temperature rise at laser-irradiated spot in a low thermal conducting film

ELSEVIER Physica B 229 (1997) 409-415

PHYSICA

Temperature rise at laser-irradiated spot in a low thermal conducting film

P. Arun*, A.G. Vedeshwar

Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India

Received 19 July 1996

Abstract

The calculation of radial and depth profile of temperature rise in a laser-irradiated film of low thermal conductivity such as some chalcogenides and oxides is carried out in a simple manner using the method of separable variables. The proposed expression for the temperature rise is easy to compute and quite general in nature. The absorbance and thermal conductivity (assumed to be independant of temperature) are taken as material characterizing parameters. The result manifests many features and situations encountered in photothermal recording experiments. The proposed expression is quite handy and helpful in selecting the proper laser and its optimal use in photothermal recording.

Keywords: Thermal conductivity; Laser irradiation

There is a growing interest in the studies of materials suitable for optical storage. The optical storage on variety of thin films is carried out by irradiating with a laser beam of suitable wavelength and power. One can burn a hole of a micron radius in the film by a laser beam of few hundred mW power. Other kind of storage deals with the local transformation of the material at the laser spot due to the heat generated. It is also called photothermal recording [l]. Many of the chalcogenide compounds with layered structure and some oxides which have low thermal conductivity have been found to be quite useful for storage applications [1]. These materials belong to the second category mentioned above. Therefore, the knowledge about the surface temperature, its profile along the thickness or radius is quite essential for understand- ing the changes occurring during irradiation. There are many attempts in the literature to compute the

* Corresponding author.

temperature profile and references therein [2-7]. Few of them were aimed to explain the observed results of photothermal recording in a particular case [2,3]. Chaudhari et al. [2] and Kivits et al. [3] have calculated the temperature profile by solving the inhomogeneous heat equation by numerical methods. Bartholomeuz et al. [4] have employed Laplace transformation to solve the heat equation. Lax [6,7] has also made similar calculations by numerical computation and Laplace transformation. It can be noted that in all the previous works in this direction a great deal of computation is required to solve the inhomogeneous heat equation (1) in one or the other form. In the present work we have attempted this problem in case of poor thermally conducting films (such as some of the chalcogenide compounds and oxides) to arrive at a simple expression for the temperature rise and its profile using the method of separable variables. We have also tried to compare our results with those of previous researchers.

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410 P. Arun, A.G. Vedeshwar/Physica B 2 2 9 (1997) 409 415

The basic equation governing the change in temperature due to laser irradiation can be expressed in cylindrical coordinates as [8]

©r ~'©2T I~T ©2T / Cv a~ -- tO{ ar 2 + r ~,-~ + az 2 J

= Aoe-(r / ro)2e-~Z ' (1)

where c~ is the volumetric specific heat, ~c is the thermal conductivity and c~ is the absorption coefficient of the material at the wavelength of the laser and r0 is the radius of the laser beam. The exponential terms describe the Gaussian nature of the laser spot and ex- ponentially falling laser intensity along the thickness of the film. A0 is given by [3]

AP A 0 - rcr ~-'fa (2)

where A is the absorptivity of the film at the laser wavelength, P is the laser power and d the thickness of the film. Bartholomeusz [4] has expressed the right-hand side of Eq. (1) as (P/rcrg)~(1 - e -~* ) ( 1 - R)e-(r/~°)"e-~ which is the same, except for the added terms to account for reflection from the surface of the film. Other authors have used (AP/rtr~):te-(~/~°)2e-~Z to account the attentuation of the laser beam in the z-direction, or if one assumes thermally thin film (AP/~r~d)e-(~'%)'-(1 - e ~) . As will be seen, whichever expression is taken the final result only requires appropriate modifications. Also we shall not be taking into consideration terms ex- pressing reflection from the surface of the films as chalcogenide compounds and oxide films show very little reflection. In the present work we make a rea- sonable assumption like previous workers [2-6] that c~ and t¢ are slowly varying functions of temperature and will be treated as constants for computation. Also, we treat the loss of heat to the atmosphere to be negligible. Further, we assume that a pulsed static laser beam is used for irradiation on a film grown on a semi-infinite substrate of negligible thermal conductivity, t¢ ~ 0.

