Modeling and Computer Simulation of the Thermal Field Induced by Ion Beam Irradiation of Bilayers
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irradiated with a monoenergetic beam of inert gas ions: the bombardment causes ion implantation at its surface and heating because of kinetic energy loss of the ions; also the substrate is then thermally affected during the process. The thickness of the coating is typically of the order of 215 Mat. Res. Soc. Symp. Proc. Vol. 389 ©1995 Materials Research Society
I + 10 gtm, whilst the mean penetration depth of the ions implanted at the accelerating voltages concerned in the present paper, 100 to 1000 V, is much smaller than 0.1 .tm /16/; then, the
thickness of the ion implanted layer is assumed to be very small with respect to that of the film since the very early beginning of the deposition, so that the ion beam is regarded in the present model as a surface heat source. This approximation holds at deposition times where the ion range is really smaller than the thickness of the film and is realistic since when the thickness of the film is > 1 tm. Further boundary conditions of the problem are the energy conservation at the interface (T, = = T2) and the thermal insulation of the target
K
T-.
=K 2 --n
1xn
and
K-
=0
aXx=xm
F
K
ax y=ym
=0
K-
=0 az z=zm
where iý is an unit vector normal to the interface plane, K the thermal conductivity and T indicates that the equalities hold for both T7and T2. The temperature range of interest during the deposition process is expected to be of the order of a few hundred degrees, so that the thermal properties of materials can be reasonably considered as constants; however, their dependence upon temperature could be easily taken into account through the procedure described in /17/. The element diffusion at the film-substrate interface during the deposition time is neglected. Then all the information about the thermal field of both film and substrate is obtained solving the respective heat diffusion equations with the pertinent boundary conditions discussed above. 3 THE MATHEMATICAL APPROACH. -The starting point is the analytical solution of the Fourier heat diffusion equation in the film V2T = M1 Ti where M, = [pC,
/ K];
p and Cp are here the density and the specific heat
respectively. Defining T1 as the product of three functions of the respective coordinates T, = T*TX.TyTz, where T* is a constant, one obtains a system of three differential equations of the
respective CX > Ax,
coordinates.
Then
T = C+
A. exp[-(7t / Xm )2 (t + t) / M1 ]cos(xR / X.),
Ty =Cy+ Ayexp[-(t /ym) 2 (t+ty)/
M]]cos(yt /y),
where
where
Cy > Ay,
T1=•b--{erf[(z +y))]-erf[(z-2z, -7)X]} where y+ 2z, >0. Ax, Ay, C, Cy, tX,
and ty
andy
are arbitrary constants and X = rM/-,-/2.f. The constants Ax and Ay control the values of the temperature gradients both in the coating and in the substrate before that the steady state be attained for t >> tx and t >> ty. Then the time length of the transient stage, the temperature gradients and the asymptotic temperature can be imposed separately as independent boundary conditions, according to desired requirements. -The substrate temperature T2 is cal
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