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An analytical modeling of heat transfer for laser-assisted nanoimprinting processes

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ORIGINAL PAPER
F. -B. Hsiao
Æ
C. -P. Jen
Æ
D. -B. Wang
Æ
C. -H. ChuangY. -C. Lee
Æ
C. -P. Liu
Æ
H. -J. Hsu
An analytical modeling of heat transfer for laser-assistednanoimprinting processes
Received: 17 November 2004/ Accepted: 7 April 2005 / Published online: 4 June 2005
Springer-Verlag 2005
Abstract
Laser-assisted direct imprinting (LADI)technique has been proposed to utilize an excimer la-ser to irradiate and heat up the substrate surfacethrough a highly-transparent quartz mold preloadedon this substrate for micro- to nano-scaled structurefabrications. While the melting depth and moltenduration are key issues to achieve a satisfactoryimprinting pattern transfer, many material propertyissues such as crystalline phase alteration, grain sizechange and induced ﬁlm stress variation are stronglyaﬀected by transient thermal response. With one-dimensional simpliﬁcation as a model for the LADItechnique, the present paper has successfully derivedan analytical solution for the arbitrary laser pulsedistribution to predict the relevant imprinting param-eters during the laser induced melting and solidiﬁca-tion processes. The analytical results agree quite wellwith the experimental data in the literature and hencecan be employed to further investigate the eﬀects of LADI technique from laser characteristics (wave-length, ﬂuence and pulse duration) and substratematerials (silicon and copper) on the molten duration,molten depth and temperature distributions. Threekinds of excimer laser sources, ArF (193 nm), KrF(248 nm) and XeCl (308 nm) were investigated in thisstudy. For the silicon substrate, the melting durationand depth were signiﬁcantly dictated by the wave-length of laser used, indicating that employing theXeCl excimer laser with longer pulse duration (30 nsin the present study) will achieve the longest moltenduration and deepest melting depth. As for the coppersubstrate, the melting duration and depth are mainlyaﬀected by the laser pulse duration; however, thewavelength of laser still plays an insigniﬁcant role inLADI processing. Meanwhile, the laser ﬂuence shouldproperly be chosen, less than 1.4 J/cm
2
herein, so as toavoid the substrate temperature exceeding the soften-ing point of the quartz mold (
1950 K) and to makesure that the mold can still maintain the originalfeatures.
Keywords
Nanoimprinting
Æ
Pulsed laser
Æ
Heattransfer
Æ
Phase change
1 Introduction
Laser-material interaction has been intensively studiedover the past few decades. It can be categorized intotwo regimes consisted of laser ablation [1] and laserannealing processes. Laser ablation is widely used for alot of applications: for instance, micromachining [2, 3],laser deposition [4], nanoparticle fabricating [5] andso on. For the higher laser ﬂuence, the laser beamevaporates and ionizes the target material, creating aplasma plume with a crater on the target surface. Theresearch focuses span from the fundamental physical
Comput Mech (2005)DOI 10.1007/s00466-005-0688-zF. -B. Hsiao
Æ
C. -P. Jen (
&
)
Æ
D. -B. Wang
Æ
H. -J. HsuDepartment of Aeronautics and Astronautics,National Cheng Kung University,1 University Rd, Tainan 701, TaiwanE-mail: cpjen@mail.ncku.edu.twTel.: +1-480-727-8168Fax: +1-480-965-8555C. -H. ChuangMicro-Nano Research Center, National Cheng Kung University,1 University Rd, Tainan 701, TaiwanY. -C. LeeDepartment of Mechanical Engineering, National Cheng KungUniversity, 1 University Rd., Tainan 701, TaiwanC. -P. LiuDepartment of Materials Science and Engineering,National Cheng Kung University, 1 University Rd.,Tainan 701, TaiwanC. -P. JenCenter for Applied NanoBioscience, Biodesign Institute at ArizonaState University, Tempe, AZ, 85287-0808
process of ablation including the ionization process,energy transport [6, 7] and material removal mechanismto the phase and microstructure changes during andafter ablation [7–9]. On the other hand, for the lowerlaser ﬂuences of normally smaller than 1 J/cm
2
