Corresponding author: Svetlana P. Kobeleva ( kob@misis.ru ) © 2019 Svetlana P. Kobeleva, Ilya M. Anfimov, Vladimir S. Berdnikov, Tatyana V. Kritskaya.
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Citation:
Kobeleva SP, Anfimov IM, Berdnikov VS, Kritskaya TV (2019) Possible causes of electrical resistivity distribution inhomogeneity in Czochralski grown single crystal silicon. Modern Electronic Materials 5(1): 27-32. https://doi.org/10.3897/j.moem.5.1.46315
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Electrical resistivity distribution maps have been constructed for single crystal silicon wafers cut out of different parts of Czochralski grown ingots. The general inhomogeneity of the wafers has proven to be relatively high, the resistivity scatter reaching 1–3 %. Two electrical resistivity distribution inhomogeneity types have been revealed: azimuthal and radial. Experiments have been carried out for crystal growth from transparent simulating fluids with hydrodynamic and thermophysical parameters close to those for Czochralski growth of silicon single crystals. We show that a possible cause of azimuthal electrical resistivity distribution inhomogeneity is the swirl-like structure of the melt under the crystallization front (CF), while a possible cause of radial electrical resistivity distribution inhomogeneity is the CF curvature. In a specific range of the Grashof, Marangoni and Reynolds numbers which depend on the ratio of melt height and growing crystal radius, a system of well-developed radially oriented swirls may emerge under the rotating CF. In the absence of such swirls the melt is displaced from under the crystallization front in a homogeneous manner to form thermal and concentration boundary layers which are homogeneous in azimuthal direction but have clear radial inhomogeneity. Once swirls emerge the melt is displaced from the center to the periphery, and simultaneous fluid motion in azimuthal direction occurs. The overall melt motion becomes helical as a result. The number of swirls (two to ten) agrees with the number of azimuthally directed electrical resistivity distribution inhomogeneities observed in the experiments. Comparison of numerical simulation results in a wide range of Prandtl numbers with the experimental data suggests that the phenomena observed in transparent fluids are universal and can be used for theoretical interpretation of imperfections in silicon single crystals.
Czochralski single crystal growth, single crystal silicon, electrical resistivity, electrical resistivity distribution maps, growth process simulation
The growth of silicon single crystals with homogeneous impurity distributions using the Czochralski technique requires maintaining growth conditions providing for a flat crystallization front (CF), minimum radial temperature gradients at the crystallization front and absence of solutal undercooling of the melt under the crystallization front [
We studied electrical resistivity distribution maps for 17 wafers cut out at the distance X from the beginning of five boron-doped p-type 1 Ohm·cm (KDB 1) Grade single crystal silicon ingots grown using the Czochralski technique in REDMET type growth plants [
The wafer diameter was 154 ± 2 mm, the thickness being 2.4 ± 0.2 mm. The measurements were conducted on a VIK UES-A instrument using the four-probe method with a linear probe setup. Electrical resistivity calculations were carried out taking into account all the corrections as per the applicable standards [
Average wafer parameters and inhomogeneity type.
Ingot No. / Specimen No. | Х, mm | Inhomogeneity type | ρav, Ohm × cm | σ, % | d/σ | d/ρav, % |
---|---|---|---|---|---|---|
1/2 | 470 | radial | 1.07 | 1.24 | 3.77 | 4.67 |
1/3 | 520 | radial | 1.149 | 1.034 | 6.73 | 6.96 |
1/4 | 570 | azimuthal(11) | 1.058 | 2.35 | 2.82 | 6.62 |
2/17 | 500 | radial | 1.2906 | 1.051 | 3.91 | 4.11 |
2/16 | 560 | radial | 1.17 | 1.12 | 5.34 | 5.98 |
2/1 | 635 | azimuthal(6) | 1.25 | 1.70 | 4.71 | 8 |
2/6 | 710 | azimuthal(3) | 1.07 | 1.99 | 4.23 | 8.41 |
3/8 | 0 | edge drop | 1.50 | 9.15 | 3.13 | 28.7 |
3/9 | 610 | azimuthal(5) | 0.96 | 1.59 | 3.93 | 6.25 |
3/5 | 660 | azimuthal(7) | 0.91 | 1.67 | 3.95 | 6.59 |
3/10 | 780 | radial | 0.79 | 1.60 | 3.96 | 6.33 |
3/11 | 790 | radial | 0.85 | 1.14 | 5.16 | 5.88 |
4/13 | 630 | radial | 0.967 | 2.151 | 4.33 | 9.31 |
4/14 | 680 | radial | 1.15 | 2.83 | 4.92 | 13.91 |
4/7 | 730 | radial | 1.096 | 1.732 | 3.69 | 6.39 |
5/12 | 585 | radial | 1.16 | 1.61 | 6.43 | 10.34 |
5/15 | 635 | radial | 1.11 | 1.47 | 4.90 | 7.21 |
Table
(a) electrical resistivity distribution map for Specimen No. 8 and (b) its section at Y = 20 mm.
