INFLUENCE OF TEMPERATURE ON THE COEFFICIENT
OF THERMAL EXPANSION OF MONOCRYSTALS OF SILICON
A. V. Mazur1 and L. P. Stepanova2 UDC 669.13:669.715
Using specimens of monocrystalline silicon with strict crystallographic orientation and insignifi- cant (1.5°) off-orientation, we have studied changes in the value (up to 400%) and sign of the coefficient of thermal expansion for the crystallographic directions 〈111〉, 〈110〉, and 〈112〉. Ex- periments were carried out under conditions of permanent change in the temperature from 323 to 1473K and under normal pressure. We have corroborated that the monomorphism of silicon (Fd3m, i.e., a diamond-type lattice) is retained over the entire temperature range. Specimens of crystallographic direction 〈110〉 are much less subjected to the influence of changes in the tem- perature than the others.
With increase in the working temperatures of electronic chips, silicon monocrystalline substrates, micro- springs made of monocrystalline silicon, and other microarticles, the frequency of failures of electronic circuits grows. One of the causes of this phenomenon is connected with the self-destruction of articles or their detach- ment from the substrate material [1]. To produce these articles, it is customary to use monocrystalline silicon with cutting of the plates along the planes {111} or {100} [2]. However, in many cases, the influence of cut- ting and crystallographic off-orientation of articles on their thermomechanical characteristics is not taken into account.
As established in recent investigations [3, 4], polymorphic phase transformations can take place in mono- crystalline silicon with change in the temperature under normal pressure. They were recorded by the tempera- ture dependence of the coefficient of thermal expansion (CTE), and the polymorphic types of silicon crystal lat- tices were identified by the appearance of additional maxima in X-ray diffractograms at temperatures higher than 553K.
The aim of the present work is to investigate the thermal expansion of monocrystalline silicon in a wide temperature range depending on the crystallographic orientation in order to determine the thermomechanical compatibility of monocrystals of silicon with the components of electronic devices.
Material and Experimental Procedure
We studied a monocrystal of silicon of semiconductor purity, grown by the Czochralski method on a mono- crystalline substrate with cutting {111}. It represented a cylindrical rod with axis [111] 50mm in height and 16mm in diameter, cut along its generating line by the natural crystallographic planes of an octahedron.
We determined the orientation of crystallographic planes and directions in the monocrystal by the Laue method of inverse shooting with the obtaining of epigrams. The photographic film was located in front of the crystal transversely to the primary beam at a distance of 40.5mm from the surface of the crystal. The epigrams were interpreted using the standard procedure [5].
1 Helsinki University of Technology, Helsinki, Finland.
2 Zaporizhzhya National Technical University, Zaporizhzhya.
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol.41, No.4, pp.86–92, July–August, 2005. Original article submitted March 17, 2005.
1068–820X/05/4104–0531 © 2005 Springer Science+Business Media, Inc. 531
We manufactured specimens of the monocrystal by a diamond tool in a jet of cooling liquid. They had the shape of an elongated rectangular parallelepiped of length 14mm with a square cross section 2.5×2.5mm transversely to the principal axis. The surfaces of the specimens were polished (except those that are formed by the natural faces of the monocrystal).
The first group of specimens was oriented in such a way that the two opposite end surfaces of each speci- men are opposite with respect to the symmetry axis of the initial monocrystal, fragments of its natural mutually parallel faces. Each pair of such faces belongs to the assigned family of crystallographic planes, which was con- trolled by the Laue method. The second group of specimens had an off-orientation of the end surfaces of 1°30′
relative to the end surfaces of specimens of the first group. For the sake of comparison, we also used a specimen made of polycrystalline silicon of chemical purity with the same sizes.
Dilatometric measurements were carried out on a “Netzsch DIL 402C” instrument in a flowing argon at- mosphere with a rate of temperature change of 10K/min in the range from 323 to 1473K. In the course of measurements, a constant nondestructive force of 0.3N acted on the end surfaces of specimens formed by their natural faces. The procedure of calibration of the instrument and measurements of the temperature dependence of the CTE of monocrystalline silicon was described earlier [6].