, thelaser beam anneals the targets, which are involved withmany material issues including crystalline phase alter-ation (such as amorphous to microcystalline, amor-phous to polycrystalline) [10, 11], grain size change [11,12], induced ﬁlm stress variation [13–15], and defectannihilation [16].Nano-imprinting lithography (NIL) has been devel-oped over a decade [17–21] and is now a promisingmethod for nano-patterning and nano-fabrication.Figure 1a illustrates the basic concepts of conventionalnano-imprinting, which includes a mold, an etchingresist layer and a sample substrate. The mold has somenano-scale features on the surface fabricated by eitherE-beam lithography (EBL) or focus ion bean (FIB)techniques. The resist layer usually used thermo-plasticpolymer such as poly-methylmethacrylate (PMMA). Byheating up the resist layer above its glass transitiontemperature (T
g
), the mold can impinge into the resistlayer and form a pattern. Following by reaction ionetching (RIE or ICP), the nano-pattern is transformedto the resist layer and the substrate. It is then followedby standard lithography processes to achieve nano-structures on the substrate surface. However, theheating mechanism of conventional nanoimprinting isslow and usually causes misalignment due to the dif-ferent thermal expansion between the mold and thesubstrate. Recently, laser assisted direct imprinting(LADI) method was proposed by Chou et al. [22]. Thisnew imprinting method shares some similar concepts of the nano-imprinting but with a much more straight-forward approach for nano-pattern transformation. Asdepicted schematically in Fig. 1b, LADI process utilizesa laser pulse to radiate on the sample surface which isin contact with and pre-loaded by a mold (fusedquartz) with pre-fabricated nano-scale features on itscontact side. Upon radiating the laser pulse on thesample surface, the near-surface materials melt and alaser-induced molten layer is formed, which allows themold to impinge into the sample directly. Subsequentcooling and solidiﬁcation of the molten layer will thencomplete the transformation of the nano-patterns fromthe mold to the sample. This LADI method has severalobvious advantages over the nano-imprinting techniquein terms of simplicity and eﬃciency, and thereforeshows a great potential for future nano-patterning andfabrication of nano-structures. The main purpose of the present study is to perform an analytical modelingof laser induced melting and solidiﬁcation duringLADI process and to predict the melting of the sub-strate theoretically.
2 Theoretical Modeling
Laser-material interaction between a pulse laser and asolid substrate is the key element of laser-assisted na-noimprinting. During the heating process of the sub-strate by the pulsed laser, one-dimensional transientheat-diﬀusion equation can be employed herein. Thephysical domain for the heat-transfer system consideredhere is illustrated in Fig. 1c. Assuming that the laser spotis larger than the mold and all materials involved ishomogeneous, therefore this system can be simpliﬁed asone-dimensional. The source term of heat-generation
Fig. 1a–c
Schematic diagramof nanoimprinting processes:
a
Conventional nanoimprintinglithography (NIL) and
b
laser-assisted direct Imprinting(LADI); and
c
illustration of LADI imprinting process andone-dimensional simpliﬁcation
srcinating from the energy absorption of a laser pulsecan be described as:
S
ð
x
;
t
Þ¼ð
1
R
Þ
b
exp
ð
b
x
Þ
I
ð
t
Þ ð
1
Þ
where
x
and
t
are the spatial variable and time, respec-tively.
R
is the reﬂectance,
b
is the optical absorptioncoeﬃcient and
I
ð
t
Þ
is the power density function of theincident laser pulse (power per unit area). A Gaussian-shaped laser pulse is assumed in the present study andthe power density function can be expressed as:
I
ð
t
Þ¼
I
max
ﬃﬃﬃﬃﬃﬃ
2
p
p
r
exp
ð
t
t
m
Þ
2
2
r
2
!
ð
2
Þ
where
I
max
is the maximum power density of the laserpulse at
t
¼
t
m
and
r
the standard deviation of thefunction and can be determined by the full-width at half-maximum (FWHM,
t
0
) of the pulse,
t
0
¼
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ln2
p
r
.
t
m
ischosen as the triple of the standard deviation (
r
) and theirradiating time is the double of
t
m
. For the LADI pro-cess, the incident laser propagates through two inter-faces, air(vacuum)/quartz and quartz/substrate. Thepulsed laser is assumed as normal incidence, thus, theoverall reﬂectance of the incident laser pulse can beevaluated by the following equation [23]:
R
¼ð
R
1
þ
R
2
2
R
1
R
2
Þð
1
R
1
R
2
Þ ð
3
Þ
where
R
1
¼
ð
n
0
n
1
Þ
2
ð
n
0
þ
n
1
Þ
2
and
R
2
¼
ð
n
1
n
2
Þ
2
þ
j
22
ð
n
1
þ
n
2
Þ
2
þ
j
22
.
n
0
(equals to 1.0)and
n
1
(is 1.46) are the refractive indices of air(vacuum)and quartz mold, respectively.