We consider the temperature rise T as a function of r, z and time t and is capable of being seperable, i.e.,

T(r,z, t) = f ( t )9(r)h(z) . (3)

On substituting this in Eq. ( 1 ) and rearranging, we get

~f t¢ / l~2g 1 ~g 1 ~2h'[ A' at c ~ _ ~ + ~ 9 ~ r + h a z 2 J f = c~'

A I = Aoe-(r/r°)'-e -(~z) gh (4)

We shall treat the term inside the bracket as a constant and seek the solution as the following. Rewriting Eq. (4) as

~ f A 1 ~- + ~ A f = - - , (5)

C v Ct:

where

{ l ~2 g 1 ~g l ~2 h } A = - ? T A + +

We can solve the differential equation (5) to get f as

f = ~ 1 - e -(K'/c')~x¢ , (6)

where R is an integration constant. Now substituting for A ~ and rearranging we get

T = fgh = A°e-(r/r°)Ze-xzKA (1 - ~7-Rcc e_(~/c ' )at']j. (7)

R can be evaluated by setting t = 0, i.e., before irradiation starts film is at room temperature To. Therefore, we get

Rc~gh To,cA z 1 --

Aoe_(r /r o)2 e_~Z Aoe_(r /r o)2 e_~z

and the expression for temperature at position (r,z) can be written as

Aoe-(r/ro )~ e - :~z Tr, z(t) - 1cA (1 - e -(K''c' )zx')

+Toe -(~/c' )~t. (8)

It can be seen that the second term on the right-hand side of Eq. (8) serves as the constant background and contributes very little to the temperature rise. There- fore, it can be neglected for all practical purposes. Now, we can see that on substituting the first term on the right-hand side of Eq. (8) for T at any position (r,z) in Eq. (1), the left-hand side is consistent with the right-hand side.

P. Arun, A. G. Vedeshwar / Physica B 229 (1997) 409-415 411

Now, let

1 ~2g 1 ~ g - - - + - - n . ( 9 ) g ~r 2 rg ~r

Multiplying Eq. (9) by rZg and rearranging we get

r 2 ~2g ~g - j r 2 + r~r + r2ng = 0. (10)

Eq. (10) can be identified as the Bessel's differential equation with known solutions given by [9]

g(r) = AJo(x /nr) 4- BYo(v/nr) .

For a non-singular solution, we assume B = 0 to get

g(r ) = AJo( v/nr ).

The merit of this choice can be understood by the fact that at r = 0, this solution gives a finite value for (1 / r )~g /~r . Jo(x) has a wavy behaviour and goes to zero atx = 2.405,x = 5.52 and so on [9]. We consider Jo(x) only upto x = 2.405 because beyond this Jo(x) is negative and is meaningless in the present case. Also, the power of the laser beam falls to zero beyond r0. Therefore, we take the first root to evaluate n,

x/~r0 = 2.405,

i.e. n = 5.784/r02. (11 )

Similarly, the other variable z can be separated to get

1 ~2h - - ~o 2 ( 1 2 )

h Oz 2

giving the solution h = Q cos(o~z) where co = 1/d. Thus, our constant A can be evaluated as

A = n +~o 2

5.784 1 - - - 4 - ( 1 3 )

r 2 d 2"

Our final expression for temperature rise can be written a s

/ 2 Aoe--(r/ro) e - ~ z

(1 --e -~z At/c' ) 4- Toe -~ At/c' . Tr, z( t ) = tea

(14)

time, yields an expression for the decay of temperature as

Tr, z(t) = Tme-KAt/c' 4- To, (15)

where Tm is the maximum temperature attained before switching off the laser as can be computed by Eq. (14).