For the sake of independent control of the possible phase transformations in monocrystalline silicon, we carried out differential scanning calorimetry of the specimens on a Netzsch STA 449 “Jupiter” instrument in the same temperature range under study. The sensitivity of this method with respect to enthalpy is ±2%. The spec- imen was heated in a preliminarily calibrated Alundum crucible with a flowing atmosphere of pure argon. The heating/cooling rate was 10–20K/min, and the flow rate of the inert gas was 20cm3/min. The frequency of scan of the detectors was 20sec–1 in both experiments.
We carried out crystallographic control of the type and parameters of the crystal lattice of specimens over the entire temperature range under study on a DRON-UM1 instrument in monochromatic CuKα-radiation.
A graphite monocrystal was used as a monochromator. Measurements were performed by the method of step- by-step scanning every 0.05° in the angle interval 2θ = 10–160°, and the exposure time at each point was 3–9sec. We calculated the lattice spacing of silicon with the help of the “CaRIne 3.1” program for full-profile analysis of X-ray spectra using the relation
a(T) = a i T T i
i
0 0
1 3
1+ ( − )
⎛
⎝⎜
⎞
= ⎠⎟
∑
α ,where a0 = 0.543047nm is the period of the elementary cell of Si at T0 = 293K and αi are the coefficients of thermal expansion for silicon (α1 = 1.887⋅10–6K–1, α2 = 1.934⋅10–9K–1, and α3 = –4.544⋅10–13K–1 [7]). Experimental Results and Discussion
Analysis of the diffraction patterns taken from the upper and lateral faces of the specimens shows that the symmetry axis of the monocrystal corresponds to the crystallographic direction [111], and six rays in its cross section coincide with the crystallographic directions 〈110〉. The group of lateral faces of the monocrystal per- pendicular to these rays is cut by {110}-type planes. Another group of lateral faces of the monocrystal located between the mentioned faces of the first group coincides with the crystallographic planes of the family {112}.
Thus, we determined the temperature dependence of CTE of the monocrystal of silicon by dilatometric measure- ments in the directions perpendicular to its crystallographic planes {111}, {110}, and {112}, whereas, in the literature, one can find the corresponding characteristics in the “technologically conditioned” directions 〈111〉,
〈100〉, and 〈110〉 [7].
Fig. 1. A calorimetric curve of the monocrystal of silicon: (1) heating, (2) cooling.
Fig. 2. Diffractograms of monocrystalline silicon at different temperatures.
Calorimetric analysis of the specimens for a rate of temperature change of 20K/min shows (Fig.1) that, in the high-temperature region, there exist two peaks describing endothermic and exothermic processes, i.e., the melting and crystallization of silicon. We did not detect any other thermal effects that could be ascribed to the polymorphic transformations of silicon in the solid state.
In the diffractograms of the specimens at different temperatures, there are reflections from the diamond- type crystal lattice of silicon (Fig.2). At 293K, the spacing of this lattice is equal to 0.543047nm, its holo- hedral class is m3m of cubic syngony, and its space group is Fd3m. The width and intensity of peaks vary pro- portionally to changes in the temperature of the specimens; however, interference from different types of crystal lattices, superstructures, and other phases are absent in the diffractograms at all temperatures under study.
The temperature dependence of CTE of the specimens of monocrystalline silicon is nonmonotonic (Fig.3).
For the specimens of group 1 loaded in the directions 〈111〉 and 〈112〉, it is similar and differs from the curve for the direction 〈110〉 by a CTE jump in the range 323–553K (Fig.3a). The CTE value for specimens of the directions 〈111〉 and 〈112〉 decreases from (5.4–5.8)⋅10–6K–1 at 323K to 2.85⋅10–6K–1 at 448K and grows to (4.3–4.7)⋅10–6K–1 at 543K. The CTE value for the specimen of the direction 〈110〉 changes not so intensely: it is equal to 3.8⋅10–6K–1 at 323K, decreases to 3.65⋅10–6K–1 at 393K, and grows to 5.2⋅10–6K–1 at 543K. For all three specimens, the CTE values are negative at the beginning of heating, which corresponds to the data presented in [8]. In the course of subsequent heating, the curves for the specimens of group 1 are similar and oscillate in the range (3.8–4.8)⋅10–6K–1.