n
2
and
j
2
are the real andimaginary parts of the complex refractive index of thesubstrate. The absorption coeﬃcient is related to
j
2
, theimaginary part of refractive index, and the laser wave-length,
k
, by
b
¼
4
pj
2
k
[24].First of all, convective and radiative heat losses at thesurface of the substrate and heat absorption by thequartz mold are neglected herein. The thermal proper-ties and optical parameters are listed in Table 1. Sincethe constant parameters (independent of temperature)are employed in the present analytical modeling whichinvolves a wide range of temperature, the parametersshould be set reasonably. For instance, the thermalconductivity of silicon is about 130 W/mK at roomtemperature, but is around 10 W/mK at 1500 K. Thevalues of thermal properties are chosen based on theirtemperature-dependent curves. Based on the thermalconductivities of silicon (20 W/mK) and copper(352 W/mK) used in this study are much larger than thatof fused quartz which is 1.05 W/mK, the insulationcondition on the top surface of substrate can be as-sumed. The fused quartz mold is treated as highlytransparent to laser wavelength without energy absorp-tion. As a result of this assumption, the mold could bereasonably excluded from the heat transfer system andthe problem could be simpliﬁed as a semi-inﬁnite slab
ð
0
<x<
1Þ
as shown in Fig. 1c. Typically, the softeningpoint of the fused quartz is 1950 K (from the data sheetprovided by the manufacturer). At this point it will startto deform and slump under its own weight. The actualmelting point is not clearly deﬁned for fused quartz as itis for ‘‘softer glasses’’. Instead, fused quartz simply be-comes more and more soft and liquid, as its temperatureis raised over a broad range above the softening point.For imprinting, the temperature of the substrate shouldnot exceed the softening point of mold to maintain thesrcinal features on the mold. The superposing eﬀects of a large number of instantaneous sources (one-shot) areadopted herein to obtain the time- and spatial-depen-dent temperature proﬁle analytically. First, consider the
x
-direction through an inﬁnite domain with constantproperties. The equation for time-dependent heat con-duction in the
x
-direction is expressed as following:
o
h
o
t
¼
a
@
2
h
@
x
2
ð
4
Þ
where
h
ð
x
;
t
Þ¼
T
ð
x
;
t
Þ
T
i
is the excess temperaturerelative to the initial (far-ﬁeld) temperature of the do-main (
T
i
).
a
¼
k
q
C
is the thermal diﬀusivity of the mate-rial.
q
denotes the density,
C
represents the speciﬁc heat,and
k
is the material’s thermal conductivity. The excesstemperature distribution of the inﬁnite domain in whichan instantaneous plane source,
q
00
(J/m
2
), is releasedonce at t=0 is [27]:
h
ð
x
;
t
Þ¼
q
00
2
q
C
ð
pa
t
Þ
1
=
2
exp
x
2
4
a
t
ð
5
Þ
The initial and boundary conditions in the modelingdomain can be expressed as:
T
ð
x
;
t
¼
0
Þ¼
T
i
;
T
ð
x
¼1
;
t
Þ¼
T
i
;
o
T
o
x
x
¼
0
¼
0
ð
6
Þ
where
T
i
is the initial temperature of the substrate. Sincethe gradient of temperature at x = 0 equals to zero, thecalculated domain can be extend to an inﬁnite medium
ð1
<x<
1Þ
which is symmetric at x = 0. According tothe concept of the super position procedure, the mediumwhich is heated by the heat-generation sources,
S
ð
x
;
t
Þ
,the time-dependent temperature proﬁle is solvedanalytically and expressed as:
h
ð
x
;
t
Þ¼
Z
t
p
0
Z
11
S
ð
n
;
s
Þ
2
q
C
pa
ð
t
s
Þ½
1
=
2
exp
ð
x
n
Þ
2
4
a
ð
t
s
Þ
" #
d
n
d
s
ð
7
Þ
where
t
p
is the irradiating time of a laser pulse. Thus, thetemperatureinsidethe substrate isrewrittenasfollowing:
T
ð
x
;
t
Þ¼
Z
t
p
0
Z
11
S
ð
n
;
s
Þ
2
q
C
pa
ð
t
s
Þ½
1
=
2
exp
ð
x
n
Þ
2
4
a
ð
t
s
Þ
" #
d
n
d
s
þ
T
i
for
x
0
;
0
t
t
m
;
t
t
s
ð
8
Þ
where
t
m
and
t
s
are the times when melt and solidiﬁcationof the substrate material takes place, respectively. Thecalculated domain has been extended to an inﬁnitemedium; hence, the virtual heat sources have to imposein the negative region of the medium to conserve thetotal energy absorbed in the substrate. The source term
S
ð
n
;
s
Þ
should be written as:
S
ð
n
;
s
Þ¼ð
1
R
Þ
b
exp
ð
b n
j jÞ
I
ð
s
Þ
for
1
<
n
<
1ð
9
Þ
