silicon wafer during rapid thermal processing. A one-dimensional thermal model and temperature-dependent thermal properties of a silicon wafer are adopted in this study. Our results show that the thermal nonuniformity can be reduced considerably if the incident heat fluxes on the wafer are dynamically controlled according to the inverse-method results. An effect of successive temperature measurement errors on thermal uniformity is discussed. The resulting maximum temperature differences are only 0.618, 0.776, 0.981, and 0.326 C for 4-, 6-, 8- and 12-in wafers, respectively. The required edge heating compensation ratio for thermal uniformity in 4-, 6-, 8- and 12-in silicon wafers is also evaluated.
Index Terms—12-in silicon wafer, inverse heat-transfer method, rapid thermal processing, thermal uniformity.
I. INTRODUCTION
A
S DEVICE dimension shrinks to the submicrometer range, reduction of thermal budget during microelec-tronic processing is becoming a crucial issue. Single-wafer rapid thermal processing (RTP) has become an alternative to the conventional furnace-based batch processing in many processes [1], [2]. To obtain uniform processing across the wafer and to prevent the creation of slip defects due to thermal stresses, the temperature must be nearly uniform on the wafer throughout the process cycle [3].It is known that the incident heat flux profiles (the distribu-tion of energy) from a RTP system must be nonuniform over a wafer to obtain uniform temperature at all times, the reason being heat loss by the edge of the wafer. Hill and Jones [4] in-vestigated thermal uniformity with a uniform intensity field and one in which the intensity was linearly enhanced to a maximum of 8% vertically over the last 15 mm of a 6-in wafer. Kakoschke et al. [5] evaluated enhanced illumination intensities at wafer peripheries vertically and laterally for a compensation of edge heat losses during processing. Gyurcsik et al. [6] introduced a two-step procedure for solving an inverse optimal-lamp-con-tour problem to achieve temperature uniformity in steady state. Sorrel et al. [1] determined the increase in perimeter radiation
Manuscript received November 22, 1999; revised April 14, 2000. This work was supported by the National Science Council, R.O.C., under Grant NSC 88-2218-E-009-003.
The authors are with the Department of Mechanical Engineering, Na-tional Chiao Tung University, Hsinchu, 300, Taiwan R.O.C. (e-mail: [email protected]).
Publisher Item Identifier S 0894-6507(00)09494-X.
based on linear programming for minimization of worst case error during temperature trajectory following. Zöllner et al. [8] compensated for radial temperature decreases using an adjustable lamp arrangement with optimized power settings calculated from wafer heat losses. Riley and Gyurcsik [9] determined the amount of lateral heating needed to counteract edge cooling during RTP. Cho et al. [10] optimized the incident heat flux profile over a wafer by determining the heat loss profiles using Lord’s thermal model [3], which simulates radial temperature gradients by assuming uniform temperature through the wafer thickness. Following the work of Riley and Gyurcsik [9], Perkins et al. [2] used their special wafer-edge node analysis to show that idealized intensity profiles can maintain thermal uniformity at steady-state temperatures, and that dynamic continuously changing profiles are required to maintain temperature uniformity during thermal transients.
The works mentioned above describe quantifying incident heat flux over a wafer to achieve the necessary thermal unifor-mity requirement during RTP. However, the question is whether there is a more efficient way than a purely trial-and-error approach to determine the incident heat flux over a wafer to ensure thermal uniformity. The inverse heat transfer problems (IHTPs) deal with the determination of the crucial parameters in analysis such as the internal energy sources, surface heat fluxes, thermal properties, etc., and have been widely applied in many design and manufacturing problems [11]–[14]. The inverse source problem is practical in thermal uniformity of RTP systems in which the heat source strength required to achieve temperature uniformity is undetermined. The one-di-mensional inverse problem with two unknown sources has been investigated, and satisfactory results are reported [15].