Therefore, Eqs. (13)-(15) can be used to calculate the temperature rise and its decay at any position (r ,z) . The radial and along thickness profile can be computed by calculating the temperature at various positions (r ,z) . We have calculated the rise and decay of temperature taking the same values used by Chaudhari et al. [2] for the purpose of comparison and is shown in Fig. 1. The parameters used were cv = 1.65 x 106j/m3°C, •=6 .15 x 10-SW/m°C, d = 6 0 0 A , r 0 = 3 g m , c ~ = 5 x 105cm -1 and laser power P = 1 W. It can be seen from Fig. 1 that the agreement between the two is excellent. However, the difference between the two grows for increasing distance away from the centre. This may be due to the stability factor employed in the numerical solution of the inhomogeneous heat equation (1) in Ref. [2]. Anyway, the difference is not significant. We have also tried to compare the temperature values obtained by numerical solution of Eq. (1) with those calculated using Eq. (14) for various values of ~c and ~. The expression tends to agree for low values of ~c as seen in case of chalcogenides.

Our expression can be used to predict the temperature rise and its decay in case of films having low thermal conductivity grown on a substrate of negligible thermal conductivity. However, the influence of the substrate via its thermal conductivity and thickness can only be incorporated by taking A ~ A(r , z ) , i.e. a function of r and z and not as a constant. Devi- ation between values obtained by numerical solution and that from Eq. (14) would also be removed by taking A ~ A(r ,z ) . The characterizing parameter of the film is its thermal conductivity and the absorptivity at the wavelength of the laser assumed to be independent of temperature. However, if A is a linear function of temperature as in Ref. [10],

Similar calculations applied for the case of heat source being switched off after heating for a given A = m T + Ai.

412 P. Arun, A.G. Vedeshwar/ Physica B 229 (1997) 409~415

I CJ001-- - - PRESENT WORK

] A - - - R e f ' { 2 ]

~oo l /~ , - , , ,o o I I 1 " \ \ . - r : ' l pm

- / // \ \ c-,:2,m

I/A\\:,-. " '

0 /+ 8 12 16 20

t(n sec )

Fig. 1. Comparison of present calculation of temperature rise and decay with that or Ref. [2]. The calculated temperature is at a distance of 100 A below the surface. We have taken the same parameters used in Ref. [2].

Then

Tr, z(t) = AiPe-(r/ro)2e-~Z

rcrgdtcA

x(1 - e -~AUc') + Toe -KAUc', (16)

where now A is given by

5.784 1 mPe-(r/r°)2e-~Z A - +

4 d~ ~gd~

Therefore, temperature-dependent absorptivity pro- duces a positive feedback which increases the temperature rise and takes longer to reach saturation. We have shown the dependence o f temperature profile on the thermal conductivity o f the film in Fig. 2. As can be seen from the figure, the temperature rise is less and the decay is faster for the films having a larger thermal conductivity. This result can be understood easily due to the loss of heat by conduction. Important result is the large decay time as predicted for poor thermal conducting films. The final temperature is maintained upto the order o f few milliseconds. Therefore, this suggests that the metal films require a higher power o f laser than for a chalcogenide film (having low t<) to burn a hole if we take the melting temperature o f both the films as comparable. The result also suggests that

300,

20(]

lO(] j

0 I -g -8

. . . . K : ] x l0 -7W/m °C

r I l I [ I I -7 -6 -5 -/., -3 -2 -1

LOft t ( s )

Fig. 2. Dependence of temperature rise and decay on the thermal conductivity 1< of the film. The maximum temperature shown is at the beam centre (r = 0) and film surface (z = 0). We have taken A =0.5, P - 0.2W, r0 = 2.51am, d = 1000A and Cv = 10 6 J/m 3 °C which were kept constant for all profiles.

a single laser pulse of few nanoseconds is sufficient for making local changes on a poor thermal conducting film and hence the faster recording. Therefore, our expression is quite simple to predict the suitability of a given laser and its power for photothermal recording on the films if its ~c and A are known. Also, it is quite possible to predict the suitability o f the material for recording applications if its t< and A are known.