When the surface temperature of the substrate reachesthe melting point
T
m
(when
t
¼
t
m
), the boundary be-tween the solid and liquid phase begins to move from thesurface into the depth of the substrate. The equilibriummodel (i.e. the isothermal one-dimensional Stefanproblem [28]) is considered in this work. The tempera-ture at the solid/liquid interface (
x
¼
h
ð
t
Þ
) is constantand equal to the melting temperature
T
m
. However, themoving boundary can be treated as a moving heat sinkinside the substrate [25]. The temperature induced by theheat sink is approximated as a constant value of
T
q
¼
q
=
C
, where
q
is the latent heat of fusion. Therefore,the resultant temperature in the substrate can besuperposed after the onset of melting:
T
ð
x
;
t
Þ¼
Z
t
p
0
Z
11
S
ð
n
;
s
Þ
2
q
C
pa
ð
t
s
Þ½
1
=
2
exp
ð
x
n
Þ
2
4
a
ð
t
s
Þ
" #
d
n
d
s
þ
T
i
T
q
for
x
0
;
t
m
t
t
s
ð
10
Þ
where
t
m
and
t
s
are the times when melt and solidiﬁca-tion of the substrate material occur. The analyticalsolutions of the temperature distribution above can beintegrated using 104-point Gauss-Legendre quadrature.The calculation of the melt depth and melting durationcan be performed by tracking the position of the movingboundary numerically.
3 Validation and Uncertainty of Modeling
The present study ignores the radiation and convectionheat dissipation at boundaries. The energy absorption of the substrate induced by the irradiated laser is calculatedbased on the entire system, i.e., air/quartz/substrate.However, the energy absorption and the existence of quartz mold in modeling of heat transfer are neglected.The boundary on the surface of the substrate is assumedas insulated due to the smaller thermal conductivity of quartz mold compared with that of silicon and coppersubstrates. It should be notice that nonequilibriumbetween electrons and phonons is not speculated due tothat the time scale of pulses investigated herein isnanosecond that is still much longer than the time scale(i.e. picosecond) in which the nonequilibrium occurs.Moreover, the nonequilibrium process of amorphization[29–31] is not taken into consideration in this study.Meanwhile, the vaporization of the substrate is notconsidered herein. The temperature distribution isanalytically expressed by Eqs. (8–10) and the numericalerror only occurs at the integration of Gauss-Legendrequadrature. The following is the part which integratedusing Gauss-Legendre quadrature.
F
ð
x
;
t
Þ¼
Z
t
p
0
Z
11
f
ð
n
;
s
Þ
d
n
d
s
¼
Z
t
p
0
Z
11
S
ð
n
;
s
Þ
2
q
C
pa
ð
t
s
Þ½
1
=
2
exp
ð
x
n
Þ
2
4
a
ð
t
s
Þ
" #
d
n
d
s
ð
11
Þ
When integrating a speciﬁc function, for example
g
ð
x
Þ
,the error estimate of
n
-point Gauss-Legendre formulaitself is expressed as:
E
GL
¼
2
2
n
þ
1
½
n
!
4
ð
2
n
þ
1
Þ½ð
2
n
Þ
!
3
g
ð
2
n
Þ
ð
1
Þ
1
<
1
<
1
ð
12
Þ
where
n
is the number of points in the formula and
g
ð
2
n
Þ
ð
1
Þ
is the (2
n
)th derivative after the change of vari-able. Furthermore, Eq. (11) is integrated from negativeinﬁnity to positive inﬁnity due to the source term.
Table 1
Thermal propertiesand optical parameters of sub-strates. Thermal properties (i.e.density; speciﬁc heat, thermalconductivity, and latent heat of fusion) of silicon are taken fromRef. 25; the parameters of cop-per and the wave-dependentoptical parameters for bothmaterials are Ref. 26Substrate Si CuDensity (kg/m
3
) 2420 8940Thermal conductivity (W/mK) 20 352Heat capacity (J/kgK) 1030 384.9Melting Temperature (K) 1688 1358Latent heat (J/kg) 1.802
·
10
6
2
:
035
10
5
Refractive index (
n
2
;
j
2
), ArF, 193 nmAbsorption coefﬁcient
b
¼
4
pj
2
k
(nm
)
1
)(0.872, 2.757)0.1795(0.972, 1.403)0.0914Refractive index (
n
2
;
j
2
), KrF 248 nmAbsorption coefﬁcient
b
¼
4
pj
2
k
(nm
)
1
)(1.570, 3.565)0.1806(1.470, 1.780)0.0902Refractive index (
n
2
;
j
2
), XeCl 308 nmAbsorption coefﬁcient
b
¼
4
pj
2
k
(nm
)
1
)(5.013, 3.689)0.1505(1.350, 1.710)0.0698
However,
f
ð
n
;
s
Þ
is exponentially decreasing as theabsolute value of
n
, i.e.
n
j j
, increasing. In the numericalintegration, suitable upper and lower limits of
n
are se-lected to substitute for inﬁnity limit, and the criterion of truncating is set as 10
)
10
.