In this paper, a finite-difference solution to a one-dimen-sional (radius, assuming uniform temperature through the wafer thickness) thermal model in which both surfaces of a 12-in silicon wafer are heated is studied for application to RTP systems. The wafer is subjected to a steady uniformly distributed heat flux [2], [5] (uniform heat flux, i.e., intensity mode during processing). The temperature-dependent thermal properties of the silicon wafer are considered. Then, using the inverse heat transfer method [14], [15], the incident heat fluxes over the wafer required for thermal uniformity during ramp-up and steady-state phase of RTP are determined from known (desired or measured) wafer-center temperature trajectory. 0894–6507/00$10.00 © 2000 IEEE
Fig. 1. Schematic representation of energy flux in a silicon wafer subjected to two-sided incident heat flux and heat losses emitted from all wafer surfaces. The measurement-error effects on thermal uniformity are also discussed.
II. THERMALMODEL
Consider a thin axially symmetrical circular silicon wafer, as shown in Fig. 1. Let and be the radius and thickness, respec-tively. is the initial temperature of the wafer, and the temper-ature of the surrounding medium is . Symmetric heating on both sides of the wafer is assumed. The total incident heat fluxes on the top and the bottom surfaces of the wafer are represented by and , respectively. The heat losses occur at all wafer surfaces. Assume that the temperature is uniform through the wafer thickness. Thus a one-dimensional thermal model is adopted.
The governing equation for an axially symmetric cylindrical coordinate system with its origin at the wafer center is
(1) with
where Wcm K is Stefan–Boltzmann
constant; wafer temperature is a function of radius and time
; and , , , , , , is wafer
den-sity, thermal conductivity, specific heat capacity, absorptivity of the top side, emissivity of the top side, absorptivity of the bottom side, and emissivity of the bottom side, respectively.
Note that the absorptivity and emissivity may depend on wafer temperature, position, and radiating spectral wavelength [16], [17]. And, because of the large temperature variations during processing, the temperature dependence of wafer thermal conductivity as well as specific heat capacity must be considered as follows [18]:
T W cm K K
(2a)
Fig. 2. Temperature-dependent thermal properties of a silicon wafer.
T J g K K
(2b) while the wafer density is assumed to be constant and equal to 2.33 g-cm . Since the silicon wafer is considered to be homo-geneous in the present study, the dependence of on spa-tial position is introduced implicitly only by the spaspa-tial depen-dence of the temperature. Because is weakly dependent on temperature (see Fig. 2), spatial temperature variations across wafers at a certain time are expected to be small enough ( 200 K) so that spatial variations in thermal conductivity may be ig-nored [5]. Equation (1) is thus reduced to
(3) The initial and boundary conditions for the system mentioned above are
at (4)
at (5)
at (6)
where is the emissivity for radiant heat loss emitted from the wafer edge. We may assume without loss of generality that the incident heat flux on both sides during processing is equal,
i.e., , and that the
absorp-tivity in all wafer surfaces is the same as the emissivity of these surfaces. For simplicity, the emissivity in all surfaces is assumed to be the same and only temperature dependence as [19]
We can approximate the governing equation and the ini-tial condition, as well as the boundary conditions, using with equidistant grid and the temporal coordinate increment . After the nonlinear radiant fourth-power terms in (8) and (9) have been simulated using a linear scheme and the successive over-relaxation (SOR)-by-lines method has been adopted, the unknowns in the subgroups to be modified simultaneously are set up such that the matrix of coefficients will be tridiagonal in form permitting use of the Thomas algorithm as follows:
(10) The superscript is denoted as the index of the temporal grid, and the subscript is denoted as the index of the spatial grid. Given the incident heat flux, we can obtain the wafer temper-ature distributions.
III. INVERSEHEATTRANSFERMETHOD
The inverse heat transfer problem in application to RTP is given a wafer temperature-distribution history to determine the incident heat flux profiles on the wafer required for achieving thermal uniformity during processing. The given wafer temper-ature-distribution history is just our desired temperature trajec-tory required for thermal uniformity. Without loss of generality, we may assume that the desired temperature trajectory is the temperature history of the wafer center as calculated from the thermal model given above using constant incident heat flux (uniform heat flux), i.e., intensity mode during processing [2], [5].
The finite-difference method in the thermal model above at is used to construct the following matrix equation [14], [15]:
(11) Then the temperature distribution can be derived as fol-lows:
(12)
where and .