The temperature profile along the radius o f the laser spot and along the thickness of the film is shown in Fig. 3 for two different cases o f thermal conductivity. The major contribution to the radial distribution

of temperature is from e -(r/r°)2 term which describes the Gaussian nature of the laser beam. Similarly, the term e -~z which accounts for the falling laser intensity along the thickness is the major contributor for the temperature profile along the thickness. Tempera- ture fall is more rapid along the thickness compared to that along the radius of the laser spot. Therefore, in hole burning or any storage process the area o f the hole or stored region will be smaller than the radius o f the laser beam if the temperature at r = 0 is just about the melting point of the film.

Important result is the dependence of temperature rise on the laser beam radius as depicted in Fig. 4. The final temperature of the film surface is larger for smaller radius of the laser beam. This is also observed by Abraham et al. [11]. Normally, the laser beam is used both in the focussed and defocussed geometries

P. Arun, A.G. Vedeshwar/ Physica B229 (1997) 409-415 413

CENTER ro~+ + r o f 2 3 t o / & ro o

_ T ~'}K =3 . lOl"Tw/m *C

0 : t t I 500 1000

~U~FACE E ( *A )

Fig. 3. Radial and along-thickness profile of the temperature rise

for irradiation time of 5 ns for two different thermal conductivities.

The direction of arrows indicates the scale of reference. The other parameters used in the calculations are A = 0.5, P = 0.2 W, r 0 = 2.5 pm, d = 1000/~. and c~ = 106 J /m 3 °C.

for recording. Reducing the radius of the laser beam refers to focussing and large radius refers to the de- focussing. In Fig. 4 we use a continuous laser beam. The temperature shoots up instantaneously for smaller radius of the beam. As can be seen in the figure for the case o f r = 0.6 ram, the temperature saturates after reaching a maximum value. It takes longer time to go to final temperature if the beam radius is increased. Therefore, the temperature does not rise further even if the irradiation time is more then the rise time. The important consequence is that the beam diameter should be sufficiently small for pulsed lasers used in case o f good thermally conducting films. In any case temperature rise saturates. However, the final temperature will be very large in case o f laser beam of very small radius. This way one can use the desired beam size in a focussed geometry to get the required temperature. This is indeed the situation encountered in many photothermal recording. We have tried to use these results in chalcogenide amorphous thin films. Most of our ex- perimental results are tallying with these calculations semi-quantitatively. When we used the laser beam of 2 mm size we could not get any change. When we reduced the beam size, a circular mark with radius less than half the beam radius was observed.

The SEM photographs displayed in Fig. 5 show the real-time photothermal recording in a 5300 ,~ thick

800

60(

o ~ +0(

I -

Oi

20C

ro=Zp+

/2 I I _17 I

- 9 -8 -6

ro= 20

-~, LOG t l s )

3= z00 pm

to= 0-6 mm

= mm

-3 -1

Fig. 4. Dependence of temperature rise on the radius of laser beam used for irradiation.

