case of strong light absorption (α−1 = 0.2 µm),
practically all radiation is absorbed in the first layer. In
the first and second cases, we will estimate also a role of
the temperature dependence of the coefficient of thermal
diffusivity (D(T )).
3. Thermal Diffusivity does not Depend on
Temperature
3.1. Homogeneous sample (D = const)
Let us analyze the temperature distribution at the irra-
diation of a structurally homogeneous sample. We will
consider that the coefficient of thermal diffusivity does
not depend on the temperature (D = const). In this
case, a solution of Eq. (1) can be obtained analytically
in the form
T1(z, t) =
∞∑
n=0
fn(t) cos(anz), t ≤ τ,
∞∑
n=0
fn(τ)
exp(−anDt)
exp(−anDτ)
cos(anz), t ≥ τ,
fn(t) = An(1− exp(−a2
nDt)),
An =
2
zmax
I(1−R)α
cρDa3
n
(−1)n exp(−αzmax) + α/an
1 + (α/an)2
,
an =
(π
2
+ πn
) 1
zmax
.
In Fig. 1, we present the calculated temperature pro-
files at the irradiation of a Si sample by a laser impulse
with the duration τ = 20 ns at various moments of the
temporal cycle ‘ ‘heating — the heating end — cooling”.
Such temporal intervals are chosen to show the general
tendency of process of distribution of heat to show that
such a tendency holds during the whole cycle with heat-
ing and cooling.
3.2. Structurally inhomogeneous sample
(D = D(z))
We now analyze the temperature distribution at the ir-
radiation of an ion-implanted monocrystalline Si sam-
ple (two-layer structure Sip+p+ , the thickness dp+ = 0.6
µm). We will model the given structure by a system
318 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
EVOLUTION OF TEMPERATURE DISTRIBUTION
Fig. 1. Temperature distributions for a homogeneous sample at
various moments of the temporal cycle
which consists of two layers. It is possible to write the
coefficient of thermal diffusivity as
D(z) =
{
D1 = Dp+ , z < dp+ ,
D2 = D0, z ≥ dp+ ,
(2)
where D1 and D2 are the coefficients of thermal diffu-
sivity of the top and bottom (crystal substrate Si) lay-
ers, respectively. In Fig. 2, we present the relevant
temperature profiles calculated by the finite-element
method. As seen from Fig. 2,a, it is possible to describe
the temperature profiles in the sample by the function
T2(z, t) = F (z, t,D1, D2, dp+). The given function at a
point z = dp+ has the simple discontinuity of the first
derivative. It is a result of the “sharp border” model 2
and physically arises from the continuity of a heat flux
through the boundary between the first and second lay-
ers (D1(∂T/∂z)|z=p++0 = D2(∂T/∂z)|z=p+−0). Under
the condition D1 = D2, the shape of the given curve
passes in that of a curve which corresponds to a ho-
mogeneous sample (Section 3.1). The presence of such
a break has unforeseen consequences in the case where
one needs to conduct the subsequent calculations (e.g.,
the use of the Laplace transformation in calculations of
a PA signal leads to the appearance of “boundary fre-
quencies”). This break can be removed, if we replace the
“sharp border” model by the “transient layer” model, in
which the coefficient of thermal diffusivity changes not
by jump, but, for example, according to a linear law
D(z) = a + bz (see Fig. 2,b, insertion). Here, a and b
are constants which can be found from the condition of
a
b
Fig. 2. Temperature distributions for a two-layer structure Sip+p+
at various moments of the temporal cycle; the insert shows a model
dependence D(z) (a). Temperature distributions for a two-layer
structure Sip+p+ in the “transient layer” model at various moments
of the temporal cycle; the insert insert shows a model dependence
D(z) (b)
continuity of the function D(z):
D|z=p++0 = Dp+ ,
D|z=p++l−0 = D0,
where l is the transient layer thickness. In Fig. 2,b,
we give the temperature distributions for the “transient
layer” model. Evidently, the break is removed in this
case.