The vector contains values of the initial dis-tribution or the temperature disdis-tribution for the preceding time step. The vector is composed by the unknown incident
(a)
(b)
(c)
Fig. 3. (a) Desired temperature trajectory and inverse results for measurement errors = 0:0, = 0:001, and = 0:005. (b) Desired temperature ramp-up rate and inverse results for measurement errors = 0:0, = 0:001, and = 0:005. (c) Thermal distortions during uniform heat flux processing.
Fig. 4. Incident heat flux profile calculated by inverse method for measurement errors = 0:0.
heat fluxes for , , . As well, is the
grid number of the location of the estimated heat flux func-tion . The vector includes any known variables of the problem, and the vector contains the coefficients for the unknown variables . A time-sequential procedure is used to determine the unknown incident heat flux parameters. The time domain is divided into analysis intervals, each of length , where is the number of future time steps [11]. The parameters are determined simultaneously for each analysis intervals.
For the next time step , we arrive at
(13) In the same way, the temperature distribution at successive future times, , can be represented as follows:
(14) To stabilize the estimated results in the inverse algorithms, a temporary assumption that the incident heat flux is constant over future time steps is used
for (15)
Then, the temperatures at each -spatial grid
( , ) for each analysis interval
(a)
(b)
(c)
Fig. 5. Incident heat flux calculated by inverse methods at several times for measurement errors (a) = 0:0, (b) = 0:001, and (c) = 0:005.
This equation provides a sequential algorithm that can be used to estimate the unknown incident heat fluxes by increasing the value of by one for each time step. Thereafter, the inci-dent heat fluxes can be obtained iteratively along the temporal coordinate. The incident heat flux profiles on the silicon wafer required to maintain thermal uniformity during RTP can be de-termined from the desired temperature trajectories.
IV. RESULTS ANDDISCUSSION
The numerical solution techniques described above were used in a typical 12-in (775- m thickness) silicon wafer. A simula-tion from initial temperature C (300 K) transition over 20 s to a steady state of 1097 C (1370 K) is demonstrated at the surrounding temperature C (300K) under the
uniform incident heat flux W/cm . The
assump-tions of intensity mode and no reabsorption by the wafer itself were made during processing [2], [5]. The wafer center temper-ature and ramp-up rate were calculated by the finite-difference scheme during this temperature transition, as shown in Fig. 3(a) and (b), respectively, denoted by desired temperature, which was taken as the desired uniform temperature for the inverse calculation. Since there are no losses, also no intensity is re-quired for compensation initially, and therefore all the absorbed energy is used to increase the wafer temperature. But, in this study, since the temperature-dependent absorptivity and emis-sivity are 0.3 at the initial lower temperature, and 0.68 from 800 K to 1700 K (see Fig. 2), there is a sharp increase at the tem-perature range 600 K–800 K, which is approximately 400 C. The wafer is more efficient in energy absorption above this tem-perature. Thus, the wafer temperature rises more rapidly at this jump. As the wafer temperature increases with the increasing energy absorption, heat losses also increased as the wafer tem-perature increased and part of the absorbed energy is consumed for compensation. Accordingly, less absorbed energy is left for ramping. The ramp-up rate is decreased gradually. During the steady state, all the absorbed energy is consumed for compen-sating heat losses and nothing is left for ramping [5].
Fig. 3(c), a three-dimensional graph, shows temperature dif-ference during uniform heat-flux processing. The axis “Radial Position” shows the distance from the wafer center in centime-ters. The axis “Time” represents the time during this temper-ature transition. The vertical axis represents thermal nonuni-formity graphed according to the temperature differences be-tween points on the wafer and the wafer’s center. The signif-icant thermal gradients in the wafer undergoing this
temper-(a)
(b)
Fig. 6. Inverse results of (a) temperature distributions and (b) thermal distortions for measurement errors = 0:0.
ature transition were similar to the results reported by other authors [1]–[3]. Initially, the temperature difference developed near the wafer perimeter is small. Gradually, it becomes signif-icant firstly at the wafer edge and increases with the time. Fi-nally, when the wafer temperature approaches steady state, the greatest temperature difference of 25 C occurs during this tem-perature transient.
