- Email: [email protected]
Solidification front stability during zone-melting recrystallization of thin silicon films
Journal of Crystal Growth 126 (1993) 275—284 North-Holland
~
o,
CRYSTAL GROWTH
Solidification front stability during zone-melting recrystallization of thin silicon films Sharon M. Yoon and loannis N. Miaoulis
1
Thermal Analysis of Materials Processing Laboratory, Mechanical Engineering Department, Tufts University, Medford, Massachusetts 02155, USA Received 14 April 1992; manuscript received in final form 25 August 1992
The effects of constitutional supercooling and scanning speed on the stability of the solidification front during zone-melting recrystallization of thin silicon films were investigated. We found that constitutional supercooling did not effect the stability of the solid/liquid interface for typical concentrations of oxygen, nitrogen, and carbon found in zone-melting recrystallized films. Interface growth was stable for scanning speeds less than 250 p~m/s.The critical impurity concentration for which unstable growth is induced for scanning speeds smaller than 250 ~tm/s was identified. Carbon and nitrogen diffused into the silicon film seem more likely to cause unstable solidification than oxygen.
1. Introduction Multilayer film structures are currently used in the development of advanced electronic devices. One type of film structure, the silicon-on-insulator (S DI) structure, is often used to obtain lower junction and metallization capacitances and to avoid latch-up, a phenomenon in which a parasitic circuit draws a large current which can potentially damage the integrated circuit. A typical SOl structure consists of a substrate of silicon (-~400 p~m),topped with a layer of silicon dioxide (—~1 /Lm), a layer of polycrystalline silicon (—~1 jsm) and an insulating layer of silicon dioxide (—~1 ~.Lm)(fig. 1). Electronic devices are built in the encapsulated silicon film in the SO! structure. In order for the silicon film to be usable, it is recrystallized to a single-crystal using the zone-melting recrystallization (ZMR) technique. In ZMR processing, the wafer is preheated with a bottom heater/susceptor to a temperature below the film’s melting point. A line heat source,
1
Author to whom correspondence should be addressed,
0022-0248/93/$06.00 © 1993
—
such as a graphite-strip heater (—~2 mm x 2 mm), scans the surface of the film, creating a moving narrow molten zone as shown in figs. 1 and 2. The film solidifies into a single crystal following the motion of the strip heater. Since the film is very thin (—~1 ~.tm),it is difficult to obtain accurate temperature measurements experimentally, and thus, necessary to model the recrystallization process analytically or numerically. Since the crystal quality is determined by the temperature profiles within the silicon film and the stability and morphology of the solidification front during recrystallization, it is necessary to monitor the temperature distribution computationally during recrystallization. The morphology of the solid/ liquid interface determines the localization of defects in the recrystallized film since defect trails and subboundaries are generated at the cusps of cellular morphologies or at the interior corners of faceted interfaces [1]. A nonplanar interface morphology caused by an unstable solidification interface [2] is a precursor to subboundary formation [1]. Stability of the interface can be altered by temperature gradients at the interface and by impurity segregation which may lead to constitu-
Elsevier Science Publishers B.V. All rights reserved
276
S.M Yoon, I.N Miaoulis
/
Solidification front stability during ZMR of thin Sifilms
I Graphite Strip Heater I ~ Silicon
____________
I ~ilicon ~iystal ~1 Silicon ~
~
silicon
~
NBottom heating susceptor
~
Fig. 1. Schematic drawing of the silicon-on-insulator structure during zone-melting recrystallization processing.
tional supercooling. An impurity in the melt will be forced towards the solid/liquid interface during solidification [3]. When an impurity concentrates at the crystallization front more quickly than diffusion can transport it, the liquidus ternperature becomes depressed [2—4].If the temperature of the liquid in front of the interface is lower than the liquidus temperature, constitutional supercooling occurs [1,5,6]. In previous studies of silicon film recrystallization, researchers hypothesized two reasons for unstable solidification: constitutional supercool-
liquid silicon. Fan, Tsaur and Chen [71 found oxygen localized in subboundaries of ZMR recrystallized SOl structures. They reported that faceting and subboundary formation was due to constitutional supercooling and that oxygen had dissolved into the molten silicon zone from adjacent silicon dioxide layers. Mertens et a!. [8] also found that impurities were absorbed into the film from adjacent layers. They found that oxygen and nitrogen introduced from a plasma enhanced CVD capping layer were saturated in the silicon film during ZMR processing of silicon on Si02
ing inflicted by impurities in the melt, and abrupt changes in absorbed radiation at the interface due to different reflectivity values for solid and
films. They proposed that nitrogen could cause dendritic growth but oxygen was unlikely to do so. Leamy et al. [61 discovered the presence of impurities which had solidified into subboundaries of processed films. This was most likely the result of constitutional supercooling since constitutional supercooling brings about a cellular or
Graphite Strip
Sal ~ Wafer ‘~<‘~ ~-~‘~lid
~
~ ~
_____
v
~ ~ ~
~
~—2” y
Susceptor
Fig. 2. Schematic drawing of the zone-melting recrystallization process.
dendritic interface morphology [2] and impurities concentrate in the cusps of a cellular formation [2]. Lee [11 concluded that constitutional supercooling led to the breakdown of the solidification front which caused subboundary formation dur ing studies of energy beam recrystallization of SOI structures According to Dutartre Haond, and Bensahel [91 impurity segregation most likely caused the instability of the interface during stud-
S.M Yoon, IN. Miaoulis
/ Solidification front stability
ies of ZMR of SOl wafers. Geis et a!. [10] studied possible constitutional supercooling during ZMR processing and found that for the impurity concentrations found in graphite-strip heater ZMR, constitutional supercooling was unlikely for scanning speeds less than 200 lLm/s. Im et al. [11] also found no supercooling effects at scanning speeds below 300 nm/s but suggested that impurities in the melt might trigger unstable growth at higher speeds. Other investigators have found that unstable growth results from the changes in reflectivity values during phase change during solidification of silicon films [12—171,when the effects of constitutional supercooling were not taken into consideration. Jackson and Kurtze [121 studied the stability of lamal!ae growth in silicon films for a stationary interface using a Mullins and Sekerka stability analysis [4] and found that the change in optical properties during phase change induced unstable growth. In Jackson and Kurtze’s analysis, heat transfer was considered in one direction with a heat source radiating the top surface of a film. Im et al. also studied lamellae growth in thin films, as well as the solidification front, during zone-melting recrystallization with a graphitestrip heater [13]. He modelled the heat conduction in the film in one dimension and employed linear approximations of the radiation to the top and bottom surfaces of the film in a heat source term. A cosine function was used as an estimate of the intensity profile. Grigoropoulos, Buckholz, and Domoto [14—161analyzed the stability of moving solidification interfaces during silicon film recrystallization using a laser heat source with a Gaussian intensity profile. They modelled the heat transfer during recrystallization as one-dimensional conduction in the silicon film which received heat from the glass substrate and the laser. Unstable solidification resulted from increasing the scanning speed of the laser [14,16]. Limanov and Musatova [17] considered the effects of constitutional supercooling using a Mullins and Sekerka [4] stability analysis of ZMR processed silicon films. In their analytical model, they solved two coupled one-dimensional heat conduction equations representing the substrate and the silicon film and assumed a linear temper-
duringZMR of thin Si films
277
ature profile throughout the substrate. Their resuits showed stable growth for scanning speeds up to 2000 pm/s neglecting effects of constitutional supercooling. However, unstable growth has been experimentally observed for velocities greater than 350 jim/s in graphite-strip ZMR processing [11]. We believe that the discrepancy between the theoretical [17] and experimental [11] findings is due to the simplifying assumptions which affected the temperature profiles at the solidification interface (i.e. linearization of radiative terms, Gaussian profile assumed for the strip intensity, and linear temperature profile assumed in the substrate). When the effects of impurity segregation were taken into account in their stability analysis [17], they found that impurities caused unstable solidification at scanning speeds above 1000 ~tm/s [17] which is still above the experimental threshold velocity to cause unstable growth. [11] This study by Limanov and Musatova differs from the stability analysis to be presented in this paper because the heat transfer during ZMR processing was modelled differently, and because different boundary conditions were used to determine the impurity concentration distribution. Although previous studies on the stability of the solidification interface of thin silicon films undergoing ZMR with an infrared heat source, such as a graphite-strip, provide useful qualitative insight in the mechanisms involved, the simplified thermal models limit the ability to deduce quantitative information. Recently developed numerical models accurately estimate the temperature profiles as the solidification interface by taking into consideration all the significant thermal phenomena: optical property variation during phase change [18,19], two-dimensional heat conduction, non-linear treatment of radiative boundary conditions, exact derivation of the intensity profile [191, and heat strip motion [20]. This paper presents a numerical stability analysis including constitutional supercooling effects that incorporates thermal profiles obtained for a moving solidification interface using these numerical models. An overview of the Mullins and Sekerka stability analysis [4] as it applies to the present situation is given, and results for typical ZMR processing
278
SM. Yoon, IN. Miaoulis
/ Solidification front stability during ZMR
conditions are presented. This analysis includes the effects of three cases of constitutional supercooling triggered by different impurities (oxygen, carbon, and nitrogen) under typical ZMR processing conditions. This analysis differs from previous analyses which did not take constitutional supercooling into consideration [12—16]because impurities present in the melt control the motion of the solidification front.