amorphous Sb2S3 film. The Sb2S3 films were grown on thin glass substrates held at room temperature by resistive heating using a molybdenum boat at a vac- uum better than 10 -6 Torr. An argon ion laser beam (power = 2 0 0 m W and 2 = 514 nm) was used in focussed geometry for photothermal recording. Fig. 5(a) shows the recorded spot. Atleast three or four concen- tric rings having different contrast can clearly be seen in the figure. This is due to the radial profile of the temperature rise. Even though the radius o f the laser beam was 220 Ixm the recorded spot has the radius of about 160 ~tm. This is because, beyond this radius temperature falls below 160°C at which S b 2 S 3 transforms to crystalline phase as seen in DTA studies of the amorphous film. We have taken ~ = 8.2 × 10 -3 W/re°C, a representative value for chalcogenide compounds and A = 3.2 from our absorbance data o f the film for temperature calculations. The calculated temperature rise at the centre of the spot is about 270°C and falls off to 250°C at r = 60 pm, 220°C at r = 100 pm and finally to 160°C at r = 160 pm (detailed radial and depth profile has been given in Table 1 ). Therefore, we have observed different contrast and grain sizes as a function o f r . Fig. 5(b) shows the enlarged area near the centre o f the spot while Fig. 5(c) shows the enlarged area at r = 60 pm. Fig. 5(d) shows the magnified area near the outer edge (r = 100 pm) of the white spot. Similar grain structures were observed when

414 P. Arun, A.G. Vedeshwar/ Physica B 229 (1997) 409~t15

Table 1 Calculated temperatures (in °C) using Eq. (14) as a function of

radius and depth at the laser-irradiated spot in Sb2S3 film. z (A) is measured from the surface towards substrate

r(pm) z = 0 z = 250 z = 500 z = 750

0 272.1 234.0 201.2 173.0

20 269.8 232.0 199.5 171.6

40 263.2 226.4 I94.6 167.4

60 252.5 2t7.2 186.7 160.6

80 238.3 205.0 176.3 151.6 100 221.2 190.0 163.7 140.7

120 202.0 173.8 150.0 128.4

140 181.5 t56.0 134.2 115.5 160 160.3 137.8 118.5 102.0

we heated the film at the temperatures calculated at various r. For r greater than 160/am temperature falls below 160°C. Therefore, one can see the amorphous background in Fig. 5(a) for r greater than 160/am.

In conclusion, we have obtained a simple expression for the temperature rise and its profile in a laser- irradiated film of low thermal conductivity grown on thermally non-conducting substrates using the method of separable variable. It manifests many features encountered in photothermal recording experiments. The single expression is capable of calculating radial and along-thickness temperature profiles. The expression can also be used to select the proper laser and to use it optimally. The striking agreement between the results of our actual photothermal recording in Sb2S3 films and the calculation of temperature profile indicates the utility of the temperature calculation in analysing the photothermally recorded spot. At present, Eq. (14) presents a back of the envelope method of calculating temperature rise due to laser irradiation on a film of low thermal conductivity. Work is in progress to generalise A ~ A ( r , z ) and account for good thermally conducting films and also the substrate.

The financial assistance in the form of scholar- ship (S.R.F) to P. Arun by U.G.C, India is gratefully acknowledged.

Fig. 5. SEM photograph of (a) laser-irradiated spot in amorphous Sb2S 3 film and the magnified portion of the spot at (b) the centre, (c) r = 60 pm and (d) r = 100 ~tm.

P. Arun, A.G. Vedeshwar/ Physica B 229 (1997) 409~I15 415

References

[1] K. Schwartz, The Physics of Optical Recording (Springer, Berlin, 1993).

[2] P. Chaudhari and R.J. von Gutfeld, J. Appl. Phys. 43 (1972) 4688.

[3] P. Kivits, de Bont and P. Zalm, Appl. Phys. 24 (1981) 273. [4] B. Bartholomeusz and G. Olin, Opt. Eng. 29 (1990) 988. [5] B. Bartholomeuz, J. Appl. Phys. 64 (1988) 3815.

[6] M. Lax, J. Appl. Phys. 48 (1977) 3919. [7] M. Lax, Appl. Phys. Lett. 33 (1978) 786. [8] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids

(Oxford Univ. Press, Oxford, 1954). [9] G. Stephanson, Partial Differential Equations for Scientist and

Engineers (Longman, London, 1968). [10] J.F. Ready, Proc. IEEE 70 (1982) 533. [11] E. Abraham and J.M. Halley, Appl. Phys. A 42 (1987) 279.

temperature rise at laser-irradiated spot in a low thermal conducting film

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