On the whole, by comparing the results of Sections
3.1 and 3.2, it is clear that the presence of a modified
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 319
R. BURBELO, M. ISAIEV, A. KUZMICH
Fig. 3. Temperature distributions in homogeneous Si for D(T )
(physical nonlinearity); the insert shows the model dependence
D(T (z, t)) at various moments of the temporal cycle
layer (structural heterogeneity) leads to a reduction of
the area of thermal energy localization.
4. Thermal Diffusivity Depends on
Temperature
4.1. Homogeneous sample (D = D(T ))
In this case, the situation becomes essentially more com-
plicated, as the coefficient of thermal diffusivity becomes
temperature-dependent. To find the temperature pro-
files, we use a modified finite-element method as in [7].
With regard for the fact that the thermal conductivity
in hyperpure silicon has the phonon character and using
experimental data [5], we get the explicit dependence of
the coefficient of thermal diffusivity on the temperature:
D(T ) = D0/(1 + aT ).
In Fig. 3, we give the results of calculations of the tem-
perature profiles in a homogeneous sample in the case
where the coefficient of thermal diffusivity depends on
the temperature.
Comparing results of Sections 3.1 and 4.1, it is clear
that the presence of the physical nonlinearity (depen-
dence D on T ) also leads to a reduction of the localiza-
tion area of thermal energy.
4.2. Structurally inhomogeneous sample
(D((T, z))
Let us consider a two-layer structure Sip+p+ (the thick-
ness dp+ = 0.6 µm). We consider that the top (mod-
Fig. 4. Temperature distributions for a two-layer structure Sip+p+ ;
the insert shows the dependence D(z, T (z, t)) at various moments
of the temporal cycle
ified) layer has a constant coefficient of thermal diffu-
sivity, because the scattering of phonons by crystal de-
fects (in the presence of impurities in the concentra-
tion indicated in Section 2) prevails over the phonon-
phonon scattering. For the bottom layer (substrate Si),
we take the coefficient of thermal diffusivity in the form
D(T ) = D0/(1 + aT ).
The scheme for calculations of temperature profiles
does not differ essentially at this point from that used
in Section 4.1. We note that, in this case (as in Section
3.2), the temperature profile curve has a discontinuity of
the first derivative, resulting from a difference in values
of the coefficients of thermal diffusivity of the first and
second layers. But it is smoothed out as a result of
the descending dependence of the coefficient of thermal
diffusivity on the temperature. In some cases (e.g., at
t � τ and t � τ), this break can become substantial.
That is why, for its diminution, we will use the “transient
layer” model with a linear law D(T, z) = a(T ) + b(T )z,
as it was made in Section 3.2. Here, a(T ) and b(T ) are
functions of the temperature, which can be found from
the conditions of continuity of the function D(z):
D|z=p++0 = Dp+ ,
D|z=p++l−0 = D(T ).
In Fig. 4, we present the temperature distributions
within the “transient layer” model.
Comparing the results given in Section 4, we will pay
attention to the presence of layers in a sample (struc-
tural heterogeneity D = D(z)) in the case where the
coefficient of thermal diffusivity depends on a temper-
320 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
EVOLUTION OF TEMPERATURE DISTRIBUTION
ature (physical non-linearity D = D(T )) and does not
lead to substantial differences in the curves of tempera-
ture distributions as distinct from the results in Section
3, where the dependence D = D(z) leads to a reduction
of the localization area of thermal energy.
We note that, in all cases, the amount of the absorbed
energy of a laser impulse is the same. Therefore, by
the law of energy conservation, a reduction of the area
of heat localization in the process of energy conversion
“light-heat” and the further diffusion of heat in a material
lead to an increase of the temperature in the surface layer
of the sample.
5. Conclusions
Here, we have considered the process of formation of
temperature profiles in spatially inhomogeneous silicon-
based structures at their irradiation by a short (τ = 20
ns) laser impulse. A modified finite-element method was
applied to the analysis of solutions of the nonlinear equa-
tion of thermodiffusion. This has given an opportunity
to find an approximate solution for systems that have a
layered structure with arbitrary ratios between the ther-
mal parameters of layers and values of their thicknesses.
We have demonstrated the difference of the processes
of formation of temperature distributions in cases where
the coefficients of thermal diffusivity depend or do not
depend on the temperature in homogeneous and inho-
mogeneous doped Si-based structures.