2. Stability theory
of thin Sifilms
polynomials expressed as functions of distance from the interface. Due to the small thickness of the silicon film, the temperature variation in the y-direction is negligible so the temperature distribution can be considered one-dimensional in the x-direction. The second part of the temperature solution accounts for the change in temperature due to the perturbation inflicted on the morphology of the solidification interface. The solution technique can be found in the original analysis [4]. A sinusoidal perturbation, 111, is introduced at the interface:
In this stability analysis, the complete temperature distribution for a perturbed interface morphology is comprised of two parts. The first part of the temperature distribution, T10(x), is solved assuming that the interface morphology is nonperturbed. The temperature distributions in the solid and liquid regions of the interface within the silicon film are determined by solving the heat conduction equation:
where ~ is the original amplitude of the perturbation (1 X i0” cm), w is the growth rate of the amplitude of the perturbation (~1), t is time (s), and q is the wave number of the perturbation (cm i) The complete temperature distribution including the contribution of the perturbed interface has the following form:
d21°
Tj(~~’z) = T~.°(x) + Ti(x) ~ e’°~’ sin qz.
d2T° +
=
~
y
o,
=
(‘s, if film is solid ~l, if film is liquid’ (1)
cP = ~ eu” sin qz,
(2)
—
(3)
T,°(x)is the basic-state temperature field and where T 1 is the film temperature (K). This two-dimensional heat conduction equation is solved using a numerical model which employs finite-difference techniques to simulate thermal processing with a graphite strip [19,20]. Radiative exchanges between the film structure and the graphite strip heater above or the susceptor below the film structure are implemented as heat sources in the boundary conditions for the control volumes on theGraphite upper and surfaces of the film structure. striplower motion is modelled [20] by re-evaluating the heat sources in the boundary control volumes as a function of the relative position of the control volume to the graphite strip heater. The temperature distribution in the region near the solidification interface was separated into two equations representing
is represented by the fourth order polynomial expression for the temperature profile found using our numerical model. I[~(x)~ec~~t sin qz is the leading order perturbation. This form of solution is a regular normal-mode small amplitude expansion for the temperature field [4,15,16]. Ti(x) can be determined from the following perturbation differential equation [15,16]: 2I~ v dT~ I 0) J ddx2 + I q2 + ~ + = 0. (4) adx \ a, —
—
Ti is the film temperature (K), v is the velocity of the strip heater (cm/s), a is the thermal diffusivity (0.093 cm2/s for solid and 0.281 cm2/s for liquid), x is the distance from the interface (cm), k 1 is the thermal conductivity (0.2 W/cm K for solid and 0.6 W/cm K for liquid), h is the thickness of silicon film (1 x2). iO~ and J is Thecm) heat source the sourceheat termemitted (W/cmfrom the top of the termheat includes .
the solid and liquid domains. In order to obtain a continuous expression of the computed temperature distribution, theare temperature distributions in the solid and liquid represented as 4th order
SM. Yoon, I.N Miaoulis
/ Solidification front stability during ZMR
film, the radiative interaction between the film and the bottom susceptor, and heat absorbed from the graphite strip heater [13]. The form of solution of the perturbed temperature differential equations is: T~(x)=a eA,1, (5) where a is a constant to be determined and A 1 is the eigenvalue of the perturbation equation. Assuming that the temperature far from the perturbed interface is equal to the temperature at the same point in the case of an unperturbed interface, the eigenvalues for the perturbation differential equations are evaluated:
_______________________ V 2
V
2a 1
~ ~2a1j
a1
(6)
/
T
0(x) +a e~ 8 e°~t sin qz. (7) 1(x,z) = T1 Temperature at the interface is affected by the curvature of the interface and a depressed liquidus temperature due to constitutional supercooling. The decrease in temperature due to the curvature, Zh7~urvature, can be determined by the Gibbs—Thomson equation [4,12]: —TmpF6q2 ewt sin qz,
279
K/at% for carbon and —600 K/at% for nitrogen) [101,and c4 is the concentration of impurity at the interface which is calculated at a later point. Including these effects, the temperature at the interface, T4, is: 2e~’tsin qz. (10) T4=T~p+mc4—T~pF6q The distribution of the impurity was solved in a similar manner to the temperature distribution. First, the concentration distribution is solved for the case of a nonperturbed interface using the one-dimensional steady-state equation for diffusion [4,21]: 2c° v dC° d ~ (11) x where c is the concentration of the impurity
where C is a constant that depends on the heat source domain term, J,has of eq. (1). Thesecond eigenvalue the liquid a negative term,for while the solid domain has a positive term. Thus, the complete temperature distribution within the silicon film will be of the form:
LtTcurvature =
of thin Sifilms
(8)
where Tmp is the melting point of silicon (1685.15 K) [12—16] and F is the capillary constant of silicon (3 x 108 cm) [12—161.The temperature will also decrease due to the depression of the melting point from constitutional supercooling, ~1I~S [1,5,6]:
(at%) and D~is the diffusion coefficient of the 2/s for oxygen, in silicon x iO~ cm7 X 10~0cm2/s 1impurity X iO~ cm2/s for (8 carbon, and for nitrogen) [22]. The impurity distribution in the melt was solved for the case of a nonperturbed interface for known values of concentration in the solid, C~,far from the interface, and a concentration in the melt, CL, at the interface. The value CL, may be found by the segregation coefficient, k, for the impurity [21]: CL
=
Cs/k.
(12)
The following values for impurity concentration were estimated from secondary ion mass spectrometry [10,23] for graphite-heater recrystallized films: (3 x iO~ at% for oxygen, 2 x iO~ at% for carbon, and 2 x 10-8 at% for nitrogen). The segregation coefficients used in this analysis are 0.9 for oxygen [8], 0.06 for carbon [10], and 7 X iO~ for nitrogen [10]. For a nonperturbed interface, the impurity concentration as a function of distance from the interface is [21]: c°(x) = C~+
(CL
—
C~)ex1’~)c.
(13)
(9) where m is the slope of the liquidus line (freezing point) in the phase diagram between the impurity and silicon (—420 K/at% for oxygen, —520
The same perturbation introduced to the interface morphology which was used to solve for the complete temperature distribution is also used in solving for the concentration distribution (eq.