Fig. 3. Dilatometric curves (ΔL/L0) and change in the CTE (α) for the specimens of groups 1 (a) and 2 (b) and for polycrystal- line silicon (c).
The curves for the specimens of group 2 (Fig.3b) can be divided into two kinds differing in their shape and intensity. The curves of the first kind are monotonic with insignificant oscillations, and the coefficient α in the direction 〈110〉 + 1.5° takes values (3.5–4.5)⋅10–6K–1 over the entire temperature range under study. These data are in good agreement with the results [3, 9]. The curves of the second kind are nonmonotonic and can be divided into two parts by temperature ranges. The curves α = f(T) for the specimens 〈111〉 + 1.5° and
〈112〉 + 1.5° have a maximum at 433K and pass to the part of monotonic change at temperatures higher than 493K. We observe a vast difference in the CTE values for these specimens at the low-temperature part (from –0.5⋅10–6 to 1 . 5⋅1 0–6K–1, i.e., by 400% in absolute value) and their approximate constancy (0.5–0.75)⋅10–6K–1 at the high-temperature.
The CTE for polycrystalline silicon drops sharply from 3.8⋅10–6 to 1.1⋅10–6K–1 in the temperature range below 423K, then grows to 2.0⋅10–6K–1 with heating to 573K, and, with subsequent heating, remains in the range (1.7–2.0)⋅10–6K–1 (Fig.3c).
Monocrystalline silicon is an anisotropic material, and, hence, its CTE is a tensor of second rank. The value and sign of its components (αi j) depend on the crystallographic orientation of the monocrystal at any temperature. To pass from the direction [110] to other crystallographic directions in a perfect monocrystal of sil- icon, it is necessary to deviate by an angle of 6.3° for obtaining the direction [540], by 8° for [551], by 10° for [441], by 11.3° for [320], by 13.3° for [331], by 14° for [530], by 18.4° for [210], by 19.5° for [221], by 26.6° for [310], by 35.3° for [111], and by 45° for the direction [100] (see [10]). Therefore, we may consider the specimens with an off-orientation of 1°30° crystallographically similar to the corresponding specimens with strict orientation.
Over the entire temperature range, X-ray diffraction analysis detects only lines from a diamond-type lattice, which is evidence of the fact that the polymorphism of silicon is absent. In [11], one can find mention of an ad- ditional peak for 2θ = 43° with intensity on the background level in the diffractogram of a specimen of mono- crystalline silicon (about 1cm3 in volume), recrystallized on a crystalline substrate above 1573K. However, in this paper, the additional peak is considered to be rather an artifact than a new reflection from silicon, especially since the other (traditional) reflections from silicon and their relative intensities in this diffractogram remained
without changes. In the literature, there is no information on the detection of polymorphic modifications of sili- con under normal pressure induced only by an increase in the temperature of specimens, except the data of [3, 4].
Over the entire temperature range, calorimetric control recorded neither heat flows caused by phase trans- formations of the first kind (to which the polymorphic transformation is ascribed to) nor thermal effects from phase transformations of the second kind. Hence, the assumption on the influence of a hypothetical polymorphic transformation in silicon under normal pressure on the temperature dependence of its CTE is not confirmed ex- perimentally. The crystallographic direction characterized by the minimum electric resistance corresponded to the direction 〈111〉 after recrystallization at 1573K and to 〈100〉 before it [11]. This is connected with the density of defects in monocrystalline silicon in the direction 〈100〉 but not with changes in the type of crystal lattice of the specimen [11]. The dilatometric curves ΔL
/
L0 for monocrystalline silicon in the direction 〈111〉 obtained in our experiments (Fig.3) are similar to those established earlier [11].The contrast introduced by the minor crystallographic off-orientation of specimens of the second group to the overall picture of our experiment enables us to detect confidently two temperature regions on the CTE poly- therms of monocrystalline silicon, which border each other in a narrow temperature range 473–493K. In all cases studied in this work and in [8], the temperature dependence of the CTE of monocrystalline silicon does not coincide with that for polycrystalline silicon over the entire temperature range from 293 to 1473K.