In general, the temperature at the wafer center spot was mon-itored during the rapid thermal processing. The calculated tem-perature trajectory of the wafer center as shown in Fig. 3(a) was adopted as our desired (or measured) uniform tempera-ture tracking required for thermal uniformity during processing. Since our desired (or measured) temperature trajectory was gen-erated from the “exact” calculated finite-difference solutions de-scribed in Section II, it is presumed to contain errors for succes-sive temperature measurement if there is an active temperature control. Random measurement errors may be added to the de-sired temperature trajectory in simulations, as described else-where [15]
(18) where the subscript “1” is the grid number of the spatial-co-ordinate at the wafer center and the superscript denotes the
(a) (b)
(c)
Fig. 7. Desired thermal uniformity and inverse results at several times for measurement errors (a) = 0:0, (b) = 0:001, and (c) = 0:005.
grid number of the temporal-coordinate. is the “exact” cal-culated temperature, is the “measured” temperature, is the standard deviation of measurement errors, and is a random number. The value of is calculated using the IMSL subroutine
DRNNOR and chosen over the range 2.576 2.576,
which represents the 99% confidence bound for the
measure-ment temperature. For the cases of and , in
this study, the respective measured temperatures C
and C are simulated.
We set
(19)
as the requirement of thermal uniformity during rapid thermal processing, for our known temperature distribution used in the inverse heat-transfer method described in Section III to eval-uate the incident heat flux profiles on the wafer. After the
incident heat flux profiles for thermal uniformity were deter-mined, the radial temperature distribution across the wafer was computed again using the finite-difference method described in Section II to make a comparison between the desired ture distribution and the inverse-method results. The tempera-ture trajectory and ramp-up rate calculated by inverse methods at the wafer center for measurement errors (means
“exact”), , and are also shown in Fig. 3(a)
and (b). From this figure, we can see that estimation errors re-sulting from the measured errors are reasonable. The greater the measurement errors, the less accurate the estimated results. In the case of , there is a good accuracy of estimated re-sults through such an inverse heat transfer method.
Fig. 4 shows the three-dimensional graph of the incident heat-flux profiles calculated by inverse methods for mea-surement errors during processing. It is found that the heat compensation for thermal uniformity is only needed near the wafer edge. The result is similar to those reported
Fig. 8. Desired and inverse temperature-difference results between wafer edge and wafer center for measurement errors = 0:0, = 0:001, and = 0:005.
by Perkins et al. [2]. During processing, the edge heating compensation increases with the increasing wafer temperature due to the increasing heat losses emitted from the wafer edge by fourth-power relationships with the temperature and temperature-dependent emissivity of the wafer. As the steady state is reached, the edge heating compensation is also to be constant. Fig. 5(a)–(c) shows the incident heat-flux profiles calculated by inverse methods at several times for measurement
errors of , , and , respectively.
The only edge heating compensation for is also seen in Fig. 5(a). Since the measurement errors affect the given tem-perature trajectory during processing, in the case of
[Fig. 5(b)] and [Fig. 5(c)], the incident heat flux profiles must be dynamically modulated with the measurement errors to maintain thermal uniformity in RTP systems. If it were not for the cooler edge of a wafer, thermal uniformity would be achieved by applying uniform heat flux profiles to the top or bottom surfaces of a wafer. However, because of the edge, the additional amounts of energy are directed to the edge for thermal uniformity, as shown in Figs. 4 and 5.
Fig. 6(a) shows a three-dimensional graph of the wafer temperature distributions calculated by inverse methods
for measurement errors during processing. The
thermal nonuniformity of temperature difference from wafer center calculated by inverse methods for measurement errors is demonstrated on Fig. 6(b). Initially, the thermal distortion developed at the wafer edge is small. As the wafer temperature increases, the thermal nonuniformity near the edge is increased with time. But the thermal nonuni-formity can be decreased by the modulation of incident heat fluxes, and the wafer returns to thermal uniformity at the higher steady-state temperature of 1097 C. Comparing with the uniform heat flux case in Fig. 3(c), we see that the maximum thermal distortion is reduced from 25 C to
Fig. 9. Maximum temperature difference calculated by inverse method in the 4-, 6-, 8-, and 12-in silicon wafers for measurement errors = 0:0, = 0:001, = 0:003, and = 0:005.