280
S.M Yoon, IN. Miaoulis
/
Solidification front stability during ZMR of thin Sifilms
(2)). The concentration distribution for a perturbed interface morphology was solved using a variation of the Mullins and Sekerka analysis [4] similar to previous work by Pamplin [21]. As time approaches oc, the solute concentration at every point in the liquid varies with time as e~t.The form of the concentration distribution is similar to the form of the complete temperature distribution; it is a regular normal-mode small amplitude expansion for the impurity field: c(x,z)
c~(x)+ c~(x)8 e”t sin qz.
=
(14)
The following perturbation differential equalion is solved for impurity diffusion within the melt [21]:
(
v dCi
d2ci ~+—~—
dx2
D~ dx
constant a can be solved, where b is an unknown constant still to be determined: a
=
mb
—
D~)
D~
where b is a constant to be determined. The eigenvalue, A~,of the differential equation for the impurity diffusion is: / V
—
2
2+
V
2D~— ~(2D~)
— 0)
.
—
1)
d
)
(20)
Using eqs. (7), (18), and (19), and the second part of eq. (20), b can be solved:
(16)
e~,
‘dc\
c
=
4( k
ci(x) =b
KLIdTL\ E~-~i-)
—)—
The form of the solution of the concentration distribution for the perturbed differential equation of impurity diffusion is:
(19)
—
e’°tsin qz— K~(d1~\ pL dx
(15)
q2+_~ci=O.
mV “ñ~~(CL Cs).
—
Since the constant a, part of the temperature solution, is dependent on constant b, part of the impurity distribution, the thermal distribution is related to the impurity distribution. In the following expression, thermal diffusion is coupled to impurity diffusion since the solidification front moves at the same speed, due to the diffusion of the temperature and the motion of the impurity in the melt [4]: V+w6
~
TmpFq2
b
=
I
D~pL V k
+
(17)
(
V
1 (CL C~) ~ A~) 2V ~(CL— C~)(KsT~ KLTL) + —
~
—
xI(KSTS
—
CL~]
—KLTL) +mCL(KSAS—KLAL)
+q
The complete concentration distribution including the effect of the perturbed interface morphology is:
—
A~D~pL 1 ~ k 1
j
—
—
V
~
—
C~)~
(21)
and c(x,z)
= C~+ (CL—
C~)ex~’~)c ~=
+b e~kcx6 e~’tsin qz.
—2(KsT~’—KLTfl
(18) 2mV
The concentration at the interface, c 4, which is used in the boundary condition for the temperature at the interface (eq. (9)) can be determined from eq. (18), by using the position of the interface, x = 6e~’sin qz, as the position in this equation. Using the boundary condition for the ternperature at the interface (eqs. (7) and (10)), the
+ (Tmp~2 + ~(CL
X (K~A5 KLAL) —
—
+
CS)~ /
(K~A~T~KLALTL). (22) —
From eqs. (7) and (19), along with the first part of eq. (20), the following expression can be de-
S.M Yoon, I.N Miaoulis
/ Solidification front stability during ZMR of thin Sifilms
rived which enables us to determine the value of 0) and the interface stability:
pLw
=
(—k~T~A+8 kLTLAL) 2(kLAL k + TmpFq 1A8)
281
fication front on the liquid side is greater than the rate on the solid side. The fourth term contains the effects of constitutional supercooling. If impurity segregation were not included in this
—
B
+
(2k5T~’ 2kLT~)
—
mB( K5A5
V = ~-(CL
analysis, this last term would be omitted, and the remaining stability equation would be unaltered. Previous researchers who have performed stabil-
—
—
C8)
—
—
KLAL),
b,
(23) (24)
where T~and T~’are the rates of change of the thermal gradients at the interface obtained from Tnumj(X). The interface is stable if the rate of growth of the perturbation, w, is negative, and the perturbed interface will return to the original interface morphology. If the calculated value of w is positive, the perturbation will continue to grow from the interface and growth in unstable. To evaluate stability, all frequencies of the sinusoidal function must be used in eq. (23) to represent Fourier terms of the continuous perturbation function. Only when w is negative for all perturbation wave numbers, q, is the solidification growth stable. If w is positive for any perturbation wave number, the interface growth is unstable, since the growth is characterized by the term, e~’. For typical processing conditions, of strip ternperature 2300°C, and susceptor temperature 1000°C,the stability of the solid/liquid interface was studied for processing speeds up to 1000 jim/s. For each processing speed, the effects of impurity segregation on the stability were determined.
ity analyses neglecting constitutional supercooling have found expressions similar to eq. (23), excluding the fourth term [12,13,16]. The stability equation found by Limanov and Musatova [17] has a similar form; however, their constant, B, differs from eq. (24) because their solution for the impurity concentration was based on assuming a specific concentration gradient at the interface. The last term in eq. (23) will be stabilizing when the constant, B, is negative since m is a negative value and the term, (KSAS KLAL), is always positive. From eq. (24), we conclude that the first term is destabilizing since K4 is always positive. In a previous analysis, we only studied the effects of scanning speed on the stability of the solidification front [25]. Since supercooling effects were not considered, the decrease in melting point at the interface was not taken into account (eq. (9)), and only the first equality was of interest in eq. (20). We found that stable solidification occurred until a scanning speed of 250 /Lm/s. For faster scans, the temperature gradient at the interface in the solid material became less steep because there was less time for radiative cooling. A decrease in this gradient caused unstable growth because the first term in eq. (23) became positive and dominated the stability expression. By observing the destabilizing term due to impurity segregation in the melt which is 2V(CL CS)/DC, one can easily see that a higher processing speed or a higher impurity concentration in the melt could induce an instability. Using typical concentrations of oxygen measured in ZMR processed silicon films [10,23], we found that the interface growth was stable for scanning speeds below 250 jim/s. Stability results for oxygen are shown in fig. 3. In this figure, the growth rate remains below zero for velocities of 150 and 200 jim/s. For higher velocities of 250 and 400 j.tm/s, the growth rate, to, exceeds zero for cer—
—
3. Results and discussion The four terms in eq. (23) describe the three criteria for interface stability. The first term is stabilizing when the temperature gradient of the interface on the solid side is steeper than the gradient on the liquid side. The second term, a capillary term, is always negative and stabilizing, The third term is stabilizing when the rate of change of the temperature gradient at the solidi-
282
S.M Yoon, IN. Miaoulis
/ Solidification front stability during ZMR of thin Si films
to
to measure the amount of impurity in a film, which could actually be higher in a molten phase, it is necessary to determine the critical concentralion of impurity which would trigger instability. To find the concentration of impurities needed to induce unstable solidification at scanning speeds below 250 jim/s, we varied the concentration. The critical concentration of oxygen that would result in unstable solidification is shown in fig. 4a. As the velocity increases, less solute is necessary to cause unstable growth. As long as the term V(CL C5)/D~is greater than the value of b, the
~
0.0
~>-.. ~
~
—to
‘~‘~
\,,,
-
—zo
I
0.0 300.0 600.0 PERTIJRBA11ON WAVE NUMBER, q (1/cm)
—
Fig. 3. Rate of growth of perturbation when oxygen is present in the solidifying melt for the following scanning speeds: )15O,am/s,( )200 am/s (———)250~sm/s, an — i~m/s.
