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NUMERICAL CALCULATION OF THE SURFACE TEMPERATURE OF SEMICONDUCTOR TIPS IN
PULSED-LASER AP-FIM
M. Tomita, T. Kuroda
To cite this version:
M. Tomita, T. Kuroda. NUMERICAL CALCULATION OF THE SURFACE TEMPERATURE OF SEMICONDUCTOR TIPS IN PULSED-LASER AP-FIM. Journal de Physique Colloques, 1989, 50 (C8), pp.C8-59-C8-63. �10.1051/jphyscol:1989811�. �jpa-00229909�
COLLOQUE DE PHYSIQUE
Colloque C8, supplement au n o 11, Tome 50, novembre 1989
NUMERICAL CALCULATION OF THE SURFACE TEMPERATURE OF SEMICONDUCTOR TIPS IN PULSED-LASER AP-FIM
M. TOMITA' and T. KURODA
The Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567, Japan
A b s t r a c t
-
The surface temperature of semiconductor tips irradiated by pulsed-laser is calculated. This calculation is based on new idea because the idea used in calculation of temperature of metal tips cannot be applied to that of semiconductor tips. The radius of the tip apex is not an important factor in estimating the temperature of semiconductor tips irradiated by the pulsed-laser. The maximum surface temperature of the semiconductor tips apex reached depends almost linearly on the incident energy density of the pulsed-laser beams. The apparent optical absorption coefficient closely related to the angle of cone of tip also plays an important role in the temperature rise of the semiconductor tip. Here is a complementation of the previous paper, Surface Science 203 (1989) 295. From the numerical result the simplified equation is proposed and a example using modified Einstein's specific formula for specific heat is used.1. Introduction
The process of filed evaporation has been extensively studied both experimentally and theoretically because of its importance in field ion microscopy and atom probe method /I/. Very few of these studies, however, have involved semiconductors. Sakurai et al. /2/ found that the low temperature field evaporation of silicon in vacuum was anomalous and concluded that cluster field evaporation resulted from microscale rupture due to Maxwell stress from the high electric field. Later, Kellogg /3,4/ found that at high temperature, cluster ions of silicon field evaporated and reported that the anomalous field evaporation (random cluster formation ) observed at low temperature was replaced by a uniform and layer-by-layer field evaporation at a certain temperature. Above all, the introduction of pulsed-laser unit brought about discovery of a great interest in semiconductors. In pulsed-laser stimulated field evaporation of silicon, the photo excitation was found but not in the case of metals by Tsong /5,6/. He also found that the species of cluster ions was related to the highly symmetric, small units of atoms existing in silicon. On the other hand, we found /7/ that in pulsed-laser stimulated field 'l)Present address: Shindengen Electric Mfg. Co., Ltd, 10-13 Minami, Hannno, Saitama 357,
Japan.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989811
evaporation of gallium phosphide, the ion distribution varied with field strength and laser energy density and could be clearly dived into three regimes. Later, this phenomenon was found to be closely related to the surface reaction and temperature 181.
As Liu e t al. 19,101 pointed out, it is fundamentally necessary to understand how a tip can be heated by a laser pulse, how fast it can cool down after the laser pulse irradiation, what is the temperature profile of the tip, and how this profile evolves with time. In short, it is important to obtain more information on the temperature of the tip surface irradiated by pulsed-laser to clarify the field evaporation of -semiconductors. However, the answers will never be gained by experimental measurements but theoretical calculations. The calculation method of the semiconductor tips carried out by us 1111 is different from that of metal tips carried out by Liu et al. because of the penetration of a laser pulse is not negligible. That is, we chose a finite differential method and calculated the surface temperature rise of semiconductor tips. Here, we also propose a further simplified method to calculate the surface temperature based on the numerical re'sult.
2. Equation and solution of tip temperature
As shown in the previous paper /11/, the equation of temperature of a semiconductor tip is as follows:
where c i s the specific heat, p is the density, K is the thermal conductivity, IL is the incident laser energy density, R, is the optical reflectively, rt is the tip radius, and a is the optical absorption coefficient. This equation implies that the tip is irradiated in one dimension by the incident laser beam and can be translated into a finite differential equation for numerical calculations as follows:
where Tjn+l is the temperature of the j th slice at time t = (n+l)At, (Qd)j is the amount of heat slice j received by thermal conduction from neighboring slices, (Q,)j is the input laser energy directly absorbed by slice j
.
These terms are as follows:where Kj+1/2 and K,(j_l12 are the thermal conductivities at the two boundaries of the slices, expressed as follows:
The stability condition is as follows /12/:
Only if the parameters satisfy eq. (7), numerical calculation of the differential equation is stable in spite of the dependency of the parameters on the temperature. This method also has merits in point of programming and computation time.