0.132 C. Fig. 7(a)–(c) shows the temperature differences between points on the wafer and the wafer’s center at several
times for measurement errors of , , and
, respectively. For , in Fig. 7(a), it also can be seen that the temperature difference is first developed at the wafer edge, and the evidence of the wafer’s returning the thermal uniformity is also shown at the time of 20 s (near steady state). In Fig. 7(b) and (c), the thermal nonuniformity is developed at all surfaces, not from the edge. However, the maximum temperature difference occurs at the wafer edge. Fig. 8 shows the temperature differences between wafer edge and wafer center for several measurement errors during the processing. From these figures, we can find that the thermal nonuniformity increases with increasing measurement errors but remained under 0.3 C when the incident heat fluxes on the wafer were dynamically varied according to the results calculated by inverse methods, although the dimensional measured error did reach 3.864 C (in the case of ). The dynamic incident heat flux results yielded by the present inverse method show that thermal uniformity could be achieved efficiently during rapid thermal processing.
The dynamic incident heat flux calculated by inverse methods on thermal uniformity are also studied in 4-in (600- m thick-ness), 6-in (675- m thickthick-ness), and 8-in (725- m thickness) silicon wafers. Fig. 9 shows the absolute value of maximum temperature difference from wafer center for these wafers in the temperature transients for measurement errors of ,
, , and , respectively. The
most maximum temperature differences for measurement er-rors of are 0.618, 0.776, 0.981, and 0.326 C for the 4-, 6-, 8-, and 12-in silicon wafers, respectively. The max-imum temperature difference of 0.981 C occurs in the case of
Fig. 10. Dynamic edge heating compensation scaling factors calculated by inverse method for 4-, 6-, 8-, and 12-inch silicon wafers.
the 8-in wafer. The 12-in wafer has the lowest maximum tem-perature difference. In the transient periods with measurement errors, the maximum temperature differences from wafer center remain under 1 C.
Furthermore, as mentioned above, in the exact measurement errors of , only edge heat compensation is needed to maintain thermal uniformity during processing. Following the works of Perkins et al. [2], we define the edge heat flux-scaling factor as the ratio of required edge compensation to uni-form heat applied at central region. Fig. 10 shows the edge heat flux-scaling factor from the present results calculated by inverse methods for 4-, 6-, 8-, and 12-in wafers, respectively. Because edge heat radiant emission increased with the increasing wafer temperature, in the initial transient, the scaling factor increased with time. But when the energy of the incident heat fluxes finally balanced with the total energy emitted from the wafer, the wafer reached a constant temperature, referred to as the steady state, the scaling factor is independent of time. Since wafer thickness has increased much less than wafer diameter, the amount of ra-diant edge emission has not increased much, so the required edge heating ratio has decreased. If the edge heat flux can be controlled as shown in Fig. 10, thermal nonuniformity during processing can be reduced considerably. The maximum tem-perature differences in our study were 0.184, 0.385, 0.655 and 0.132 C for 4, 6, 8, and 12 in, respectively, as shown in Fig. 9
for .
V. CONCLUSION
Through an inverse heat-transfer method, this paper presents a finite difference formulation for the detection of unknown in-cident heat fluxes for achieving thermal uniformity in silicon wafer during RTP. A simulated 12-in silicon wafer subjected to a uniform heat flux of 20 W/cm from 27 C transition to a steady state of 1097 C was studied at the surrounding temperature 27
C. Our results show that the thermal nonuniformity can be re-duced considerably if the incident heat fluxes on the wafer can be dynamically controlled according to the results calculated by inverse methods. Measurement error effects on the thermal uni-formity were also discussed. The most maximum temperature differences in our study were 0.618, 0.776, 0.981, and 0.326 C for 4-, 6-, 8-, and 12-in wafers, respectively. The maximum temperature difference occurred at the 8-in wafers. The 12-in wafer has the lowest maximum temperature difference. In the transient periods with measurement errors, the maximum tem-perature differences from wafer center were remained under 1 C. The required edge heating compensation for thermal unifor-mity in 4-, 6-, 8-, and 12-in silicon wafers was also calculated using the inverse heat-transfer method. The maximum tempera-ture differences in our study were only 0.184, 0.385, 0.655, and 0.132 C for the 4-, 6-, 8-, and 12-in wafers, respectively.