fourth term of the stability equation (eq. (23)) will be destabilizing. If the velocity of the strip heater 1), increases, (CL— C~) can decrease and still produce the same effect. For velocities above 150
tam wave numbers, q, and the perturbation continues to grow away from the interface. This figure is consistent with work by Pamplin [21], who found that short wavelengths are stabilized by capillary and long wavelengths by the slow diffusion of the solute. However, since unstable growth was also instigated at a scanning speed of 250 jim/s for the case without supercooling, we believe that constitutional supercooling due to oxygen in the film was not the cause of this instability. The decrease in temperature gradient of the solid material at the interface induced unstable growth in this case. Previous researchers have suggested that the level of impurities found in ZMR processes films do not contribute to unstable growth [11]. However, since it is difficult
jim/s, the critical impurity remains at —~0.02 at%. At this point, although the velocity increases, the value of b also increases and the concentration remains at this level. The amount of impurity needed to force the interface growth to be unstable ranges between 0.02 and 0.07 at%, which is three orders of magnitude larger than the measured amount of impurity in ZMR recrystallized silicon films [10,23]. This would suggest that oxygen would be unlikely to trigger instabilities during processing. For a typical carbon concentration diffused in solidifying melt, solidification is stable for scanning speeds below 250 jim/s, suggesting that this concentration of carbon does not induce unstable growth. The concentration of carbon needed to trigger instabilities in the solidification front for
0.10
3.0
10.0
0 0
0
‘
a 0.00
50.0
z
2.0
150.0
VELOCflI
(wrVs)
250.0
to
b 50.0
0~
150.0
o
o
VEI.QCT1Y (tLrrVs)
o
~ 250.0
o 0.0
50.0
150.0
VtLOCfl~(ani/s)
250.0
Fig. 4. Critical concentration of (a) oxygen, (b) carbon and (c) nitrogen (at%) needed which would induce unstable solidification for scanning speeds below 250 ~sm/s.
S.M. Yoon, IN. Miaoulis
/ Solidification front stability during ZMR of thin
low scanning speeds varies from 0.00013 to 0.00023 at% (fig. 4b). The critical concentration decreases slightly from 50 to 200 jim/s. At 230 jim/s, there is a sharp decrease in the amount of carbon needed to start an instability, At this velocity, the value for b decreases for the specific diffusion coefficient of carbon in silicon. The concentration of solute required to destabilize the interface is only one magnitude higher than the values measured experimentally [10,23]. In films containing nitrogen, unstable solidification occurred for scanning speeds above 250 jim/s, which similarly suggests that this concentration of nitrogen found in ZMR processed films does not cause instabilities during growth. In fig. 4c, the critical concentration of nitrogen for low scanning speeds is shown. This concentration ranged between 2 x i0~ and 8 x iO~ at%, which is only one order of magnitude above the actual measured values. For increasing scanning speeds, the critical concentration decreased as expected. Minute changes in temperature gradients cannot be captured due to the size of the nodes in the numerical grid. This results in calculated critical concentrations that are similar for different velocities. The amount of constitutional supercooling at the interface was calculated for each impurity at measured concentration levels. It was found to be 0.01°Cfor oxygen impurities, 0.2°Cfor nitrogen, and 0.17°Cfor carbon. Our results show that oxygen seems unlikely to induce unstable growth. Carbon and nitrogen impurities could induce unstable growth if the concentration were increased by an order of magnitude from typical concentrations measured. Actually, the concentration of solute in the melt may be higher than the experimental values, since the impurity was measured in solid material. It is also possible that a combination of impurities is present during recrystallization. In such cases constitutional supercooling may have been a cause of instability.
Sifilms
283
constitutional supercooling and scanning speed. The contribution to unstable growth of typical impurity concentrations for oxygen, carbon, and nitrogen was investigated. For each of these impurities, solidification was stable up to a scanning speed of 250 jim/s which is the same velocity where unstable growth occurred when constitutional supercooling effects were neglected. Concentration levels for each of the impurities were varied to find the critical concentration which would induce unstable solidification. To trigger instabilities at slow scanning speeds ( < 250 jim/s) in the solidification front, the concentration of oxygen would have to be three orders of magnitude higher than experimentally measured values which suggests that oxygen would not cause unstable growth. However, if concentrations of carbon and nitrogen were increased one order of magnitude from the experimentally measured vatues, growth would be unstable at scanning speeds below 250 jim/s. The actual impurity concentration is most likely greater than the experimentally measured values since the solute measurement was taken in solid material.
Acknowledgement This research was sponsored by the National Science Foundation under grant CTS-9 157278.
References [1] E. Lee, Mater. Res. Soc. [2] B. Chalmers, Principles
Symp. Proc. 35 (1985) 563. of Solidification (Wiley New York 1964) pp 150—157 [31D.P. Woodruff, The Solid—Liquid Interface (Cambridge University Press, London, 1973) p. 83. [4] W.W. Mullins and R.F. Sekerka, J. AppI. Phys. 35 (1964) [5] CA. Knight, The Freezing of Supercooled Liquids (Van Nostrand, Toronto, 1967) p. 87. [6] H.J. Leamy, CC. Chang, H. Baumgart, R.A. Lemons and
4. Conclusions
J. Cheng, Mater. Letters 1 (1982) 33. [7] i.C.C. Fan, B.-Y. Tsaur and C.K. Chen, Mater. Res. Soc. Proc. 23 D.J. (1984) 477. H.E. Maes, A. Dc Veirman [8] Symp. P.W. Mertens, Wouters,
A stability analysis was conducted on ZMR processed silicon films including the effects of
and J. Van Landuyt, J. AppI. Phys. 63 (1988) 2660. [9] D. Dutartre, M. Haond and D. Bensahel, J. AppI. Phys. 59 (1986) 632.
284
SM. Yoon, IN. Miaoulis
/ Solidification front stability
[10] M.W. Geis, C.K. Chen, Henry I. Smith, R.W. Mountain and C.L. Doherty, Mater. Res. Soc. Symp. Proc. 35 (1985) 575. [11] J.S. Im, C.K. Chen, C.V. Thompson, M.W. Geis and H. Tomita, Mater. Res. Soc. Symp. Proc. 107 (1988) 169. [12] K.A. Jackson and D.A. Kurtze, J. Crystal Growth 71 (1985) 385. [131 J.S. Im, Doctor of Philosophy Dissertation, Massachusetts Institute of Technology (1989). [14] C.P. Grigoropoulos, R.H. Buckholz and G.A. Domoto, J. Appl. Phys. 62 (1987) 474. [15] C.P. Grigoropoulos, R.H. Buckholz and G.A. Domoto, J. Heat Transfer 109 (1987) 841. [16] C.P. Grigoropoulos, RH. Buckholz and G.A. Domoto, J. AppI. Phys. 59 (1986) 454. [171 A.B. Limanov and L.V. Musatova, Mater. Letters 9(1990) 456. [181 J.S. Im, J.D. Lipman, IN. Miaoulis, C.K. Chen and C.V. Thompson, Mater. Res. Soc. Symp. Proc. 157 (1990) 455.
during ZMR of thin Sifilms
[19] IN. Miaoulis, P.Y. Wong, iD. Lipman and J.S. Im, J. Appl. Phys. 69 (1991) 7273. [201 SM. Yoon and I.N. Miaoulis, J. Mater. Res. 7 (1992) 124. [211 R.T. Delves, in: Crystal Growth, Ed. B.R. Pamplin (Pergamon, Oxford, 1975) pp. 40—70. [221 M. Schultz, Semiconductors: Impurities and Defects in Group IV Elements and Ill—V Compounds (Springer, Berlin, 1989) pp. 254—262. [23] R.F. Pinizzotto, F.Y. Clark, S.D.S. Maihi and R.R. Shah, in: Comparison of Thin Film Transistors and SOl Technology, Eds. H.W. Lam and M.J. Thompson (North-Holland, New York, 1984). [241 W. Zulehner and D. Huber, in: Crystals: Growth, Properties and Applications, Vol. 8 (Springer, Berlin, 1982) pp. 1—143. [25] S.M. Yoon and I.N. Miaoulis, J. Appl. Phys. 72 (1992) 316.
~
o,
CRYSTAL GROWTH
Solidification front stability during zone-melting recrystallization of thin silicon films Sharon M. Yoon and loannis N. Miaoulis
1
Thermal Analysis of Materials Processing Laboratory, Mechanical Engineering Department, Tufts University, Medford, Massachusetts 02155, USA Received 14 April 1992; manuscript received in final form 25 August 1992
The effects of constitutional supercooling and scanning speed on the stability of the solidification front during zone-melting recrystallization of thin silicon films were investigated. We found that constitutional supercooling did not effect the stability of the solid/liquid interface for typical concentrations of oxygen, nitrogen, and carbon found in zone-melting recrystallized films. Interface growth was stable for scanning speeds less than 250 p~m/s.The critical impurity concentration for which unstable growth is induced for scanning speeds smaller than 250 ~tm/s was identified. Carbon and nitrogen diffused into the silicon film seem more likely to cause unstable solidification than oxygen.
1. Introduction Multilayer film structures are currently used in the development of advanced electronic devices. One type of film structure, the silicon-on-insulator (S DI) structure, is often used to obtain lower junction and metallization capacitances and to avoid latch-up, a phenomenon in which a parasitic circuit draws a large current which can potentially damage the integrated circuit. A typical SOl structure consists of a substrate of silicon (-~400 p~m),topped with a layer of silicon dioxide (—~1 /Lm), a layer of polycrystalline silicon (—~1 jsm) and an insulating layer of silicon dioxide (—~1 ~.Lm)(fig. 1). Electronic devices are built in the encapsulated silicon film in the SO! structure. In order for the silicon film to be usable, it is recrystallized to a single-crystal using the zone-melting recrystallization (ZMR) technique. In ZMR processing, the wafer is preheated with a bottom heater/susceptor to a temperature below the film’s melting point. A line heat source,
1
Author to whom correspondence should be addressed,
0022-0248/93/$06.00 © 1993
—
such as a graphite-strip heater (—~2 mm x 2 mm), scans the surface of the film, creating a moving narrow molten zone as shown in figs. 1 and 2. The film solidifies into a single crystal following the motion of the strip heater. Since the film is very thin (—~1 ~.tm),it is difficult to obtain accurate temperature measurements experimentally, and thus, necessary to model the recrystallization process analytically or numerically. Since the crystal quality is determined by the temperature profiles within the silicon film and the stability and morphology of the solidification front during recrystallization, it is necessary to monitor the temperature distribution computationally during recrystallization. The morphology of the solid/ liquid interface determines the localization of defects in the recrystallized film since defect trails and subboundaries are generated at the cusps of cellular morphologies or at the interior corners of faceted interfaces [1]. A nonplanar interface morphology caused by an unstable solidification interface [2] is a precursor to subboundary formation [1]. Stability of the interface can be altered by temperature gradients at the interface and by impurity segregation which may lead to constitu-
Elsevier Science Publishers B.V. All rights reserved
276
S.M Yoon, I.N Miaoulis
/
Solidification front stability during ZMR of thin Sifilms
I Graphite Strip Heater I ~ Silicon
____________
I ~ilicon ~iystal ~1 Silicon ~
~
silicon
~
NBottom heating susceptor
~
Fig. 1. Schematic drawing of the silicon-on-insulator structure during zone-melting recrystallization processing.
tional supercooling. An impurity in the melt will be forced towards the solid/liquid interface during solidification [3]. When an impurity concentrates at the crystallization front more quickly than diffusion can transport it, the liquidus ternperature becomes depressed [2—4].If the temperature of the liquid in front of the interface is lower than the liquidus temperature, constitutional supercooling occurs [1,5,6]. In previous studies of silicon film recrystallization, researchers hypothesized two reasons for unstable solidification: constitutional supercool-
liquid silicon. Fan, Tsaur and Chen [71 found oxygen localized in subboundaries of ZMR recrystallized SOl structures. They reported that faceting and subboundary formation was due to constitutional supercooling and that oxygen had dissolved into the molten silicon zone from adjacent silicon dioxide layers. Mertens et a!. [8] also found that impurities were absorbed into the film from adjacent layers. They found that oxygen and nitrogen introduced from a plasma enhanced CVD capping layer were saturated in the silicon film during ZMR processing of silicon on Si02
ing inflicted by impurities in the melt, and abrupt changes in absorbed radiation at the interface due to different reflectivity values for solid and
films. They proposed that nitrogen could cause dendritic growth but oxygen was unlikely to do so. Leamy et al. [61 discovered the presence of impurities which had solidified into subboundaries of processed films. This was most likely the result of constitutional supercooling since constitutional supercooling brings about a cellular or
Graphite Strip
Sal ~ Wafer ‘~<‘~ ~-~‘~lid
~
~ ~
_____
v
~ ~ ~
~
~—2” y
Susceptor
Fig. 2. Schematic drawing of the zone-melting recrystallization process.
dendritic interface morphology [2] and impurities concentrate in the cusps of a cellular formation [2]. Lee [11 concluded that constitutional supercooling led to the breakdown of the solidification front which caused subboundary formation dur ing studies of energy beam recrystallization of SOI structures According to Dutartre Haond, and Bensahel [91 impurity segregation most likely caused the instability of the interface during stud-
S.M Yoon, IN. Miaoulis
/ Solidification front stability
ies of ZMR of SOl wafers. Geis et a!. [10] studied possible constitutional supercooling during ZMR processing and found that for the impurity concentrations found in graphite-strip heater ZMR, constitutional supercooling was unlikely for scanning speeds less than 200 lLm/s. Im et al. [11] also found no supercooling effects at scanning speeds below 300 nm/s but suggested that impurities in the melt might trigger unstable growth at higher speeds. Other investigators have found that unstable growth results from the changes in reflectivity values during phase change during solidification of silicon films [12—171,when the effects of constitutional supercooling were not taken into consideration. Jackson and Kurtze [121 studied the stability of lamal!ae growth in silicon films for a stationary interface using a Mullins and Sekerka stability analysis [4] and found that the change in optical properties during phase change induced unstable growth. In Jackson and Kurtze’s analysis, heat transfer was considered in one direction with a heat source radiating the top surface of a film. Im et al. also studied lamellae growth in thin films, as well as the solidification front, during zone-melting recrystallization with a graphitestrip heater [13]. He modelled the heat conduction in the film in one dimension and employed linear approximations of the radiation to the top and bottom surfaces of the film in a heat source term. A cosine function was used as an estimate of the intensity profile. Grigoropoulos, Buckholz, and Domoto [14—161analyzed the stability of moving solidification interfaces during silicon film recrystallization using a laser heat source with a Gaussian intensity profile. They modelled the heat transfer during recrystallization as one-dimensional conduction in the silicon film which received heat from the glass substrate and the laser. Unstable solidification resulted from increasing the scanning speed of the laser [14,16]. Limanov and Musatova [17] considered the effects of constitutional supercooling using a Mullins and Sekerka [4] stability analysis of ZMR processed silicon films. In their analytical model, they solved two coupled one-dimensional heat conduction equations representing the substrate and the silicon film and assumed a linear temper-
duringZMR of thin Si films
277
ature profile throughout the substrate. Their resuits showed stable growth for scanning speeds up to 2000 pm/s neglecting effects of constitutional supercooling. However, unstable growth has been experimentally observed for velocities greater than 350 jim/s in graphite-strip ZMR processing [11]. We believe that the discrepancy between the theoretical [17] and experimental [11] findings is due to the simplifying assumptions which affected the temperature profiles at the solidification interface (i.e. linearization of radiative terms, Gaussian profile assumed for the strip intensity, and linear temperature profile assumed in the substrate). When the effects of impurity segregation were taken into account in their stability analysis [17], they found that impurities caused unstable solidification at scanning speeds above 1000 ~tm/s [17] which is still above the experimental threshold velocity to cause unstable growth. [11] This study by Limanov and Musatova differs from the stability analysis to be presented in this paper because the heat transfer during ZMR processing was modelled differently, and because different boundary conditions were used to determine the impurity concentration distribution. Although previous studies on the stability of the solidification interface of thin silicon films undergoing ZMR with an infrared heat source, such as a graphite-strip, provide useful qualitative insight in the mechanisms involved, the simplified thermal models limit the ability to deduce quantitative information. Recently developed numerical models accurately estimate the temperature profiles as the solidification interface by taking into consideration all the significant thermal phenomena: optical property variation during phase change [18,19], two-dimensional heat conduction, non-linear treatment of radiative boundary conditions, exact derivation of the intensity profile [191, and heat strip motion [20]. This paper presents a numerical stability analysis including constitutional supercooling effects that incorporates thermal profiles obtained for a moving solidification interface using these numerical models. An overview of the Mullins and Sekerka stability analysis [4] as it applies to the present situation is given, and results for typical ZMR processing
278
SM. Yoon, IN. Miaoulis
/ Solidification front stability during ZMR
conditions are presented. This analysis includes the effects of three cases of constitutional supercooling triggered by different impurities (oxygen, carbon, and nitrogen) under typical ZMR processing conditions. This analysis differs from previous analyses which did not take constitutional supercooling into consideration [12—16]because impurities present in the melt control the motion of the solidification front.
2. Stability theory
of thin Sifilms
polynomials expressed as functions of distance from the interface. Due to the small thickness of the silicon film, the temperature variation in the y-direction is negligible so the temperature distribution can be considered one-dimensional in the x-direction. The second part of the temperature solution accounts for the change in temperature due to the perturbation inflicted on the morphology of the solidification interface. The solution technique can be found in the original analysis [4]. A sinusoidal perturbation, 111, is introduced at the interface:
In this stability analysis, the complete temperature distribution for a perturbed interface morphology is comprised of two parts. The first part of the temperature distribution, T10(x), is solved assuming that the interface morphology is nonperturbed. The temperature distributions in the solid and liquid regions of the interface within the silicon film are determined by solving the heat conduction equation:
where ~ is the original amplitude of the perturbation (1 X i0” cm), w is the growth rate of the amplitude of the perturbation (~1), t is time (s), and q is the wave number of the perturbation (cm i) The complete temperature distribution including the contribution of the perturbed interface has the following form:
d21°
Tj(~~’z) = T~.°(x) + Ti(x) ~ e’°~’ sin qz.
d2T° +
=
~
y
o,
=
(‘s, if film is solid ~l, if film is liquid’ (1)
cP = ~ eu” sin qz,
(2)
—
(3)
T,°(x)is the basic-state temperature field and where T 1 is the film temperature (K). This two-dimensional heat conduction equation is solved using a numerical model which employs finite-difference techniques to simulate thermal processing with a graphite strip [19,20]. Radiative exchanges between the film structure and the graphite strip heater above or the susceptor below the film structure are implemented as heat sources in the boundary conditions for the control volumes on theGraphite upper and surfaces of the film structure. striplower motion is modelled [20] by re-evaluating the heat sources in the boundary control volumes as a function of the relative position of the control volume to the graphite strip heater. The temperature distribution in the region near the solidification interface was separated into two equations representing
is represented by the fourth order polynomial expression for the temperature profile found using our numerical model. I[~(x)~ec~~t sin qz is the leading order perturbation. This form of solution is a regular normal-mode small amplitude expansion for the temperature field [4,15,16]. Ti(x) can be determined from the following perturbation differential equation [15,16]: 2I~ v dT~ I 0) J ddx2 + I q2 + ~ + = 0. (4) adx \ a, —
—
Ti is the film temperature (K), v is the velocity of the strip heater (cm/s), a is the thermal diffusivity (0.093 cm2/s for solid and 0.281 cm2/s for liquid), x is the distance from the interface (cm), k 1 is the thermal conductivity (0.2 W/cm K for solid and 0.6 W/cm K for liquid), h is the thickness of silicon film (1 x2). iO~ and J is Thecm) heat source the sourceheat termemitted (W/cmfrom the top of the termheat includes .
the solid and liquid domains. In order to obtain a continuous expression of the computed temperature distribution, theare temperature distributions in the solid and liquid represented as 4th order
SM. Yoon, I.N Miaoulis
/ Solidification front stability during ZMR
film, the radiative interaction between the film and the bottom susceptor, and heat absorbed from the graphite strip heater [13]. The form of solution of the perturbed temperature differential equations is: T~(x)=a eA,1, (5) where a is a constant to be determined and A 1 is the eigenvalue of the perturbation equation. Assuming that the temperature far from the perturbed interface is equal to the temperature at the same point in the case of an unperturbed interface, the eigenvalues for the perturbation differential equations are evaluated:
_______________________ V 2
V
2a 1
~ ~2a1j
a1
(6)
/
T
0(x) +a e~ 8 e°~t sin qz. (7) 1(x,z) = T1 Temperature at the interface is affected by the curvature of the interface and a depressed liquidus temperature due to constitutional supercooling. The decrease in temperature due to the curvature, Zh7~urvature, can be determined by the Gibbs—Thomson equation [4,12]: —TmpF6q2 ewt sin qz,
279
K/at% for carbon and —600 K/at% for nitrogen) [101,and c4 is the concentration of impurity at the interface which is calculated at a later point. Including these effects, the temperature at the interface, T4, is: 2e~’tsin qz. (10) T4=T~p+mc4—T~pF6q The distribution of the impurity was solved in a similar manner to the temperature distribution. First, the concentration distribution is solved for the case of a nonperturbed interface using the one-dimensional steady-state equation for diffusion [4,21]: 2c° v dC° d ~ (11) x where c is the concentration of the impurity
where C is a constant that depends on the heat source domain term, J,has of eq. (1). Thesecond eigenvalue the liquid a negative term,for while the solid domain has a positive term. Thus, the complete temperature distribution within the silicon film will be of the form:
LtTcurvature =
of thin Sifilms
(8)
where Tmp is the melting point of silicon (1685.15 K) [12—16] and F is the capillary constant of silicon (3 x 108 cm) [12—161.The temperature will also decrease due to the depression of the melting point from constitutional supercooling, ~1I~S [1,5,6]:
(at%) and D~is the diffusion coefficient of the 2/s for oxygen, in silicon x iO~ cm7 X 10~0cm2/s 1impurity X iO~ cm2/s for (8 carbon, and for nitrogen) [22]. The impurity distribution in the melt was solved for the case of a nonperturbed interface for known values of concentration in the solid, C~,far from the interface, and a concentration in the melt, CL, at the interface. The value CL, may be found by the segregation coefficient, k, for the impurity [21]: CL
=
Cs/k.
(12)
The following values for impurity concentration were estimated from secondary ion mass spectrometry [10,23] for graphite-heater recrystallized films: (3 x iO~ at% for oxygen, 2 x iO~ at% for carbon, and 2 x 10-8 at% for nitrogen). The segregation coefficients used in this analysis are 0.9 for oxygen [8], 0.06 for carbon [10], and 7 X iO~ for nitrogen [10]. For a nonperturbed interface, the impurity concentration as a function of distance from the interface is [21]: c°(x) = C~+
(CL
—
C~)ex1’~)c.
(13)
(9) where m is the slope of the liquidus line (freezing point) in the phase diagram between the impurity and silicon (—420 K/at% for oxygen, —520
The same perturbation introduced to the interface morphology which was used to solve for the complete temperature distribution is also used in solving for the concentration distribution (eq.
280
S.M Yoon, IN. Miaoulis
/
Solidification front stability during ZMR of thin Sifilms
(2)). The concentration distribution for a perturbed interface morphology was solved using a variation of the Mullins and Sekerka analysis [4] similar to previous work by Pamplin [21]. As time approaches oc, the solute concentration at every point in the liquid varies with time as e~t.The form of the concentration distribution is similar to the form of the complete temperature distribution; it is a regular normal-mode small amplitude expansion for the impurity field: c(x,z)
c~(x)+ c~(x)8 e”t sin qz.
=
(14)
The following perturbation differential equalion is solved for impurity diffusion within the melt [21]:
(
v dCi
d2ci ~+—~—
dx2
D~ dx
constant a can be solved, where b is an unknown constant still to be determined: a
=
mb
—
D~)
D~
where b is a constant to be determined. The eigenvalue, A~,of the differential equation for the impurity diffusion is: / V
—
2
2+
V
2D~— ~(2D~)
— 0)
.
—
1)
d
)
(20)
Using eqs. (7), (18), and (19), and the second part of eq. (20), b can be solved:
(16)
e~,
‘dc\
c
=
4( k
ci(x) =b
KLIdTL\ E~-~i-)
—)—
The form of the solution of the concentration distribution for the perturbed differential equation of impurity diffusion is:
(19)
—
e’°tsin qz— K~(d1~\ pL dx
(15)
q2+_~ci=O.
mV “ñ~~(CL Cs).
—
Since the constant a, part of the temperature solution, is dependent on constant b, part of the impurity distribution, the thermal distribution is related to the impurity distribution. In the following expression, thermal diffusion is coupled to impurity diffusion since the solidification front moves at the same speed, due to the diffusion of the temperature and the motion of the impurity in the melt [4]: V+w6
~
TmpFq2
b
=
I
D~pL V k
+
(17)
(
V
1 (CL C~) ~ A~) 2V ~(CL— C~)(KsT~ KLTL) + —
~
—
xI(KSTS
—
CL~]
—KLTL) +mCL(KSAS—KLAL)
+q
The complete concentration distribution including the effect of the perturbed interface morphology is:
—
A~D~pL 1 ~ k 1
j
—
—
V
~
—
C~)~
(21)
and c(x,z)
= C~+ (CL—
C~)ex~’~)c ~=
+b e~kcx6 e~’tsin qz.
—2(KsT~’—KLTfl
(18) 2mV
The concentration at the interface, c 4, which is used in the boundary condition for the temperature at the interface (eq. (9)) can be determined from eq. (18), by using the position of the interface, x = 6e~’sin qz, as the position in this equation. Using the boundary condition for the ternperature at the interface (eqs. (7) and (10)), the
+ (Tmp~2 + ~(CL
X (K~A5 KLAL) —
—
+
CS)~ /
(K~A~T~KLALTL). (22) —
From eqs. (7) and (19), along with the first part of eq. (20), the following expression can be de-
S.M Yoon, I.N Miaoulis
/ Solidification front stability during ZMR of thin Sifilms
rived which enables us to determine the value of 0) and the interface stability:
pLw
=
(—k~T~A+8 kLTLAL) 2(kLAL k + TmpFq 1A8)
281
fication front on the liquid side is greater than the rate on the solid side. The fourth term contains the effects of constitutional supercooling. If impurity segregation were not included in this
—
B
+
(2k5T~’ 2kLT~)
—
mB( K5A5
V = ~-(CL
analysis, this last term would be omitted, and the remaining stability equation would be unaltered. Previous researchers who have performed stabil-
—
—
C8)
—
—
KLAL),
b,
(23) (24)
where T~and T~’are the rates of change of the thermal gradients at the interface obtained from Tnumj(X). The interface is stable if the rate of growth of the perturbation, w, is negative, and the perturbed interface will return to the original interface morphology. If the calculated value of w is positive, the perturbation will continue to grow from the interface and growth in unstable. To evaluate stability, all frequencies of the sinusoidal function must be used in eq. (23) to represent Fourier terms of the continuous perturbation function. Only when w is negative for all perturbation wave numbers, q, is the solidification growth stable. If w is positive for any perturbation wave number, the interface growth is unstable, since the growth is characterized by the term, e~’. For typical processing conditions, of strip ternperature 2300°C, and susceptor temperature 1000°C,the stability of the solid/liquid interface was studied for processing speeds up to 1000 jim/s. For each processing speed, the effects of impurity segregation on the stability were determined.
ity analyses neglecting constitutional supercooling have found expressions similar to eq. (23), excluding the fourth term [12,13,16]. The stability equation found by Limanov and Musatova [17] has a similar form; however, their constant, B, differs from eq. (24) because their solution for the impurity concentration was based on assuming a specific concentration gradient at the interface. The last term in eq. (23) will be stabilizing when the constant, B, is negative since m is a negative value and the term, (KSAS KLAL), is always positive. From eq. (24), we conclude that the first term is destabilizing since K4 is always positive. In a previous analysis, we only studied the effects of scanning speed on the stability of the solidification front [25]. Since supercooling effects were not considered, the decrease in melting point at the interface was not taken into account (eq. (9)), and only the first equality was of interest in eq. (20). We found that stable solidification occurred until a scanning speed of 250 /Lm/s. For faster scans, the temperature gradient at the interface in the solid material became less steep because there was less time for radiative cooling. A decrease in this gradient caused unstable growth because the first term in eq. (23) became positive and dominated the stability expression. By observing the destabilizing term due to impurity segregation in the melt which is 2V(CL CS)/DC, one can easily see that a higher processing speed or a higher impurity concentration in the melt could induce an instability. Using typical concentrations of oxygen measured in ZMR processed silicon films [10,23], we found that the interface growth was stable for scanning speeds below 250 jim/s. Stability results for oxygen are shown in fig. 3. In this figure, the growth rate remains below zero for velocities of 150 and 200 jim/s. For higher velocities of 250 and 400 j.tm/s, the growth rate, to, exceeds zero for cer—
—
3. Results and discussion The four terms in eq. (23) describe the three criteria for interface stability. The first term is stabilizing when the temperature gradient of the interface on the solid side is steeper than the gradient on the liquid side. The second term, a capillary term, is always negative and stabilizing, The third term is stabilizing when the rate of change of the temperature gradient at the solidi-
282
S.M Yoon, IN. Miaoulis
/ Solidification front stability during ZMR of thin Si films
to
to measure the amount of impurity in a film, which could actually be higher in a molten phase, it is necessary to determine the critical concentralion of impurity which would trigger instability. To find the concentration of impurities needed to induce unstable solidification at scanning speeds below 250 jim/s, we varied the concentration. The critical concentration of oxygen that would result in unstable solidification is shown in fig. 4a. As the velocity increases, less solute is necessary to cause unstable growth. As long as the term V(CL C5)/D~is greater than the value of b, the
~
0.0
~>-.. ~
~
—to
‘~‘~
\,,,
-
—zo
I
0.0 300.0 600.0 PERTIJRBA11ON WAVE NUMBER, q (1/cm)
—
Fig. 3. Rate of growth of perturbation when oxygen is present in the solidifying melt for the following scanning speeds: )15O,am/s,( )200 am/s (———)250~sm/s, an — i~m/s.