By the way, the result of numerical calculation mentioned above provides a helpful and important clue to the problem of laser pulse heating of the semiconductor tips. In fact, the maximum surface temperature depends almost on (Q,)j but not (Qdj
.
That is,the surface temperature is not almost changed by the neighboring slice. Even if a temperature change by the neighboring slice exists, such a change can be eliminated by variation of an optical absorption coefficient. Therefore, the equation of surface temperature Ts can be approximately simplified such as follows:
This equation is probably ultimate and will not be further simplified. Moreover, an analytical solution on proper assumption can be obtained eliminating nuisances in numerical calculation methods with a computer. For instance, in the previous work 1111, it could not be helped but assume that the specific heat is constant in order to save computation time and to realize a lowering of the costs of computer use.
It is well known that the specific heat varies significantly at low temperature. For instance, from Einstein's formula for specific heat, Wolf 1131 considers the specific heat as follows:
where R is the gas constant, x is an auxiliary parameter expressed as follows:
where TD is the Debye temperature. By the way, if x is defined as follows:
Einstein's gives better agreement with Debye's when temperature has relation with Debye's temperature such as follows 1141:
This may not be so well known, but can be easily confirmed. Therefore, assuming that all parameters except for specific are constant during the irradiation of laser pulse, an analytical solution of eq. (8) is as follows:
LASER ENERGY DENSITY (lo6 ~ l c r n ~ )
Fig. 1. The maximum surface temperature of the tip apex. as a function of the laser energy density for Silicon based on eq. (2). dashed curves, and eq. (13). solid curves, with the initial temperature
= 100 K and 300 K.
where t~ is the irradiation time of laser, C1 and C2 are constants as follows:
3. Results a n d discussion
The result of calculations of the maximum surface temperature of the semiconductor tip based on eq. (2) and eq. (13) is shown in fig. 1. For the parameters in case of eq. (2), a density p = 2.33 g/cm3, a specific heat c = 0.7 J/g-K, an optical absorption coefficient a = 7 x 104/cm, and a reflectivity R, = 0.55 were taken. Experimental data on the thermal conductivity K exist limitedly. Therefore, we assumed that the thermal conductivity decreases as KRIT. KR is the value of the thermal conductivity at room temperature. For calculation, KR was taken 400 W/cm*K. In case of eq. (13), c = 3R, t~ = 10 ns, p = 2.33 g/cm3, a = 7xl04/cm, R, = 0.55, TD = 658 K /13,15/. As shown in fig. 1, the maximum surface temperature is almost in proportion to the laser energy density. This result indicates that the eq. (13) is not so bad approximation and the factor defined in the previous work /11/ is essentially important. Therefore, eq. (13) shows the same results as numerical calculation: (1) the radius of the tip apex is not an important factor in estimating the temperature of semiconductor tips irradiated by the pulsed-laser (2) the maximum surface temperature of the semiconductor tips apex reached depends almost linearly on the incident energy density of the pulsed-laser beams. (3) the apparent optical absorption coefficient closely related to the angle of cone of tip also plays an important role in the temperature rise of the semiconductor tip.
Anyway, the most important thing is how to estimate the value of each parameter although the maximum surface temperature depends not on the radius but the optical absorption coefficient of the tip /11/.
R e f e r e n c e s
/I/ See, for example, E.W.Mueller and T.T.Tsong, in: Progress in Surface Science, Ed. S.G.Davison , New York, 1973) Vol. 4.
/2/ T.Sakurai, R.J.Culbertson and A.J.Melmed, Surface Sci. 78 (1978) L221.
/3/ G.L.Kellogg, Surface Sci.
120
(1982) 319.141 G.L.Kellogg, Surface Sci.
124
(1982) L55./5/ T.T.Tsong, Appl. Phys. Letter
45
(1984) 1149./6/ T.T.Tsong, Phys. Rev. &X! (1984) 4946.
/7/ M.Tomita and T.Kuroda, Surface Sci. 201 (1989) 385.
/8/ M.Tomita and T.Kuroda, Proceeding of 36 1.F.E.S (1989) /9/ H.F.Liu and T.T.Tsong, Rev. Sci: Instr.
55
(1984) 1779./ l o / H.F.Liu, H.M.Liu and T.T.Tsong, J. Appl. Phys. 59 (1986) 1334.
/11/ M.Tomita and T.Kuroda, Surface Sci. 203 (1988) 295.
1121 Richtmyer and K.W.Morton, Differential methods for initial value problems, 2nd Ed.
(Interscience-Publishers, New York, 1967).
1131 H. Wolf, Semiconductors (Wiley, New York,1971).
1141 C.Hamaguchi, Solid State Physics (Maruzen, Tokyo, 1976) in Japanese.
/15/ K.Seeger, Semiconductor Physics (Springer, Wien, 1973).