REFERENCES
[1] F. Y. Sorrell, M. J. Fordham, M. C. Öztürk, and J. J. Wortman, “Temper-ature uniformity in RTP furnaces,” IEEE Trans. Electron Devices, vol. 39, pp. 75–79, Jan. 1992.
[2] R. H. Perkins, T. J. Riley, and R. S. Gyurcsik, “Thermal uniformity and stress minimization during rapid thermal processes,” IEEE Trans.
Semi-conduct. Manufact., vol. 8, pp. 272–279, Aug. 1995.
[3] H. A. Lord, “Thermal and stress analysis of semiconductor wafers in a rapid thermal processing oven,” IEEE Trans. Semiconduct. Manufact., vol. 1, pp. 105–114, Aug. 1988.
[4] C. Hill, S. Jones, and D. Boys, “Rapid thermal annealing—Theory and practice,” in Reduced Thermal Processing for ULSI, NATO ASI Series
B: Physics, 1989, pp. 143–180.
[5] R. Kakoschke, E. Bubmann, and H. Foll, “Modeling of wafer heating during rapid thermal processing,” Appl. Phys. A, vol. 50, pp. 141–150, 1990.
[6] R. S. Gyurcsik, T. J. Riley, and F. Y. Sorrell, “A model for rapid thermal processing: Achieving uniformity through lamp control,” IEEE Trans.
Semiconduct. Manufact., vol. 4, pp. 9–13, Feb. 1991.
[7] S. A. Norman, “Optimization of transient temperature uniformity in RTP systems,” IEEE Trans. Electron Devices, vol. 39, pp. 205–207, Jan. 1992.
[8] J.-P. Zöllner, K. Ullrich, J. Pezoldt, and G. Eichhorn, “New lamp ar-rangement for rapid thermal processing,” Appl. Surf. Sci., vol. 69, pp. 193–197, 1993.
[9] T. J. Riley and R. S. Gyurcsik, “Rapid thermal processor modeling, con-trol, and design for temperature uniformity,” in Proc. Mater. Res. Soc.
Symp. , vol. 303, 1993, pp. 223–229.
[10] Y. M. Cho, A. Paulraj, T. Kailath, and G. Xu, “A contribution to optimal lamp design in rapid thermal processing,” IEEE Trans. Semiconduct.
Manufact., vol. 7, pp. 34–41, Feb. 1994.
[11] J. V. Beck, B. Blackwell, and C. R. St. Clair, Inverse Heat
Conduc-tion—Ill-Posed Problem. New York: Wiley, 1985.
[12] E. Hensel, Inverse Theory and Applications for Engineers. Englewood Cliffs, NJ: Prentice-Hall, 1991.
[13] O. M. Alifanov, Inverse Heat Transfer Problems. Berlin/Heidelberg, Germany: Springer-Verlag, 1994.
[14] K. Kurpisz and A. J. Nowak, Inverse Thermal Problems. Boston, MA: Computational Mechanics Publications, 1995, pp. 106–108.
[15] C.-Y. Yang, “The determination of two heat sources in an inverse heat conduction problem,” Int. J. Heat Mass Transfer, vol. 42, pp. 345–356, 1999.
[16] J. P. Hebb and K. F. Jensen, “The effect of multilayer patterns on tem-perature uniformity during rapid thermal processing,” J. Electrochem.
Soc., vol. 143, no. 3, pp. 1142–1151, Mar. 1996.
[17] P. Y. Wong, I. N. Miaoulis, and C. G. Madras, “Transient and spatial radiative properties of patterned wafers during thermal processing,” in
Proc. Mater. Res. Soc. Symp. , vol. 387, 1995, pp. 15–20.
[18] V. E. Borisenko and P. J. Hesketh, Rapid Thermal Processing of
Semi-conductors. New York: Plenum , 1997.
[19] A. Virzi, “Computer modeling of heat transfer in Czochralski silicon crystal growth,” J. Cryst. Growth, vol. 112, pp. 699–722, 1991. [20] G. Strang, Linear Algebra and its Application, 2nd ed. New York:
