CALCULATED TEMPERATURE DISTRIBUTION DURING LASER ANNEALING IN SILICON AND ...

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CALCULATED TEMPERATURE DISTRIBUTION DURING
LASER ANNEALING IN SILICON AND CADMIUM TELLURIDE
R.O.BELL °
Mobil Tyco Solar Energy Corporation
Waltham , Massachusetts ,02154
USA
M. TOULEMONDE , P. SIFFERT
Centre de Recherches Nucléaires ( PHASE)
67037 Strasbourg-cedex
FRANCE
A B S T R A CT
By solving the time dependent heat flow equation, the temperature reached
by silicon and cadmium telluride surface layers under high power density
ruby laser pulsed illumination, is calculated. The results are presented in
directly useful figures allowing the determination of the surface temperature,
its evolution towards the bulk as a function of time... In particular, it
should be noticed that for a ,25 ns . half power width pulse,0,8 j/cnv are
sufficient to melt the top of an amorphous silicon layer, this value becomes
noticeably lower for cadmium telluride.
Work performed while visiting the Centre de Recherches Nucléaires,
Groupe PHASE , STRASBOURG.
CALCULATED TEMPERATURE DISTRIBUTION DURING
LASER ANNEALING IN SILICON AND CADMIUM TELLURIDE
INTRODUCTION
The use of pulsed laser beams to produce recristallisation of amorphous
layers existing on top of single cristalline materials, especially Si and GaAs
has recently been demonstrated ~ . In fact, two regrowth mechanisms have
been observed, depending on the laser characteristics, either by solid phase
epitaxial regrowth or by a liquid phase process. Here, we will restrict
ourself to the later. The potential applications for such a technique are
quite obvious : making solar cells , localized annealing of integrated circuits,
controlling rapid quenching of alloys and doping of surface layers. Many
experiments showing that such a process can be controlled have already been
-6-7)
performed " ~ but to better understand what happens, calculations with a
model are necessary. Such evaluations gave up to now contradictory results :
(8)
some authors found that the m, :2rial does not melt , whereas other calcula­
tions indicate thnt the melting temperature of a material like silicon can easily
be reached " . Here, we present the results of our calculations for Si and
CdTe materials, taking into account the most probable values of the various
parameters. Typically, a very short pulse (25 ns) melts the surface several tenths
of a micron deep for a period of time of less than 200 ns., followed by a verv
rapid regrowth (3 m/s. ).
The general method for making the calculation is quite straight forward
and just looks at the time dependent, heat flow equation with a volume generation
term for the energy deposition by the optical source. The most difficult
aspect is making reasonable assumptions about the dependence of material para­
meters such as the thermal conductivity on temperature and photon flux density.
A good deal of information on the temeprature dependence of material pa­
rameter is available but not nearly as much on behavior at high photon excitattions.
Optical absorption can occur from(l) interband electron transitions, (2) free
carrier absorption, (3) fundamental and harmonic lattice vibrations and (4) impuri­
ties either by electrons bound to defects, localized optical impurity modes or
inner shell electronic transitions. Cf these only the first two will be important
for the case of interest. Either the frequency is too high for significant absorption
cases (3) and (4) cr the strength of the interaction is relatively small, case (4).
A l uiM;Ii ("">t iir.nl v of liit: ,*ivf ruj'.i rbntirc in t-ïiclroii c cncôiilr.-.! i t ;;, .'.:. ,
jiroflnr.cd by photon absorption can be made by ]ooï-.inj: .it iho cqui 1 ibri'::. .'it-
11 ji i on:
* . . *a • <> . o
G
at T
where the lifetime, i , is limited by Auger recombination 'and the average
optica] generation <G> is due to intcrband transitions. Band to band recom
bination and impurity recombination will be less efficient than the Auger
process for relatively high carrier densities. We also expect free carrier
absorption to be relatively unimportant for a ruby laser (0.69 ym) in Si and
CdTc although it may be significant for the longer wavelength YAG laser (1.06
pm). Using an absorption coefficient of 10 cm , An is the order of 10 cm
which, although high, is nevertheless similar to the electron or hole concen
tration in the heavily doped surface region. Such a deviation from the un-
excited state should not drastically affect the assumption that parameters
measured at low flux levels can still be used for a laser pulse length in the
2
order of 25 ns and energies of 1 J/cm , although experimental verification
is still desirable.
Since the thermal velocities of liquid is in the order of 10 cm/s
2
(3/2 kT » 1/2 mv ) and melting and growth rates are two or three orders of
magnitude less, no unusual non-equilibrium effects might be expected. Normal
segregation and diffusion coefficients for the incorporation of impurities in
the solid can thus still be used.
The first part of this paper out line.', the model and method of calcula
tion. In the second part the results for specific cases of Si and CdTe are
presented. The assumptions used to arrive at the various material properties
are outlined in the Appendix.
