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Interface structure and solute trapping in rapid crystal growth of silicon and germanium from the laser-induced melt
Volume 2, number 3
February 1984
MATERIALS LETTERS
INTERFACE STRUCTURE AND SOLUTE TRAPPING IN RAPID CRYSTAL GROWTH OF SILICON AND GERMANIUM FROM THE LASER-INDUCED MELT Jun-ichi CHIKAWA, Fumio SAT0 and Tadasu SUNADA NHK Broadcasting Science Research Laboratories, I -I O-l1, Kinuta, Setagaya-ku, Tokyo I5 7, Japan Received 14 October 1983
Surface layers of uniformly doped Si and Ge crystals were melted by irradiation with single laser pulses. Their energy densities were adjusted to resolidify the melted layer at growth rates of al m/s. The interfacial segregation coefficient was obtained from the dopant profiles in the resultant crystals by SIMS measurement. Spreading resistances were also measured for angle-lapped surfaces of the crystals. It was observed that the spreading resistance largely fluctuates in a case where part of a (111) surface is melted. Whereas, when an entire (111) surface was melted, it was smooth, and the interfacial segregation coefficient was found to be smaller than the equilibrium one. This departure from local equilibrium is opposite to those measured by many investigators and disagrees with the current theories of solute trapping.
1. Introduction Pulsed laser annealing has been used as a technique to restore the damage in surface layers caused by ion implantation. Its essence is of rapid crystal growth following laser-induced melting of the near-surface [ 11. Such rapid crystallization accompanies phenomena due to departure from local equilibrium of the solidliquid interface. For example, impurity atoms are incorporated with segregation coefficients larger than the equilibrium coefficient’[2], and their concentration greatly exceeds the equilibrium solid solubility limit [2-41. Such solute incorporation in excess has been known from observations for rapid quenching of molten alloys [5]. To understand these phenomena, “solute trapping” [6] has been introduced as an important idea. According to Chernov’s theory, solute atoms seek the interface and are subsequently buried as the next layer of the crystal solidifies. During the burying their chemical potential is raised, and, in slow growth, they make out-diffusion and remain at the interface. In sufficiently rapid growth, solute atoms at the lnterface are buried completely, and the crystal grown has the same concentration as that of the interface. Recently, the idea of solute trapping has been developed to explain the non-equilibrium dopant incorporation 202
during rapid growth following laser-induced
melting
]7,81* AU the theories on solute trapping were introduced in the light of the experimental results that the interfacial segregation coefficient k* is always larger than the equilibrium segregation coefficient k, as the growth rate V is increased (k* >k,). The present authors showed experimentally, however, that k*
2. Measurement
of impurity
segregation coefficients
The specimens were cut from uniformly doped Si and Ge crystals, and their surfaces were melted by irradiation with single pulses from a Q-switched ruby laser (20 ns pulse width, h = 0.694 pm). The energy 0 167-577x/84/$ 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
February 1984
MATERIALS LETTERS
Volume 2, number 3
densities of the irradiation were adjusted so as to give a growth rate of = 1 m/s after the melting. (The consequent melting depth reached = 1 pm.) Dopant concentrations were measured by secondary-ion mass spectrometry (SIMS) and normalized by the original concentrations C,, in the unmelted region. The depth profile of dopant distribution in the regrown region can be calculated to a good approximation by a one-dimensional diffusion: letting C(x,t) stand for the dopant concentration in the melt, the diffusion equation can be written as D a2C(x,t)/dx2
DEPTH x ti,
= K’(x >t)/at a
(1)
where D is the diffusion coefficient in the melt and x is the depth from the surface. The following assumptions can be made: (1) The interface moves at a constant growth rate Vtoward the surface and rejects the impurity atoms at a rate V(l-k*)C*(x) cme2 s-l as shown schematically in the inset of fig. la, where C*(x) is the impurity concentration in the melt at the interface. (2) Diffusion in the solid is negligible. (3) Convective mixing in the melt is negligible. The redistribution of impurities after solidification consists of the initial and terminal transient regions, as shown in fig. 1a. When the melting depth x0 is deep enough and/or k* is not far from unity, the steady state (C,/Cu = 1) is achieved between both regions. For this case, the expressions for normal freezing [lo] are applicable, i.e. C,(x)/C,
=f (1 +erf{i[(V/D)(xo
+ (2k* - l)exp[-k*(l X erfc(i(2k*
C,(x) -=
1t 3 s
-k*)(V/D)(xo
- l)[(V/D)(xu
for the initial transient,
-x)]‘/~} -x)]
-x)]‘/~}}
(2)
and
exp [-2( V/D)x]
CQ + 5 (’ -k*)(2-k*)exp[-G(V/D)x] (1 +k*)(2+k*) t(2n + 1) (1 -k*)(2-k*)...(n (1 tk*)(2 +k*)...(n X exp [-n(n
t l)( V/D)x] + . . .
t .., -k*) +k*) (3)
for the terminal transient, where C,(x) is the impurity concentration in the solid, given by C, = k*C*(x). In fig. la, C,(x)/C,, obtained from eqs. (2) and (3), is
(b)
50
100
Oo1 (V/D)
x
Fig. 1. Calculated dopant profiles resulting from resolidification after melting of uniformly doped crystals by laser irradiation. The ratio of the concentration C,(x) at a depth x to the initial concentration Cu is plotted as a function of (V/D)x, using eqs. (2) and (3). (a) The entire profiles consisting ofthe initial transient, steady state, and terminal transient. xn = depth of melting. The inset shows schematically the profile during the regrowth. DBL stands for the diffusion boundary layer. (b) Profiles in the terminal transient calculated for various values of interfacial segregation coefficients k* by eq. (3). The figure on each curve denotes thevalueof k*.The experimental data for a (100) surface of a Bdoped crystal are also plotted as an example.
plotted fork* = 0.35,0.07, and 0.02 as a function of a dimensionless parameter (V/D)x expressing the depth from the surface. The melting depthxu = 1 pm is deep enough for forming the steady state for impurities having k* 2 0.02. To determine k* , both the calculated and measured ratios, Cs/CO, for the terminal transient region were plotted for (V/D)x and x in a logarithmic scale, respectively, as shown in fig. lb. The curves calculated for various values of k* were examined by sliding the experimental curve along the abscissa so as to superpose on each other. By finding the best fit of the calculated 203
curves, we can determine
k* without knowing values
for D and K As seen from fig. lb, the accuracy of measured k* values is high for the range of k* >, 0.1, although this method cannot be applied to measurement of small k* such as for Bi or In in Si. As an example, the data for a (100) surface of a B-doped Si crystal are plotted in fig. 1b. We can determine the growth rate to be I/= 0.9 m/s from the correspondence between the depth x and (V/D)x.
3. Orientation
dependence
of impurity
incorporation
The most stable surface of Si and Ge crystals is of a (111) type which has the highest atomic density. The liquid-solid interface parallel to a (111) plane is evidenced to be atomically smooth [ 111. It is expected that growth in a [ 11 l] direction requires a high supercooling compared with that in other directions, to make two-dimensional nucleation at the interface. Therefore, an orientation dependence of impurity incorporation was observed for the three cases shown in fig. 2. In fig. 2a, whichis referred to as a “(I 1 l& case”, the entire surface was melted. While, in fig. 2b, the central part of a (111) surface is melted. As an example of growth in the other directions, a (100) or (2 11) surface was examined, as shown in fig. 2c. The dopant profiles of the terminal transient regions for these three cases of B-doped Si and Ga-doped Ge crystals are shown in figs. 3a and 3b respectively: The curves are the calculated ones with the best fit to the experimental data. From fig. 3a, the interface segregation coefficient of B in Si was found to be k* = 0.7 for the (100) case (fig. 2~). This value is equal to the equilibrium segregation coefficient k, = 0.7 for the dilute solution (although the value currently accepted is 0.8, this has been corrected to be 0.7 [12]), i.e. k* = k,. While, for the (11 l)ent case, we obtained k* = 0.3 which is
Fig. 2. Orientation terface. (b) (111)
204
February 1984
MATERIALS LETTERS
Volume 2, number 3
(b)
(V/Lax
(0) 0.1
0.2
a5
1.0
20
( V/D) x 0.1
DEPTH
0.5
1.0
x ti,
Fig. 3. Orientation dependence of depth profiles of dopants measured after melting by laser irradiation as shown in fig. 2. (a) B in Si. Energy density of the laser pulse = 3.3 J/cm2. (b) Ga in Ge. Energy density = 2.0 J/cm2. The data obtained by SIMS are plotted against depth x, and the solid curves are calculated using eq. (3). The data for the (111) case (dashed lines) do not fit any curves calculated from eq. (3).
smaller than ko(k* < ko). Whereas, for the (111) case of fig. 2b, any curves calculated from eq. (3) do not fit the experimental result which is shown by the dotted line in fig. 3. It was found by a SIMS measurement that the B concentration was high at the center and low in the peripheral region of the laser melted area. A similar orientation dependence for Ga in Ge is seen from fig. 3b: k* = ko(= 0.09) for the (211) case; for the (11 l)ent case, the values of Cs(x)/Co are lower than the curve for k* = 0.09 in the deep region and exceed the curve as the depth x decreases. This result indicates that k* is so small that the steady state is not formed completely between the initial and terminal transient regions (k* <0.02), i.e. k*
of crystal surfaces melted by laser irradiation. (a) (11 l)ent case. The entire (111) surface case. (c) (100) case. Growth steps move in the direction of the arrows.
is melted.
Rough
in-
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MATERIALS
Fig. 4. Profiles of spreading resistances (SR) measured for the surface layers regrown from the laser-induced melt. Originally the crystalswere uniformly doped with phosphorus. (a) Anglelapping for the depth profile measurement, which was made along the dotted line. (b) (211) case. (c) (11 l)ent case, (d) (111) case. The arrow corresponds to the point A in (a). The taper angle 01 = tan-’ 0.005. The scale indicates 1 ,um in the depth direction. (e) SR profile measured for the surface lapped parallel to the crystal surface of the (111) case, i.e. 01= 0.
tween the specimen surface and lapped surface was OL = tan-l 0.005. Figs. 4b, 4c and 4d are the results for the (211) (Lll)ent, and (111) cases illustrated in fig. 2, respectively. The former two cases gave smooth profiles as seen in figs. 4b and 4c. Whereas, for the (111) case, SR values fluctuate drastically in the laserirradiated region, and the boundary between the laser melted and unmolten (substrate) regions is seen clearly, as indicated by the arrow “A”. Since resistivities up to a depth of a few microns contribute to the SR value, the variations in dopant concentration may be much larger than those obtained from the fluctuation in SR. It is shown by laser Raman spectroscopy, however, that this laser-annealed layer is of single-crystal silicon. When the specimen was lapped parallel to the (111) surface (o = 0”), much smaller fluctuations of the SR profiles were obtained as shown in fig. 4e. This observation indicates that the dopant P is stratified laterally in the laser-annealed region. The differences in both the impurity (SIMS) and SR profiles between the (111) and (11 l)ent cases indicate that solidification occurs laterally in the (111) case and perpendicular to the (111) surface in the (I1 l)ent case. It should be noted that the aspect ratio of the molten layer is 1 : 3000 (melting depth = 1 pm, size = 3 mm in diameter for the (111) case; crystal
LETTERS
February
1984
size for the (11 l)ent case = 2 X 2 mm2). One can conclude that growth perpendicular to a (111) plane requires so large supercooling that, in the (111) case, only lateral growth occurs from the periphery which supplies growth steps with terraces parallel to the (111) plane. The stratified dopant distribution may be attributed to inhomogeneities in step density and velocity, i.e. step bunching [13]. The above experimental results are summarized as follows: (1) The interfacial segregation coefficient k* is still equal to the equilibrium one k, at V x 1 m/s for B or P in Si, and Ga in Ge (k* = ko). (2) The (111) cases are in the same orientation, but the and(lll)ent shapes and structures of growth interfaces are different. Consequently, the impurity redistribution and resulting properties are different between both cases. For the (111 )ent, supercooling at the interface is very large compared with the other cases. The departure from equilibrium was found to result in a smaller segregation coefficient (k* ko) by increasing growth rates.
4. Discussion In light of the current theory [ 14,151, our experimental results indicate that the interface in the (l l l )ent case is rough and/or diffuse owing to the high supercooling, and that, in the other cases, the interface is sharp, and growth occurs by passage of steps. Leaving aside the problem of the (11 l)ent case for the moment, let us first look at the (100) or (211) cases. The result k* = k, suggests that the growth interface is still in local equilibrium even at V = 1 m/s. At this growth rate, however, the time to solidify a monatomic layer is as short as a/V=lO-lo s where a is the interatomic spacing, and in this interval, atoms at the interface must exchange their positions in the liquid and interfacial states frequently enough to reach the equilibrium distribution. Since the time for diffusive translations in the liquid isa2/D = lo-l1 lo-l2 s on average, both states should exist at the interface before releasing their latent heat of solidification, i.e. solute segregation must take place between the liquid and liquid monolayer adjacent to the interface which is referred to as “the interfacial state”. We 205
Volume 2, number 3
MATERIALS LETTERS
0.3 k” 0.2 100) 0. I k,” E 0
__ 0
I
2
3
4
v (m/s)
Fig. 5. Dependence of interfacial segregation coefficient k* on growth rate V in rapid growth following laser melting of Si (100) surfaces. The data obtained for Bi in Si by Baeri et al. [ 161 were replotted. (In their original figure, k* is shown on a logarithmic scale.) k* tends to a certain value kz for V - 0, which is much larger than the equilibrium segregation coefficient k. = 0.0007.
denote the equilibrium segregation coefficient between the interfacial state and liquid by k;. Then, our experimental results means that k; = k,. In general, however, it is expected that kc # k,. In fact, this can be seen for Bi in Si: Baeri et al. [16] obtained a relation between the interfacial segregation coefficient k* and growth rate V by laser annealing of B&implanted Si crystals. Their data for the (100) surface are replotted in fig. 5. (In their original figure, k* is shown on a logarithmic scale.) Extending the curve to V = 0, we can estimate the value of k; = 0.02 which is much larger than the equilibrium segregation coefficient k. = 0.0007, i.e. k; > k,. From these experimental results, one can conclude that the interfacial state in local equilibrium with the liquid is frozen by rapid growth at I/ = 1 m/s. The solute segregation in the interfacial state is considered as follows: An approximation to reality that is of considerable use for most semiconductor solutions is that the liquid solution is ideal (the solid is a regular solution) [ 171. Then, we may assume that both the interfacial state and the liquid are ideal solutions; solute and solvent atoms act independently at the interface. It is envisaged that atoms in the interfacial state take partial ordering, e.g., the atoms vibrate similarly to those in the liquid, but their average positions are at the crystal lattice points. In other words, the entropy of the interfacial state is lower than that of the liquid. Since the enthalpy change is nearly equal to zero (we consider the state before releasing the latent heat), we can write kT, as
February 1984
where aS is the entropy difference between both the states per solute atom and k is Boltzmann’s constant*. It follows from eq. (4) that k: is independent of temperature, i.e. composition-independent segregation takes place. In the foregoing argument, the free energy per atom of the interfacial state is higher, relative to the liq uid state. The interfacial state is considered as a transition state between the liquid and solid, sharing a part of the interfacial free energy. We now consider the reason for the experimental result that k: = k. for B or P in Si and Ga in Ge. This result implies that the environment for the individual solute atoms in the interfacial state is the same as that in the crystal, i.e. in the dilute solid solution, the solute atoms in the crystal lattice are still in the liquid state near the melting point To of pure Si (141S’C)or Ge (958°C). We may envisage that solute atoms are accommodated in the drops including some solvent atoms (partially molten regions). Then, solute atoms have no enthalpy difference between them in the liquid and crystal (neglecting the interfacial energy of the drop), and it is expected that the equilibrium segregation coefficient k,(T) is independent of temperature (k, = ko) in a region just below To. Although phase diagrams which have been reported are not precise enough to confirm this inference, it is likely that such a temperature region exists for most dopants in Si and Ge: As an example, the As-Si system is shown in fig. 6a [ 17,191. The equilibrium segregation coefflcients k,(T) obtained from the liquidus and solidus curves in fig. 6a are plotted against l/T in fig. 6b. The curve consists of three parts marked “D”, “I” and “B”. In region D, k, is independent of temperature, as described above. Regions I and B may be explained as follows: Around T = 1380°C the drops solidify and the As atoms are in the fully ionized state. The steep slope is due to the energy of the dangling bonds of solvent atoms and strain around the ion. In the range B of T 2 1260°C, most As atoms form bonds with the solvent atoms, and the slope is due to strain energy around them. Details, such as the enthalpies obtained from the slopes in regions I and B, will be discussed elsewhere. The relation k; = k, in the dilute
* For the general formula of equilibrium segregation coeffik; = exp(-A,S/k)
206
,
(4)
cients, see ref. 1181.
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February
LETTERS
1984
(b)
0.051 6
As
CONCENTRATION
(atoms
/cm3
)
7
0 x 10-4
I/T
Fig. 6. S-As system. (a) Phase diagram [17,19]. The dashed curve shows As concentration of the interfacial state in equilibrium dependence of the equilibrium segregation coefficient with the liquid which is frozen at growth rates of = 1 m/s. (b) Temperature k,. The values of kc obtained from (a) were plotted against l/T. ko = 0.3.
case can be understood from the foregoing argument. For the Bi-Si system, however, it can be seen from the phase diagram [17,19] that the curve consists of the parts I and B, and contains no D region. Consequently, we have kT, # k,. The solubility limits of substitutional solid solutions achieved by laser annealing are explained by eq. (4): In the As-Si system shown in fig. 6a, by increasing the As concentration, the melting temperature decreases. Lowering of the melting point results in growth rates of V = 1 m/s even at such energy densities of laser irradiation that give higher growth rates (~4 m/s) in the dilute cases. Therefore, the solubility limit is considered to be the maximum solute concentration in the interfacial state. When the concentration C, in the melt exceeds the eutectic composition Cy , the phase of the dopant appears. Since there is no enthalpy change between the liquid and interfacial state, the segregation coefficient between both states is independent of temperature or composition. Then, the maximum solid solubility C?$,, is given as the dopant concentration of the interfacial state in equilibrium with the eutectic composition of the melt, i.e. Tax = k*Ceu. F rom the phase diagram in fig. 6a, we obtain C’$=” 6 X lo21 . This value agrees with the maximum solubility 6 X 1021 for As in Si measured by White et al. [2], This explanation for the maximum solubility can be applicable to other dopants [9]. (For Bi in Si, using kz = 0.02 (see fig. 5) and CT = 5 X 1O22 cm-3 [17], Tax = k&l’? = 0.02 X 5 X 1O22 = 1021 cme3. The measured value of Tax = 1 .I X 1021 [20] .) Let us now turn to the (11 l)ent case. In compari-
son with the (100) or (211) case, the interface is largely supercooled, but the time to solidify a monatomic layer is similar (u/V = lo-lo s). If we assume that the interfacial state is the same, the segregation coefficient k* should have been equal to k: even for large supercooling, because k; is independent of T as described before. The result of k* < ko must be explained by different interface structure. The comparison of the experimental results for (111) and (11 l),t cases indicates that, in the (111) case, the interface is discrete (sharp) and advances by a step mechanism. According to the theories of Temkin [ 141 and Cahn [ 151, the interfacial free energy is a periodic function of the average position of the interface, with the minimum at the position of (111) lattice planes and maximum between the planes. The interface cannot advance across the maximum by thermal activation because all of the liquid atoms adjacent to the interface must hop over the energy barrier; its only means of advancement is by the motion of growth steps. For the (11 l)ent case, however, there are no sources supplying steps. Therefore, the rapid growth must occur by the driving force (supercooling) which is so large that the barrier is less effective. Then, the interface tends to be rough [14] and/or diffuse [15]. In such cases, it is natural that part of the solvent atoms in the liquid monolayer adjacent to the interface form bonds to decrease their free energy, relative to that in liquid, and at the same time the chemical potential for the solute atoms is raised. Consequently, we have a low solute concentration at the interface, i.e. k*
Volume
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MATERIALS
The observation of k*
References [l] [2]
[31 [41 [51 [61 [71
[81
208
H. Kurz, J.M. Liu and N. Bloembergen, Physica 117/ 118B (1983) 1010. C.W. White, S.R. Wilson,B.R. Appletonand F.W. Young Jr., .I. Appl. Phys. 51 (1980) 738. H.J. Leamy, J.C. Bean and J.M. Poate, J. Crystal Growth 48 (1980) 379. N. Natsuaki, M. Tamura and T. Tokuyama, J. Appl. Phys. 51 (1980) 3373. J.C. Baker and J.W. Cahn, Acta Met. 17 (1969) 575. A.A. Chernov, Kristallografiya 12 (1967) 222 [English transl. Soviet Phys. Crystallography 12 (1967) 1861. K.J. Jackson, G.H. Gilmer and H.J. Leanly, in: Laser and electron beam processing of materials, eds. C.W. White and P.S. Peercy (Academic Press, New York, 1980) p.104. R.F. Wood, Appl. Phys. Letters 37 (1980) 302.
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February
1984
[9] J. Chikawa and F. Sato, Japan. J. Appl. Phys. 19 (1980) L577. (lo] V.G. Smith, W.A. Tiller and J.W. Rutter, Can. J. Phys. 33 (1955) 723. [ 111 T. Abe, J. Crystal Growth 24/25 (1974) 463. [ 121 H.R. Huff, T.G. Digges Jr. and O.B. Cecil, J. Appl. Phys. 42 (1971) 1235. [ 131 F.C. Frank, in: Growth and perfection of crystals, eds. R.H. Doremus, B.W. Roberts and D. Turnbull (Wiley, New York, 1958) p. 411. [ 141 D.E. Temkin, in: Crystallization processes, eds. N.N. Sirota, F.K. Gorskii and V.M. Varikash (Consultants Bureau, New York, 1966) p.15. [15] J.W. Cahn, Acta Met. 8 (1960) 554. [ 161 P. Baeri, G. Foti, J.M. Poate, S.U. Compisano and A.G. Cullis, Appl. Phys. Letters 38 (1981) 800. [17] C.D. Thurmond and M. Kowalchik, Bell System Tech. J. 39 (1960) 169. [18] C.D. Thurmond and J.D. Struthers, J. Phys. Chem. 57 (1953) 831. [19] F.A. Trumbore, Bell System Tech. J. 39 (1960) 205. [20] C.W. White, B.R. Appleton, B.Stritzker, D.M. Zehner and S.R. Wilson, in: Laser and electron-beam solid interactions and materials processing, eds. J.F. Gibbons, L.D. Hessand T.W. Sigmon (North-Holland, Amsterdam, 1981) p.59.
February 1984
MATERIALS LETTERS
INTERFACE STRUCTURE AND SOLUTE TRAPPING IN RAPID CRYSTAL GROWTH OF SILICON AND GERMANIUM FROM THE LASER-INDUCED MELT Jun-ichi CHIKAWA, Fumio SAT0 and Tadasu SUNADA NHK Broadcasting Science Research Laboratories, I -I O-l1, Kinuta, Setagaya-ku, Tokyo I5 7, Japan Received 14 October 1983
Surface layers of uniformly doped Si and Ge crystals were melted by irradiation with single laser pulses. Their energy densities were adjusted to resolidify the melted layer at growth rates of al m/s. The interfacial segregation coefficient was obtained from the dopant profiles in the resultant crystals by SIMS measurement. Spreading resistances were also measured for angle-lapped surfaces of the crystals. It was observed that the spreading resistance largely fluctuates in a case where part of a (111) surface is melted. Whereas, when an entire (111) surface was melted, it was smooth, and the interfacial segregation coefficient was found to be smaller than the equilibrium one. This departure from local equilibrium is opposite to those measured by many investigators and disagrees with the current theories of solute trapping.
1. Introduction Pulsed laser annealing has been used as a technique to restore the damage in surface layers caused by ion implantation. Its essence is of rapid crystal growth following laser-induced melting of the near-surface [ 11. Such rapid crystallization accompanies phenomena due to departure from local equilibrium of the solidliquid interface. For example, impurity atoms are incorporated with segregation coefficients larger than the equilibrium coefficient’[2], and their concentration greatly exceeds the equilibrium solid solubility limit [2-41. Such solute incorporation in excess has been known from observations for rapid quenching of molten alloys [5]. To understand these phenomena, “solute trapping” [6] has been introduced as an important idea. According to Chernov’s theory, solute atoms seek the interface and are subsequently buried as the next layer of the crystal solidifies. During the burying their chemical potential is raised, and, in slow growth, they make out-diffusion and remain at the interface. In sufficiently rapid growth, solute atoms at the lnterface are buried completely, and the crystal grown has the same concentration as that of the interface. Recently, the idea of solute trapping has been developed to explain the non-equilibrium dopant incorporation 202
during rapid growth following laser-induced
melting
]7,81* AU the theories on solute trapping were introduced in the light of the experimental results that the interfacial segregation coefficient k* is always larger than the equilibrium segregation coefficient k, as the growth rate V is increased (k* >k,). The present authors showed experimentally, however, that k*
2. Measurement
of impurity
segregation coefficients
The specimens were cut from uniformly doped Si and Ge crystals, and their surfaces were melted by irradiation with single pulses from a Q-switched ruby laser (20 ns pulse width, h = 0.694 pm). The energy 0 167-577x/84/$ 03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
February 1984
MATERIALS LETTERS
Volume 2, number 3
densities of the irradiation were adjusted so as to give a growth rate of = 1 m/s after the melting. (The consequent melting depth reached = 1 pm.) Dopant concentrations were measured by secondary-ion mass spectrometry (SIMS) and normalized by the original concentrations C,, in the unmelted region. The depth profile of dopant distribution in the regrown region can be calculated to a good approximation by a one-dimensional diffusion: letting C(x,t) stand for the dopant concentration in the melt, the diffusion equation can be written as D a2C(x,t)/dx2
DEPTH x ti,
= K’(x >t)/at a
(1)
where D is the diffusion coefficient in the melt and x is the depth from the surface. The following assumptions can be made: (1) The interface moves at a constant growth rate Vtoward the surface and rejects the impurity atoms at a rate V(l-k*)C*(x) cme2 s-l as shown schematically in the inset of fig. la, where C*(x) is the impurity concentration in the melt at the interface. (2) Diffusion in the solid is negligible. (3) Convective mixing in the melt is negligible. The redistribution of impurities after solidification consists of the initial and terminal transient regions, as shown in fig. 1a. When the melting depth x0 is deep enough and/or k* is not far from unity, the steady state (C,/Cu = 1) is achieved between both regions. For this case, the expressions for normal freezing [lo] are applicable, i.e. C,(x)/C,
=f (1 +erf{i[(V/D)(xo
+ (2k* - l)exp[-k*(l X erfc(i(2k*
C,(x) -=
1t 3 s
-k*)(V/D)(xo
- l)[(V/D)(xu
for the initial transient,
-x)]‘/~} -x)]
-x)]‘/~}}
(2)
and
exp [-2( V/D)x]
CQ + 5 (’ -k*)(2-k*)exp[-G(V/D)x] (1 +k*)(2+k*) t(2n + 1) (1 -k*)(2-k*)...(n (1 tk*)(2 +k*)...(n X exp [-n(n
t l)( V/D)x] + . . .
t .., -k*) +k*) (3)
for the terminal transient, where C,(x) is the impurity concentration in the solid, given by C, = k*C*(x). In fig. la, C,(x)/C,, obtained from eqs. (2) and (3), is
(b)
50
100
Oo1 (V/D)
x
Fig. 1. Calculated dopant profiles resulting from resolidification after melting of uniformly doped crystals by laser irradiation. The ratio of the concentration C,(x) at a depth x to the initial concentration Cu is plotted as a function of (V/D)x, using eqs. (2) and (3). (a) The entire profiles consisting ofthe initial transient, steady state, and terminal transient. xn = depth of melting. The inset shows schematically the profile during the regrowth. DBL stands for the diffusion boundary layer. (b) Profiles in the terminal transient calculated for various values of interfacial segregation coefficients k* by eq. (3). The figure on each curve denotes thevalueof k*.The experimental data for a (100) surface of a Bdoped crystal are also plotted as an example.
plotted fork* = 0.35,0.07, and 0.02 as a function of a dimensionless parameter (V/D)x expressing the depth from the surface. The melting depthxu = 1 pm is deep enough for forming the steady state for impurities having k* 2 0.02. To determine k* , both the calculated and measured ratios, Cs/CO, for the terminal transient region were plotted for (V/D)x and x in a logarithmic scale, respectively, as shown in fig. lb. The curves calculated for various values of k* were examined by sliding the experimental curve along the abscissa so as to superpose on each other. By finding the best fit of the calculated 203
curves, we can determine
k* without knowing values
for D and K As seen from fig. lb, the accuracy of measured k* values is high for the range of k* >, 0.1, although this method cannot be applied to measurement of small k* such as for Bi or In in Si. As an example, the data for a (100) surface of a B-doped Si crystal are plotted in fig. 1b. We can determine the growth rate to be I/= 0.9 m/s from the correspondence between the depth x and (V/D)x.
3. Orientation
dependence
of impurity
incorporation
The most stable surface of Si and Ge crystals is of a (111) type which has the highest atomic density. The liquid-solid interface parallel to a (111) plane is evidenced to be atomically smooth [ 111. It is expected that growth in a [ 11 l] direction requires a high supercooling compared with that in other directions, to make two-dimensional nucleation at the interface. Therefore, an orientation dependence of impurity incorporation was observed for the three cases shown in fig. 2. In fig. 2a, whichis referred to as a “(I 1 l& case”, the entire surface was melted. While, in fig. 2b, the central part of a (111) surface is melted. As an example of growth in the other directions, a (100) or (2 11) surface was examined, as shown in fig. 2c. The dopant profiles of the terminal transient regions for these three cases of B-doped Si and Ga-doped Ge crystals are shown in figs. 3a and 3b respectively: The curves are the calculated ones with the best fit to the experimental data. From fig. 3a, the interface segregation coefficient of B in Si was found to be k* = 0.7 for the (100) case (fig. 2~). This value is equal to the equilibrium segregation coefficient k, = 0.7 for the dilute solution (although the value currently accepted is 0.8, this has been corrected to be 0.7 [12]), i.e. k* = k,. While, for the (11 l)ent case, we obtained k* = 0.3 which is
Fig. 2. Orientation terface. (b) (111)
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(b)
(V/Lax
(0) 0.1
0.2
a5
1.0
20
( V/D) x 0.1
DEPTH
0.5
1.0
x ti,
Fig. 3. Orientation dependence of depth profiles of dopants measured after melting by laser irradiation as shown in fig. 2. (a) B in Si. Energy density of the laser pulse = 3.3 J/cm2. (b) Ga in Ge. Energy density = 2.0 J/cm2. The data obtained by SIMS are plotted against depth x, and the solid curves are calculated using eq. (3). The data for the (111) case (dashed lines) do not fit any curves calculated from eq. (3).
smaller than ko(k* < ko). Whereas, for the (111) case of fig. 2b, any curves calculated from eq. (3) do not fit the experimental result which is shown by the dotted line in fig. 3. It was found by a SIMS measurement that the B concentration was high at the center and low in the peripheral region of the laser melted area. A similar orientation dependence for Ga in Ge is seen from fig. 3b: k* = ko(= 0.09) for the (211) case; for the (11 l)ent case, the values of Cs(x)/Co are lower than the curve for k* = 0.09 in the deep region and exceed the curve as the depth x decreases. This result indicates that k* is so small that the steady state is not formed completely between the initial and terminal transient regions (k* <0.02), i.e. k*
of crystal surfaces melted by laser irradiation. (a) (11 l)ent case. The entire (111) surface case. (c) (100) case. Growth steps move in the direction of the arrows.
is melted.
Rough
in-
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Fig. 4. Profiles of spreading resistances (SR) measured for the surface layers regrown from the laser-induced melt. Originally the crystalswere uniformly doped with phosphorus. (a) Anglelapping for the depth profile measurement, which was made along the dotted line. (b) (211) case. (c) (11 l)ent case, (d) (111) case. The arrow corresponds to the point A in (a). The taper angle 01 = tan-’ 0.005. The scale indicates 1 ,um in the depth direction. (e) SR profile measured for the surface lapped parallel to the crystal surface of the (111) case, i.e. 01= 0.
tween the specimen surface and lapped surface was OL = tan-l 0.005. Figs. 4b, 4c and 4d are the results for the (211) (Lll)ent, and (111) cases illustrated in fig. 2, respectively. The former two cases gave smooth profiles as seen in figs. 4b and 4c. Whereas, for the (111) case, SR values fluctuate drastically in the laserirradiated region, and the boundary between the laser melted and unmolten (substrate) regions is seen clearly, as indicated by the arrow “A”. Since resistivities up to a depth of a few microns contribute to the SR value, the variations in dopant concentration may be much larger than those obtained from the fluctuation in SR. It is shown by laser Raman spectroscopy, however, that this laser-annealed layer is of single-crystal silicon. When the specimen was lapped parallel to the (111) surface (o = 0”), much smaller fluctuations of the SR profiles were obtained as shown in fig. 4e. This observation indicates that the dopant P is stratified laterally in the laser-annealed region. The differences in both the impurity (SIMS) and SR profiles between the (111) and (11 l)ent cases indicate that solidification occurs laterally in the (111) case and perpendicular to the (111) surface in the (I1 l)ent case. It should be noted that the aspect ratio of the molten layer is 1 : 3000 (melting depth = 1 pm, size = 3 mm in diameter for the (111) case; crystal
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1984
size for the (11 l)ent case = 2 X 2 mm2). One can conclude that growth perpendicular to a (111) plane requires so large supercooling that, in the (111) case, only lateral growth occurs from the periphery which supplies growth steps with terraces parallel to the (111) plane. The stratified dopant distribution may be attributed to inhomogeneities in step density and velocity, i.e. step bunching [13]. The above experimental results are summarized as follows: (1) The interfacial segregation coefficient k* is still equal to the equilibrium one k, at V x 1 m/s for B or P in Si, and Ga in Ge (k* = ko). (2) The (111) cases are in the same orientation, but the and(lll)ent shapes and structures of growth interfaces are different. Consequently, the impurity redistribution and resulting properties are different between both cases. For the (111 )ent, supercooling at the interface is very large compared with the other cases. The departure from equilibrium was found to result in a smaller segregation coefficient (k*
4. Discussion In light of the current theory [ 14,151, our experimental results indicate that the interface in the (l l l )ent case is rough and/or diffuse owing to the high supercooling, and that, in the other cases, the interface is sharp, and growth occurs by passage of steps. Leaving aside the problem of the (11 l)ent case for the moment, let us first look at the (100) or (211) cases. The result k* = k, suggests that the growth interface is still in local equilibrium even at V = 1 m/s. At this growth rate, however, the time to solidify a monatomic layer is as short as a/V=lO-lo s where a is the interatomic spacing, and in this interval, atoms at the interface must exchange their positions in the liquid and interfacial states frequently enough to reach the equilibrium distribution. Since the time for diffusive translations in the liquid isa2/D = lo-l1 lo-l2 s on average, both states should exist at the interface before releasing their latent heat of solidification, i.e. solute segregation must take place between the liquid and liquid monolayer adjacent to the interface which is referred to as “the interfacial state”. We 205
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0.3 k” 0.2 100) 0. I k,” E 0
__ 0
I
2
3
4
v (m/s)
Fig. 5. Dependence of interfacial segregation coefficient k* on growth rate V in rapid growth following laser melting of Si (100) surfaces. The data obtained for Bi in Si by Baeri et al. [ 161 were replotted. (In their original figure, k* is shown on a logarithmic scale.) k* tends to a certain value kz for V - 0, which is much larger than the equilibrium segregation coefficient k. = 0.0007.
denote the equilibrium segregation coefficient between the interfacial state and liquid by k;. Then, our experimental results means that k; = k,. In general, however, it is expected that kc # k,. In fact, this can be seen for Bi in Si: Baeri et al. [16] obtained a relation between the interfacial segregation coefficient k* and growth rate V by laser annealing of B&implanted Si crystals. Their data for the (100) surface are replotted in fig. 5. (In their original figure, k* is shown on a logarithmic scale.) Extending the curve to V = 0, we can estimate the value of k; = 0.02 which is much larger than the equilibrium segregation coefficient k. = 0.0007, i.e. k; > k,. From these experimental results, one can conclude that the interfacial state in local equilibrium with the liquid is frozen by rapid growth at I/ = 1 m/s. The solute segregation in the interfacial state is considered as follows: An approximation to reality that is of considerable use for most semiconductor solutions is that the liquid solution is ideal (the solid is a regular solution) [ 171. Then, we may assume that both the interfacial state and the liquid are ideal solutions; solute and solvent atoms act independently at the interface. It is envisaged that atoms in the interfacial state take partial ordering, e.g., the atoms vibrate similarly to those in the liquid, but their average positions are at the crystal lattice points. In other words, the entropy of the interfacial state is lower than that of the liquid. Since the enthalpy change is nearly equal to zero (we consider the state before releasing the latent heat), we can write kT, as
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where aS is the entropy difference between both the states per solute atom and k is Boltzmann’s constant*. It follows from eq. (4) that k: is independent of temperature, i.e. composition-independent segregation takes place. In the foregoing argument, the free energy per atom of the interfacial state is higher, relative to the liq uid state. The interfacial state is considered as a transition state between the liquid and solid, sharing a part of the interfacial free energy. We now consider the reason for the experimental result that k: = k. for B or P in Si and Ga in Ge. This result implies that the environment for the individual solute atoms in the interfacial state is the same as that in the crystal, i.e. in the dilute solid solution, the solute atoms in the crystal lattice are still in the liquid state near the melting point To of pure Si (141S’C)or Ge (958°C). We may envisage that solute atoms are accommodated in the drops including some solvent atoms (partially molten regions). Then, solute atoms have no enthalpy difference between them in the liquid and crystal (neglecting the interfacial energy of the drop), and it is expected that the equilibrium segregation coefficient k,(T) is independent of temperature (k, = ko) in a region just below To. Although phase diagrams which have been reported are not precise enough to confirm this inference, it is likely that such a temperature region exists for most dopants in Si and Ge: As an example, the As-Si system is shown in fig. 6a [ 17,191. The equilibrium segregation coefflcients k,(T) obtained from the liquidus and solidus curves in fig. 6a are plotted against l/T in fig. 6b. The curve consists of three parts marked “D”, “I” and “B”. In region D, k, is independent of temperature, as described above. Regions I and B may be explained as follows: Around T = 1380°C the drops solidify and the As atoms are in the fully ionized state. The steep slope is due to the energy of the dangling bonds of solvent atoms and strain around the ion. In the range B of T 2 1260°C, most As atoms form bonds with the solvent atoms, and the slope is due to strain energy around them. Details, such as the enthalpies obtained from the slopes in regions I and B, will be discussed elsewhere. The relation k; = k, in the dilute
* For the general formula of equilibrium segregation coeffik; = exp(-A,S/k)
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,
(4)
cients, see ref. 1181.
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(b)
0.051 6
As
CONCENTRATION
(atoms
/cm3
)
7
0 x 10-4
I/T
Fig. 6. S-As system. (a) Phase diagram [17,19]. The dashed curve shows As concentration of the interfacial state in equilibrium dependence of the equilibrium segregation coefficient with the liquid which is frozen at growth rates of = 1 m/s. (b) Temperature k,. The values of kc obtained from (a) were plotted against l/T. ko = 0.3.
case can be understood from the foregoing argument. For the Bi-Si system, however, it can be seen from the phase diagram [17,19] that the curve consists of the parts I and B, and contains no D region. Consequently, we have kT, # k,. The solubility limits of substitutional solid solutions achieved by laser annealing are explained by eq. (4): In the As-Si system shown in fig. 6a, by increasing the As concentration, the melting temperature decreases. Lowering of the melting point results in growth rates of V = 1 m/s even at such energy densities of laser irradiation that give higher growth rates (~4 m/s) in the dilute cases. Therefore, the solubility limit is considered to be the maximum solute concentration in the interfacial state. When the concentration C, in the melt exceeds the eutectic composition Cy , the phase of the dopant appears. Since there is no enthalpy change between the liquid and interfacial state, the segregation coefficient between both states is independent of temperature or composition. Then, the maximum solid solubility C?$,, is given as the dopant concentration of the interfacial state in equilibrium with the eutectic composition of the melt, i.e. Tax = k*Ceu. F rom the phase diagram in fig. 6a, we obtain C’$=” 6 X lo21 . This value agrees with the maximum solubility 6 X 1021 for As in Si measured by White et al. [2], This explanation for the maximum solubility can be applicable to other dopants [9]. (For Bi in Si, using kz = 0.02 (see fig. 5) and CT = 5 X 1O22 cm-3 [17], Tax = k&l’? = 0.02 X 5 X 1O22 = 1021 cme3. The measured value of Tax = 1 .I X 1021 [20] .) Let us now turn to the (11 l)ent case. In compari-
son with the (100) or (211) case, the interface is largely supercooled, but the time to solidify a monatomic layer is similar (u/V = lo-lo s). If we assume that the interfacial state is the same, the segregation coefficient k* should have been equal to k: even for large supercooling, because k; is independent of T as described before. The result of k* < ko must be explained by different interface structure. The comparison of the experimental results for (111) and (11 l),t cases indicates that, in the (111) case, the interface is discrete (sharp) and advances by a step mechanism. According to the theories of Temkin [ 141 and Cahn [ 151, the interfacial free energy is a periodic function of the average position of the interface, with the minimum at the position of (111) lattice planes and maximum between the planes. The interface cannot advance across the maximum by thermal activation because all of the liquid atoms adjacent to the interface must hop over the energy barrier; its only means of advancement is by the motion of growth steps. For the (11 l)ent case, however, there are no sources supplying steps. Therefore, the rapid growth must occur by the driving force (supercooling) which is so large that the barrier is less effective. Then, the interface tends to be rough [14] and/or diffuse [15]. In such cases, it is natural that part of the solvent atoms in the liquid monolayer adjacent to the interface form bonds to decrease their free energy, relative to that in liquid, and at the same time the chemical potential for the solute atoms is raised. Consequently, we have a low solute concentration at the interface, i.e. k*
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The observation of k*
References [l] [2]
[31 [41 [51 [61 [71
[81
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H. Kurz, J.M. Liu and N. Bloembergen, Physica 117/ 118B (1983) 1010. C.W. White, S.R. Wilson,B.R. Appletonand F.W. Young Jr., .I. Appl. Phys. 51 (1980) 738. H.J. Leamy, J.C. Bean and J.M. Poate, J. Crystal Growth 48 (1980) 379. N. Natsuaki, M. Tamura and T. Tokuyama, J. Appl. Phys. 51 (1980) 3373. J.C. Baker and J.W. Cahn, Acta Met. 17 (1969) 575. A.A. Chernov, Kristallografiya 12 (1967) 222 [English transl. Soviet Phys. Crystallography 12 (1967) 1861. K.J. Jackson, G.H. Gilmer and H.J. Leanly, in: Laser and electron beam processing of materials, eds. C.W. White and P.S. Peercy (Academic Press, New York, 1980) p.104. R.F. Wood, Appl. Phys. Letters 37 (1980) 302.
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[9] J. Chikawa and F. Sato, Japan. J. Appl. Phys. 19 (1980) L577. (lo] V.G. Smith, W.A. Tiller and J.W. Rutter, Can. J. Phys. 33 (1955) 723. [ 111 T. Abe, J. Crystal Growth 24/25 (1974) 463. [ 121 H.R. Huff, T.G. Digges Jr. and O.B. Cecil, J. Appl. Phys. 42 (1971) 1235. [ 131 F.C. Frank, in: Growth and perfection of crystals, eds. R.H. Doremus, B.W. Roberts and D. Turnbull (Wiley, New York, 1958) p. 411. [ 141 D.E. Temkin, in: Crystallization processes, eds. N.N. Sirota, F.K. Gorskii and V.M. Varikash (Consultants Bureau, New York, 1966) p.15. [15] J.W. Cahn, Acta Met. 8 (1960) 554. [ 161 P. Baeri, G. Foti, J.M. Poate, S.U. Compisano and A.G. Cullis, Appl. Phys. Letters 38 (1981) 800. [17] C.D. Thurmond and M. Kowalchik, Bell System Tech. J. 39 (1960) 169. [18] C.D. Thurmond and J.D. Struthers, J. Phys. Chem. 57 (1953) 831. [19] F.A. Trumbore, Bell System Tech. J. 39 (1960) 205. [20] C.W. White, B.R. Appleton, B.Stritzker, D.M. Zehner and S.R. Wilson, in: Laser and electron-beam solid interactions and materials processing, eds. J.F. Gibbons, L.D. Hessand T.W. Sigmon (North-Holland, Amsterdam, 1981) p.59.