The general tendency of the electrical resistivity map evolution can be accounted for based on the results of physical and numerical simulations of hydrodynamics and heat exchange and their effect on the shape of the crystallization front as follows. CF curvature and local impurity distributions in the crystal depend largely on the melt hydrodynamics and the hydrodynamics-controlled local thickness of the heat and concentration boundary layers. The azimuthal melt inhomogeneity is predominant in the free convection mode. Starting from uniform crystal rotation and with the transition to the mixed convection allows controlling melt hydrodynamics and local heat and mass exchange at the CF by varying the relative contributions of free and forced convection [
Both numerical and physical simulations for axial symmetry problem statement suggest that there are some specific ratios of the thermodynamic parameters in the free convection mode, where the shape of the sub-crystal isotherm is the most planar and the radial distribution of the local heat flow in the boundary layer is the most homogeneous [
Real crystallization experiments with the model fluid being heptadecane having Pr = 45.6 showed that at dynamic parameters of the melt flow close to those for the real single crystal growth conditions in the Czochralski technique, the CF shape under single phase convection conditions is almost perfectly flat. The experiment also showed that the axially symmetrical laminar 3D shape of the flow only exists until the threshold Gr, Ma and Rec numbers are reached. Once the Gr, Ma and Rec numbers exceeded the threshold level at the same ratio between parameters, the boundary layers under the rotating CF may lose stability, causing a system of azimuthal swirls to develop under the crystallization front. Figure
3D flow shape for mixed convection mode: (a and b) photo and diagram of central flow section, respectively (H/Rc = 0.7, Rc/Rsc = 2.76, Rc = 42.5 mm, Gr = 8.09 · 104, Re = 203); (c–e) flow structure in the 1 mm thick layer under the growing crystal: ((b and e) diagram and (d) photo: Pr = 16, Gr = 7.9 × 104, Ma = 2.71 × 104, H/Rc = 0,17, Rc/Rsc = 2.76; c: (1) Re = 365; (2) Re = 620; (d) Re = 1170; (e) Re = 1180).
The photos and diagrams of flow patterns shown in Fig.
Photos of central flow of melt/growing crystal system (hexadecane, Pr = 45.6) in mixed convection modes (Gr = 4750, Ma = 4590): (top) Re = 63.2; (bottom) Re = 91. H/Rc = 0.7; Rc/Rsc = 2.76; Rc = 50 mm.
Thus, in a specific Gr, Ma and Rec range depending on the relative melt height H/Rc (Rc being the crucible radius) and relative radius Rc/Rx (Rx being the growing crystal radius), a system of well-developed radially oriented swirls may emerge under the rotating CF. In the absence of such swirls the melt is displaced from under the crystallization front in a homogeneous manner to form thermal and concentration boundary layers which are homogeneous in azimuthal direction but have clear radial inhomogeneity. Once swirls emerge the melt is displaced from the center to the periphery, and simultaneous fluid motion in azimuthal direction occurs. The resulting overall melt motion is helical. Heated melt impinges onto the crystallization front in upward flows and may locally melt the CF, while in downward flows the melt is cooled, so the CF may become locally convex toward the melt. The structures of the boundary concentration layers and the respective concentration flows are similar. This seems to be the case for convection modes which produce single crystal portions with azimuthal electrical resistivity inhomogeneity. The number of swirls for the preset Prandtl number depends on the Grashof, Marangoni and Reynolds numbers ratios and on the H/Rc and Rc/Rx ratios under the rotating CF. Thus, considerable inhomogeneities with azimuthal symmetry components emerge in the temperature fields, thermal stresses and concentration fields.
Simulation of crystallization front shape dependence on local hydrodynamics also showed [
Figure
Analysis of the electrical resistivity distribution maps for Czochralski grown 150 mm silicon single crystals revealed two inhomogeneity types in the wafers: radial and azimuthal ones. The experimentally observed distribution features agree with the experimental results for transparent melt-simulating fluids. This suggests that the cause of the azimuthal inhomogeneities is the emergence of a swirl flow structure in the melt under the crystallization front, and the cause of the radial inhomogeneities is the curvature of the crystallization front arising due to local features of heat transfer. . Comparison of numerical simulation results in a wide range of Prandtl numbers with the physical experimental data suggests that the phenomena observed in transparent fluids are universal and can be used for theoretical interpretation of imperfections in silicon single crystals.