MODEL AND CALCULATIONS
If a planar surface is exposed to opticni radiation at normal incidence,
the temperature distribution will be governed by the one-dimensional tine dc-
(12)
pendent, bent flow equation
c n -i'r - '"•- ( I: •£;•'--) <1 H i,;;f„t,, x)
where t: is the specific heat, p is I lie density, T is the temperature at.
c
any given time, t , or position, x , -jnd 1; is the thermal conductivity. The
G term expresses the generation of heat throughout the material by the opti­
cal source and has the form for non-uniform absorption:
G - I(t) o(x](l - K) oxp C- fX a(x' )<!*•) (2)
Here Ift) is the incident intensity on the material, R is the reflectivity
at the surface and a is the absorption coefficient. The possibility that all
the parameters such as u , k , R , p , etc. can be temperature dependent must
be considered.
The solution is obtained by converting to a difference equation in tine,
At , and position h , i.e.
+ T. . - 2T.) (T. , - T. ,)(k. , - k. J
T, (t + At) = T. ft) , (Ti-l l-l l 1 l+l l-l'1 l+l 1-1'
i I1 i 2 .,2
vh2
' v 4h
(3)
G.fT. , t
j Csp
The depth below the surface is given by the product of (i-1) times h . An
initial value for all values of T. for t < 0 is first assumed.
Starting with t = At new values of T. are calculated, t is then in-
a l
cremented and the process repeated. For the case i = 1 no heat flows across
the surface (thermal radiation or conduction and convection by the air are
negligible)in which case T. = T. .
1
For proper convergence of the thermal conduction equation, it is neces-
2
sary that At be less than C ph /2k . At and h are chosen with this con­
straint in mind.
At the melting and freezing temperature the latent heat, L , must be
accounted for. This is done by holding the temperature constant until enough
host lias been ;uidcd or subtracted to account for the latent heat. Also, at
the melting point k changes discontinuous!)' so the difference equation for
d(kdT/dx)clx was modified to
&i (Ti*l - V - ks<Ti - Ti-i^/h2 •
The biggest uncertainty in the calculation has to do with the values
of the material parameters and how they change with temperature. A descrip­
tion of the rational for choosing various values is given in the Appendix.
DISCUSSION OF RESULTS
Figures 1 and 2 show the results of plotting the temperature as a func­
tion of time at the surface and at different depths below for Si and CdTe
respectively. Unless otherwise noted the pulse width at the half power points
2
was 25 ns. The total energy in the Gaussian pulse was 1 J/cm for Si and
2
0.7 J/cm for CdTe. In the calculation the pulse was centered at a time equal
to the Inlf power pulse length and the amplitude integrated to determine in­
cident energy per unit area.
The initirl temperature increase becomes rapid when the incident power
2 2
approaches its rjaximuro value (35 MIV/cm for a 1 J/cm , 25 ns pulse). The
temperature rise is temporarily interrupted at the melting point until enough
heat is absorbed to melt the top layer. At this time the reflectivity in­
creases substantially and the optical absorption becomes so large that material
more t.han a couple of hundred A below the surface is no longer optically heated
but can only receive heat by conduction. This is the reason for the decrease
in temperature at 0.1 and 0.2 um which occurs in Si (Fig. 1) until conduction
from the surface causes a further increase in T . After melting the surface
temperature resumes its rise, although not as rapidly as before because of the
larger thermal conductivity of the liquid, until the energy from the laser is
less than the heat loss by conduction. The temperature falls to the freezing
point where now the latent heat must be removed before it can solidify. After
a substantial time the temperature falls. Inside the crystal the temperature
will continue to increase for some period of time. Compared to Si (Fig. 1)
the rate of increase for CdTe (Fig. 2) is initially much more rapid because
of the higher absorption coefficient. The lower thermal conductivity,latent
heat and melting temperature all contribute, to melting with a lower energy.
4
Another presentation of the data is to look at the temperature as a
function of position with time as a parameter (Figs. 3 and 4). The shape of
the laser pulse is shown as an insert in Fig. 3. Note that the surface is
always at the highest temperature with again discontinuities in the curves
at the melting point. With time the heat moves into the material in the form
of a wave.
Figure 5 shows the results of varying the total incident energy for Si.
The maximum surface temperature and energy absorbed are shown as functions of
the incident energy. A somewhat sublinear increase in absorbed energy occurs
since at higher energies a larger part of the pulse occurs for silicon in its
molten state where the reflectivity is higher than for the solid. Figure 6
shows how the relative amount of energy changes as the total incident energy
increases for CdTe.
If a longer pulse length with the same total energy is used the surface
does nut attain nearly as high a'temperature (Fig. 7a) because conduction has
a longer time to act even though the total power absorbed is slightly higher
for longer pulses (Fig. 7b).
The maximum depth of melting for CdTe as a function of incident energy
is shown in Fig. 8. Even with 0.2 J/cm about 200 A of material are melted.
This is consistent with experiments where a 500 A amorphous layer produced by
2
a 40 keV Ar beam is completely regrown by a single 0.6 J/cm ruby laser pulse.