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Received 08.07.2009
ФОРМУВАННЯ ТЕМПЕРАТУРНИХ ПОЛIВ
В ЛЕГОВАНИХ СТРУКТУРАХ НА ОСНОВI Si
ПРИ ЛАЗЕРНОМУ ОПРОМIНЕННI:
IМПУЛЬСНИЙ РЕЖИМ
Р.М. Бурбело, М.В. Iсаєв, А.Г. Кузьмич
Р е з ю м е
У роботi представлено результати аналiзу, якi пояснюють за-
гальну тенденцiю в особливостях процесу поширення тепла в
напiвпровiдникових структурах на основi Si з модифiковани-
ми властивостями приповерхневого шару при опромiненнi їх
коротким лазерним iмпульсом. Показано, що наявнiсть стру-
ктурної неоднорiдностi (модифiкованого шару) та врахування
впливу нелiнiйної залежностi коефiцiєнта температуропровiд-
ностi приводить до суттєвої трансформацiї областi (її зменше-
ння), локалiзацiї теплової енергiї та збiльшення температури в
приповерхневому шарi матерiалу.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 321
|
| id | nasplib_isofts_kiev_ua-123456789-13404 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | Ukrainian |
| last_indexed | 2025-12-07T18:41:50Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Бурбело, Р.М. Ісаєв, М.В. Кузьмич, А.Г. 2010-11-08T14:34:56Z 2010-11-08T14:34:56Z 2010 Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим / Р.М. Бурбело, М.В. Ісаєв, А.Г. Кузьмич // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 318-322. — Бібліогр.: 7 назв. — укр. 2071-0194 PACS 65.40.-b https://nasplib.isofts.kiev.ua/handle/123456789/13404 535.16:534.341 У роботi представлено результати аналiзу, якi пояснюють загальну тенденцiю в особливостях процесу поширення тепла в напiвпровiдникових структурах на основi Si з модифiкованими властивостями приповерхневого шару при опромiненнi їх коротким лазерним iмпульсом. Показано, що наявнiсть структурної неоднорiдностi (модифiкованого шару) та врахування впливу нелiнiйної залежностi коефiцiєнта температуропровiдностi приводить до суттєвої трансформацiї областi (її зменшення), локалiзацiї теплової енергiї та збiльшення температури в приповерхневому шарi матерiалу. В работе представлены результаты анализа, которые объясняют общую тенденцию процессов распространения тепла в полупроводниковых структурах на основе Si с модифицированными свойствами приповерхностного слоя при облучении их коротким лазерным импульсом. Показано, что наличие структурной неоднородности (модифицированного слоя) и учет нелинейной зависимости коэффициента температуропроводности приводит к существенной трансформации области (ее уменьшению) локализации тепловой энергии и увеличению температуры в приповерхностном слое образца. We present the results explaining the general tendency in peculiarities of the process of heat distribution in semiconductor structures with modified properties of the surface layer under a pulse laser irradiation. It is shown that the presence of a structural inhomogeneity (modified layer) and the influence of a nonlinear dependence of the thermal diffusivity coefficient result in both a substantial transformation of the area of localization (its decrease) of thermal energy and an increase of the surface temperature. uk Відділення фізики і астрономії НАН України Тверде тіло Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим Формирование температурных полей в легированных структурах на основе Si при лазерном облучении: импульсный режим Evolution of Tempe¬rature Distribution in Implanted Si-based Structures: Pulse Mode Laser Irradiation Article published earlier |
| spellingShingle | Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим Бурбело, Р.М. Ісаєв, М.В. Кузьмич, А.Г. Тверде тіло |
| title | Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим |
| title_alt | Формирование температурных полей в легированных структурах на основе Si при лазерном облучении: импульсный режим Evolution of Tempe¬rature Distribution in Implanted Si-based Structures: Pulse Mode Laser Irradiation |
| title_full | Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим |
| title_fullStr | Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим |
| title_full_unstemmed | Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим |
| title_short | Формування температурних полів в легованих структурах на основі Si при лазерному опроміненні: імпульсний режим |
| title_sort | формування температурних полів в легованих структурах на основі si при лазерному опроміненні: імпульсний режим |
| topic | Тверде тіло |
| topic_facet | Тверде тіло |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/13404 |
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