Dislocations in semiconductor crystals adversely affect the characteristic properties of semiconductors [1]. Dislocation-free or low-dislocation density bulk semiconductor crystals are required for substrates of high-performance electronic devices. To control and reduce the dislocation, the generation mechanism of dislocation has been broadly studied [2-4]. It is known that a dislocation can originate from several factors, such as agglomeration of point defects, surface damage, foreign particles or precipitates, chemical inhomogeneities, defective seeds, and thermal stress. Thermal stress is regarded as a primary reason for the presence of dislocations in as-grown crystals, and is mainly caused by the temperature gradient that exists during the crystal growth and cooling processes.
    During crystal growth, the cooling rate must be controlled to a small value to grow high-quality crystals, and thus, the thermal stress is generally small; however, during cooling, the cooling rate is usually set to be high to decrease the production cost, and thus, the thermal stress is generally large. Therefore, a critical issue in reducing the dislocation density is how to control the cooling process.
    Controlling the cooling process involves controlling both the radial cooling flux and the axial cooling flux. To determine the most effective method to reduce the dislocation density by adjusting the cooling flux, it is essential to determine the relationship between the activated slip system and its corresponding activation flux in the radial and axial directions (radial flux and axial flux). In this paper, we address this theoretical problem.
    Results show that the effect of cooling flux on the activation of slip systems in single-crystal silicon is dependent on the directions of the cooling flux and crystal growth. For crystal growth in the [001] direction, the 12 slip directions can be divided into an 8-fold symmetric group A_B2 and a 4-fold symmetric group B_A1. For crystal growth in the [111] direction, the 12 slip directions can be divided into three 3-fold symmetric groups C_A2, D_A2, and E_A1. Irrespective of the direction of crystal growth, radial flux only activates dislocation at the edges of slices and axial flux activates dislocation both in the interior and at the edges of slices.
    For [001] growth, radial flux can activate both eight-fold symmetry (group A_B2 at the edges of slices near both ends) and four-fold symmetry (group B_A1 at the edges of slices between the ends); however, axial flux can only activate eight-fold symmetry (group A_B2 in the interior of slices near both ends and at the edge of slices between the ends). For [111] growth, radial flux can only activate three-fold symmetry (group D_A2 at the edges of slices); however, axial flux can activate both groups C_A2 and E_A1, where group C_A2 exists in the interior of slices near both ends and at the edges of slices between the ends, and group E_A1 only exists in the interior of slices close to both ends.
    For practical crystal growth, it is essential to reduce the radial cooling flux as much as possible for the reduction of dislocation density.

Reference
[1] G. Dhanaraj, K. Byrappa, V. Prasad, M. Dudley, “Handbook of Crystal Growth”, p1335-1378.
[2] J.P. Hirth, J. Lothe: Theory of Dislocations, Krieger, Malabar, 1992.
[3] H. Alexander: On dislocation generation in semi-conductor crystals, Radiat. Eff. Defects Solids 112(1/2), 1-12 (1989).
[4] W. Zulehner: Czochralski growth of silicon, J. Cryst. Growth 65 (1-3), 189-213 (1983).

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