One difference of CdTe from Si is that because of the high vapor pressure of
the melt, sputtering or evaporation of the CdTe is important in the top 200
to 300 A of material except at the lowest energies. ,
5
APPENDIX
This section js devoted to a description of the assumptions made in
arriving at various material related parameters used in the calculation.
LATENT HEAT, L , AND MELTING TEMPERATUHE, T
' • m
Not a great deal of variation in the lat ent heat, L and the melt ing
, -1
temperature are reported and we will take for Si L * 430 cal g
T = 1417°C « 1690°K °3 and L = 50 cal g -1 and T * 1092CC = 136S°K for
m
CdVe^».
SPECIFIC HEAT - C
s_
The specific heat at high temperatures (T > 300 K) is nearly constant
varying from 0.17 at 25°C to 0,32 at 2000 K in Si and was thus assumed con­
stant as 0.2 cal g" K" . In CdTe the value is nearly an order of magni­
tude less and assumed to be 0.025 cal g" K .'
DENSITY - p
The density of Si is only weakly temperature dependent except at the /'
melting point where it increases. The solid density we have assumed to be
-3 -3 fl6\ -3
2.2 g cm and.the liquid as 2.46 g cm .A density of S.85 g cm v's
used for both the liquid and solid in CdTe.
THERMAL CONDUCTIVITY - K
A reasonably good approximation for many semiconductors is to assume
that for the solid the thermal conductivity varies inversely with temperature,
T . At the melting point k discontinuously increases by a factor of about
2.5 in Si and is virtually constant thereafter. Thus, below the melting point
k = S8/T cai cm sec" K H'J Above Si becomes a liquid metal and
k * 0.14 cal cm" sec" °K~ l,BJ. The equivalent values for CdTe are
k = 3.6/T cal cm"1 sec"1 K"1 ',9^ and k„ = 0.14 cal cm"1 sec"1 K"1 . No
s SL
measured values for liquid CdTe could be found so we arbitrarily used the
same value as for liquid Si.
ABSORPTION rOF,FFlCIl:NT - a
The optical absorption coefficient at 0.69 wm in Si depends primarily
on interactions with the electronic band structure of the semiconductor
The energy gap decreases by approximately 0.68 eV between room temperature
and the melting temperature v . One approximation is to assume that the
complete change can be accounted for just by shifting the entire room tem­
perature absorption curve by this same amount. If a linear variation of a
bbeettwwcee n 3000 cm at 300°K to 10000 cm at the melting temperature is used
then
o » 12.33T - 670 cm"1
Perhaps a better approximation - at least it fits the experimental results
between 77 and 300CK - is to assume
a » K exp (&T)
-1 -3 -1
where K = 1990 cm and B = 1.365.x 10 K for crystalline Si . In the
actual calculation little significant difference between the two approxima­
tions was found.
Similar arguments can be ui-ed for amorphous Si . From data for dc
produced films with a 415°C deposition temperature^ , a at 0.7 um is
4 5 -1
1.5 x 10 increasing to 4 x 10 cm for a 0.98 eV change in energy. The tempera^
ture coefficient of the optical band gap is about 7 x 10* eV/ C for a-Si
Thus, we could'use
a * 3488T - 8.96 x 104 cm"1
s
-1 -3 -1
For an exponential dependence we should use K = 7034 cm and $ = 2.523 x 10 K .
Above T , Si is a liquid metal so a will be some high value, but just
what is hard to say with precision. In fact, the actual value is not very im­
portant as long as most of the radiation is absorbed near to the surface. For
this calculation we have used a = 5 x 10 cm" which absorbs more than 90% of
the radiation in 500 A . Some justification for this can be derived from re­
flectivity data. If the reflectivity of the liquid is determined almost com­
pletely from the metallic character, then using
2 2
„ (n-1) + k , , aX
R = '— = where k = —
(n+1) + k 4u
gives a = 4.9 x 10 cm" . This is for R = .65 and n = 1 .
lor CdTe we us* similar arguments to that for Si. We expect only a
4-1 4 -1
small change-in a from 6 x 10 cm at room temperature to IS x 10 cm
f23l
at the melting point v ' . A linear variation gives
a = 3.54 x 104 • 83.96T cm"1
and was used for the CdTe calculations. Information about a for molten CdTe
apparently is not available so the same value as for Si was used.
REFLECTIVITY
Arguments much as above for the reflectivity would give a change of R
from 0.331 at 300 K to 0.375 at T . If a linear variation is used
m
R = 0.322 + 3.17 x 10"5T
The difference reported for a-Si is quite small between crystalline and
a-Si {22).
Above T , R will increase. One experiment gave R = 0.7 for the liquid
which is quite reasonable compared to other metals. The emissivity, E , of
Si at 1600 C is 0.48 ^ -1 and since E equals the absorptivity, A , and
R + A = 1 in metals, we would expect R = 0.52. We have used an intermediate
value
R = 0.65
At room temperature in CdTe R = 0.21 'z5>. For the calculations we
have assiuned this value to be constant up to the melting point above which
we have doubled R to 0.42.
S
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