fourth term of the stability equation (eq. (23)) will be destabilizing. If the velocity of the strip heater 1), increases, (CL— C~) can decrease and still produce the same effect. For velocities above 150
tam wave numbers, q, and the perturbation continues to grow away from the interface. This figure is consistent with work by Pamplin [21], who found that short wavelengths are stabilized by capillary and long wavelengths by the slow diffusion of the solute. However, since unstable growth was also instigated at a scanning speed of 250 jim/s for the case without supercooling, we believe that constitutional supercooling due to oxygen in the film was not the cause of this instability. The decrease in temperature gradient of the solid material at the interface induced unstable growth in this case. Previous researchers have suggested that the level of impurities found in ZMR processes films do not contribute to unstable growth [11]. However, since it is difficult
jim/s, the critical impurity remains at —~0.02 at%. At this point, although the velocity increases, the value of b also increases and the concentration remains at this level. The amount of impurity needed to force the interface growth to be unstable ranges between 0.02 and 0.07 at%, which is three orders of magnitude larger than the measured amount of impurity in ZMR recrystallized silicon films [10,23]. This would suggest that oxygen would be unlikely to trigger instabilities during processing. For a typical carbon concentration diffused in solidifying melt, solidification is stable for scanning speeds below 250 jim/s, suggesting that this concentration of carbon does not induce unstable growth. The concentration of carbon needed to trigger instabilities in the solidification front for
0.10
3.0
10.0
0 0
0
‘
a 0.00
50.0
z
2.0
150.0
VELOCflI
(wrVs)
250.0
to
b 50.0
0~
150.0
o
o
VEI.QCT1Y (tLrrVs)
o
~ 250.0
o 0.0
50.0
150.0
VtLOCfl~(ani/s)
250.0
Fig. 4. Critical concentration of (a) oxygen, (b) carbon and (c) nitrogen (at%) needed which would induce unstable solidification for scanning speeds below 250 ~sm/s.
S.M. Yoon, IN. Miaoulis
/ Solidification front stability during ZMR of thin
low scanning speeds varies from 0.00013 to 0.00023 at% (fig. 4b). The critical concentration decreases slightly from 50 to 200 jim/s. At 230 jim/s, there is a sharp decrease in the amount of carbon needed to start an instability, At this velocity, the value for b decreases for the specific diffusion coefficient of carbon in silicon. The concentration of solute required to destabilize the interface is only one magnitude higher than the values measured experimentally [10,23]. In films containing nitrogen, unstable solidification occurred for scanning speeds above 250 jim/s, which similarly suggests that this concentration of nitrogen found in ZMR processed films does not cause instabilities during growth. In fig. 4c, the critical concentration of nitrogen for low scanning speeds is shown. This concentration ranged between 2 x i0~ and 8 x iO~ at%, which is only one order of magnitude above the actual measured values. For increasing scanning speeds, the critical concentration decreased as expected. Minute changes in temperature gradients cannot be captured due to the size of the nodes in the numerical grid. This results in calculated critical concentrations that are similar for different velocities. The amount of constitutional supercooling at the interface was calculated for each impurity at measured concentration levels. It was found to be 0.01°Cfor oxygen impurities, 0.2°Cfor nitrogen, and 0.17°Cfor carbon. Our results show that oxygen seems unlikely to induce unstable growth. Carbon and nitrogen impurities could induce unstable growth if the concentration were increased by an order of magnitude from typical concentrations measured. Actually, the concentration of solute in the melt may be higher than the experimental values, since the impurity was measured in solid material. It is also possible that a combination of impurities is present during recrystallization. In such cases constitutional supercooling may have been a cause of instability.
Sifilms
283
constitutional supercooling and scanning speed. The contribution to unstable growth of typical impurity concentrations for oxygen, carbon, and nitrogen was investigated. For each of these impurities, solidification was stable up to a scanning speed of 250 jim/s which is the same velocity where unstable growth occurred when constitutional supercooling effects were neglected. Concentration levels for each of the impurities were varied to find the critical concentration which would induce unstable solidification. To trigger instabilities at slow scanning speeds ( < 250 jim/s) in the solidification front, the concentration of oxygen would have to be three orders of magnitude higher than experimentally measured values which suggests that oxygen would not cause unstable growth. However, if concentrations of carbon and nitrogen were increased one order of magnitude from the experimentally measured vatues, growth would be unstable at scanning speeds below 250 jim/s. The actual impurity concentration is most likely greater than the experimentally measured values since the solute measurement was taken in solid material.
Acknowledgement This research was sponsored by the National Science Foundation under grant CTS-9 157278.
References [1] E. Lee, Mater. Res. Soc. [2] B. Chalmers, Principles
Symp. Proc. 35 (1985) 563. of Solidification (Wiley New York 1964) pp 150—157 [31D.P. Woodruff, The Solid—Liquid Interface (Cambridge University Press, London, 1973) p. 83. [4] W.W. Mullins and R.F. Sekerka, J. AppI. Phys. 35 (1964) [5] CA. Knight, The Freezing of Supercooled Liquids (Van Nostrand, Toronto, 1967) p. 87. [6] H.J. Leamy, CC. Chang, H. Baumgart, R.A. Lemons and
4. Conclusions
J. Cheng, Mater. Letters 1 (1982) 33. [7] i.C.C. Fan, B.-Y. Tsaur and C.K. Chen, Mater. Res. Soc. Proc. 23 D.J. (1984) 477. H.E. Maes, A. Dc Veirman [8] Symp. P.W. Mertens, Wouters,
A stability analysis was conducted on ZMR processed silicon films including the effects of
and J. Van Landuyt, J. AppI. Phys. 63 (1988) 2660. [9] D. Dutartre, M. Haond and D. Bensahel, J. AppI. Phys. 59 (1986) 632.
284
SM. Yoon, IN. Miaoulis
/ Solidification front stability
[10] M.W. Geis, C.K. Chen, Henry I. Smith, R.W. Mountain and C.L. Doherty, Mater. Res. Soc. Symp. Proc. 35 (1985) 575. [11] J.S. Im, C.K. Chen, C.V. Thompson, M.W. Geis and H. Tomita, Mater. Res. Soc. Symp. Proc. 107 (1988) 169. [12] K.A. Jackson and D.A. Kurtze, J. Crystal Growth 71 (1985) 385. [131 J.S. Im, Doctor of Philosophy Dissertation, Massachusetts Institute of Technology (1989). [14] C.P. Grigoropoulos, R.H. Buckholz and G.A. Domoto, J. Appl. Phys. 62 (1987) 474. [15] C.P. Grigoropoulos, R.H. Buckholz and G.A. Domoto, J. Heat Transfer 109 (1987) 841. [16] C.P. Grigoropoulos, RH. Buckholz and G.A. Domoto, J. AppI. Phys. 59 (1986) 454. [171 A.B. Limanov and L.V. Musatova, Mater. Letters 9(1990) 456. [181 J.S. Im, J.D. Lipman, IN. Miaoulis, C.K. Chen and C.V. Thompson, Mater. Res. Soc. Symp. Proc. 157 (1990) 455.
during ZMR of thin Sifilms
[19] IN. Miaoulis, P.Y. Wong, iD. Lipman and J.S. Im, J. Appl. Phys. 69 (1991) 7273. [201 SM. Yoon and I.N. Miaoulis, J. Mater. Res. 7 (1992) 124. [211 R.T. Delves, in: Crystal Growth, Ed. B.R. Pamplin (Pergamon, Oxford, 1975) pp. 40—70. [221 M. Schultz, Semiconductors: Impurities and Defects in Group IV Elements and Ill—V Compounds (Springer, Berlin, 1989) pp. 254—262. [23] R.F. Pinizzotto, F.Y. Clark, S.D.S. Maihi and R.R. Shah, in: Comparison of Thin Film Transistors and SOl Technology, Eds. H.W. Lam and M.J. Thompson (North-Holland, New York, 1984). [241 W. Zulehner and D. Huber, in: Crystals: Growth, Properties and Applications, Vol. 8 (Springer, Berlin, 1982) pp. 1—143. [25] S.M. Yoon and I.N. Miaoulis, J. Appl. Phys. 72 (1992) 316.