As was established earlier (see, e.g., [1, 8, 12]), the structural and sensitive properties of monocrystalline silicon are anisotropic. However, information on the anisotropy of its CTE, obtained in the present work under conditions of continuous measurement in a wide temperature range, is new. In particular, we should mention here the fact of relative stability of the CTE of specimens with the crystallographic direction [110] over the entire temperature range under study. The temperature dependence of CTE has no strong variations, and the off-orien- tation (1°30′) also affects it weakly. On the contrary, specimens of the directions [111] and [112] respond strongly to temperature changes. Furthermore, if the direction [112] is not considered to be technologically sig- nificant, then the direction [111] is traditionally recognized as technologically important for monocrystalline sili- con [1].
As is well known [13], the cleavage and twinning of silicon crystals manifest themselves along the {111}- type planes, and the brittle destruction of crystals also occurs along these planes as the most close-packed. How- ever, this is correct only for low temperatures. Above 523K, the picture is quite different. Silicon crystals, heated rapidly to 973K, cooled afterwards to 853K with a rate of 2K/min, and held isothermally at this tem- perature, split along the planes {100} in the course of subsequent quenching in water, acquired twinning planes {110}, and texture in the direction [112] appeared on their surface [14]. Effects with the crystallographic orien- tation {111} were not observed in specimens after thermal treatment [14]. Thus, the temperature dependence of the properties of monocrystalline silicon in the direction [111] established in this work is corroborated by litera- ture data. If a monocrystal of silicon is heated to a temperature higher than 473–493K, passage of the domi- nant orientation (this term was introduced in [11]) from the direction [111] to [110] takes place.
We may assume that the energy supplied from without in the course of heating of the specimens is scattered in their bulk in the form of thermal phonons [15], which increases the total internal energy of the system. Here, structural changes will be determined by the principle of minimization of the internal energy. According to [16], a crystal lattice of the body-centered cubic type has to become the next structural type. However, an increase in the level of internal energy of the specimen by means of heating leads to its melting at 1683K, but, for a hypo- thetical transformation of the diamond lattice of silicon into a body-centered cubic lattice, it is necessary to reach 2100K under normal pressure.
If the internal energy of the system of monocrystalline silicon grows at the expense of pressure application, which does not induce an increase in its temperature up to melting, then, at a certain stage, the polymorphic transformation of the silicon crystal lattice into a tetragonal lattice (like metallic white tin) takes place. In prac- tice, for the polymorphic transformation of the diamond lattice of silicon into a tetragonal one, it is necessary to apply and hold an external pressure of about 8–10GPa [17, 18], and, under a pressure of 2GPa, the polymor-
phic transition R8 → BC8 takes place [19]. After unloading, the diamond type of the silicon lattice is restored.
Thus, theoretical analysis and laboratory investigations do not confirm the possibility of existence of the polymorphic modifications of silicon in a wide temperature range up to its melting point under normal pressure.
CONCLUSIONS
Under normal pressure, monocrystalline silicon is monomorphic in the range 323–1473K. The signs and values of its CTE are dissimilar for different crystallographic directions and depend substantially on the crystal temperature. The generally accepted parameters of the CTE are correct over the entire temperature range under consideration for the direction 〈110〉 and only from 323 to 473–493K for the directions 〈111〉 and 〈112〉. In this range, the difference between the CTE values of monocrystals depending on the crystallographic direction can be very significant (up to 400% of magnitude), including a change in the sign.
Insignificant (by 1°30′) off-orientation with respect to the crystallographic direction 〈111〉 in a monocrys- tal of silicon can induce alternate thermal stresses in electronic articles, substantially exceeding those in the ab- sence of off-orientation. Hence, to decrease the frequency of failure of articles, one should take into account the anisotropy of monocrystals of silicon by introducing crystallographic control of the orientation at the stage of their manufacture.
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