4 Results and Discussions
In the present work, two kinds of substrates, they aresilicon and copper, are investigated. Excimer lasers withthree diﬀerent wavelengths (ArF, 193 nm; KrF, 248 nm;XeCl, 308 nm) are considered as pulsed laser sources.Fig. 2 showed the surface temperature of copper sub-strate irradiated by XeCl excimer laser with 10 ns pulseduration. When the laser ﬂuence is 0.6 J/cm
2
, the copperdoes not reach the melting temperature due to insuﬃ-cient energy of the incident laser. As for irradiated ﬂu-ence of 1.0 J/cm
2
, the surface temperature exceeds themelting point of copper at 11 ns after irradiation. Thesurface temperature continues increasing up to 1300 Kat t = 18 ns, then decreases due to the absence of lasersource, which means the re-solidiﬁcation occurs. At 21ns, the surface temperature is back to the melting pointand the re-solidiﬁcation completed. This transient tem-perature variations at the surface of copper indicatedthat the temperature of copper kept constant at themelting point until the material become liquid phase.When the incident ﬂuence is 1.4 J/cm
2
, the highest sur-face temperature is about 1950 K, which is the softeningpoint of quartz mold. For this case, the ﬂuence of excimer laser should not be higher than 1.4 J/cm
2
toavoid the softening or damage of the mold. The exper-imental data of molten duration of silicon substrate inliterature [32] is plotted in Fig. 3 to compare with theanalytical solution presented herein. The analytical re-sults agreed very well with the experimental data asshown in this ﬁgure. The molten duration increases al-most linearly from 20 to 90 ns as the inﬂuence of laserincreases from 0.7 to 1.4 J/cm
2
. Accordingly, theimprinting process is accomplished in tens to hundrednanoseconds. Fig. 4 depicted the molten depth of thesilicon substrate with time under diﬀerent irradiations of pulsed laser sources. The molten depth increases duringthe laser heating, after the heating of a laser pulse, themolten depth decreases to zero due to solidiﬁcation of the substrate. When the laser ﬂuence is larger, the ab-sorbed energy increases and the molten zone enlarges.The duration of the pulsed laser is considered as 10, 20and 30 ns in the present study. The time of starting tomelt is postponed while the duration is longer under thesame laser ﬂuence. Additionally, based on the sameﬂuence, the molten depth decreases as the pulse durationincreased, however, the melting duration increases withthe pulse duration. The energy deposited in the substrateunder longer duration becomes slower and the energydissipates gradually. Therefore, the molten depth de-creases. Fig. 5 shows the maximum molten depth andduration of the silicon substrate under irradiation of theexcimer laser pulse. The results reveal that the moltendepth of the silicon substrate is the deepest under irra-diation of a XeCl laser pulse. The reﬂectances calculatedby Eq. (3) (the optical parameters are listed in Table 1)under irradiation of an excimer laser pulse with diﬀerentwavelengths are 48.25% for XeCl; 58.94% for KrF and61.50% for ArF, respectively. The silicon substrate ab-sorbed more energy (
10% more) from a XeCl laserpulse than those from ArF and KrF laser pulses. Thevalues of the absorption coeﬃcients (
b
) are calculatedand listed in Table 1. Smaller value of the absorptioncoeﬃcient indicates that the absorbed energy is depos-ited in the deeper depth in the substrate. About 51.75%of the incident energy (XeCl laser) is absorbed in thesilicon substrate and the absorption coeﬃcient is about0.1505 nm
)
1
. Thus, the absorbed energy induces thedeeper molten zone and prolongs the melting duration.The threshold laser ﬂuence for melting can be estimated
Fig. 2
The surface temperature of copper substrate irradiated byXeCl excimer laser with 10 ns pulse duration
Fig. 3
Analytical results and referred experimental results for themolten duration of Si substrate with diﬀerent laser ﬂuences (KrF of 248 nm, pulse duration is 38 ns)

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