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Why Do Impurities Segregate During Single-Crystal Silicon Growth?

To control the electrical properties of semiconductors, trace amounts of Group III elements (such as gallium) or Group V elements (such as phosphorus) are intentionally introduced into silicon. Group III dopants act as electron acceptors in silicon, generating mobile holes and forming positively charged centers; these are referred to as acceptor impurities or p-type dopants. Group V dopants, on the other hand, donate electrons when ionized in silicon, generating mobile electrons and forming negatively charged centers; these are known as donor impurities or n-type dopants.

In addition to the intentional introduction of dopant elements, other unintentional impurities are inevitably introduced during the crystal growth process. These impurities may originate from incomplete purification of raw materials, thermal decomposition of the crucible at high temperatures, or contamination from the growth environment. Ultimately, these impurities may enter the crystal in the form of atoms or ions. Even trace amounts of impurities can significantly alter the physical and electrical properties of the crystal. Therefore, it is essential to understand how impurities are distributed in the melt during crystal growth, as well as the key factors that influence impurity distribution. By clarifying these distribution laws, production conditions can be optimized to fabricate single-crystal silicon with a uniform impurity concentration.

Impurity Segregation and Transport in the Silicon Melt

Due to the phenomenon of impurity segregation, impurities in the silicon melt are not uniformly distributed along the length of a growing single-crystal silicon ingot. Instead, their concentration varies with spatial position along the crystal. Impurity transport in the silicon melt is mainly governed by two mechanisms:

1. Diffusive transport driven by concentration gradients, and

2. Convective transport induced by macroscopic melt flow.

A schematic illustration of phosphorus segregation is shown in the referenced figure. In Czochralski crystal growth, both natural and forced convection commonly exist in the crucible. The primary heater is typically located along the sidewall of the crucible, creating a radial temperature gradient in the silicon melt. Due to thermal expansion, density differences arise in the melt, and buoyancy forces generated by these density variations drive natural convection.

To maintain impurity uniformity and stabilize the thermal field, both the growing crystal and the crucible are rotated at specified angular velocities. The rotation produces inertial forces in the melt, and when these inertial forces overcome viscous forces, forced convection is generated. Consequently, the solute concentration distribution in the crystal is strongly affected by both natural and forced convection in the melt.

Thermodynamic Basis of Impurity Segregation

The growth of single-crystal silicon is a relatively slow process and can, to a good approximation, be treated as occurring under near-thermodynamic equilibrium conditions. Under such conditions, the equilibrium between the solid phase and the liquid phase at the solid–liquid interface can be applied.

If the equilibrium solute concentration in the solid at the interface is denoted as Cs0C_{s0}Cs0​, and that in the liquid is CL0C_{L0}CL0​, the equilibrium segregation coefficient is defined as:

k0=Cs0CL0k_0 = frac{C_{s0}}{C_{L0}}k0​=CL0​Cs0​​

This relationship always holds at the solid–liquid interface under equilibrium conditions. The segregation coefficient k0k_0k0​ may be less than or greater than 1. For example, the segregation coefficient of phosphorus is approximately 0.35, whereas that of oxygen is about 1.27.

• When $k0<1$, the solute is preferentially rejected into the melt during solidification. As crystal growth proceeds, the solute concentration in the melt CL0C_{L0}CL0​ continuously increases. Since k0k_0k0​ remains constant, the solute concentration in the crystal Cs0C_{s0}Cs0​ also increases along the growth direction. As a result, such impurities exhibit a low concentration at the head and a high concentration at the tail of the ingot. Phosphorus typically shows this distribution behavior.

• When $k0>1$, the solute is preferentially incorporated into the solid rather than remaining in the melt. As growth proceeds, the solute concentration in the melt decreases, which in turn causes the solute concentration in the crystal to decrease. In this case, the impurity distribution shows a high concentration at the head and a low concentration at the tail of the ingot.

Role of Mass Transport and Convection

The final impurity distribution in the crystal is determined by impurity transport in the silicon melt during solidification. A purely thermodynamic equilibrium model is insufficient to fully explain solute distribution; therefore, a physical model of crystal growth must also be considered.

In actual crystal growth, the interface does not advance infinitely slowly but grows at a finite rate. Under such conditions, solute diffusion occurs in the melt. Moreover, crystal growth takes place in a gravitational field and is always accompanied by natural convection. To further enhance heat and mass transfer, forced stirring is introduced by crystal and crucible rotation. As a result, both diffusion and convection must be taken into account when analyzing impurity segregation.

Melt flow during crystal growth ensures mass transport from the bulk melt to the solid–liquid interface and thus limits the amount of impurity that can be incorporated into the crystal.

Axial Impurity Distribution and the Gulliver–Scheil equation

These combined mechanisms lead to a non-uniform impurity distribution along the axial direction of the crystal. Under the assumptions of:

• a closed system with no evaporation or solid-state diffusion of dopants,

• and sufficiently strong melt mixing to ensure uniform solute concentration in the melt,

the impurity distribution along the solidified crystal is described by the Gulliver–Scheil equation:

CS=C0 keff (1−fS)keff−1C_S = C_0 , k_{text{eff}} , (1 - f_S)^{k_{text{eff}} - 1}CS​=C0​keff​(1−fS​)keff​−1

where:

• CSC_SCS​ is the impurity concentration in single-crystal silicon,

• C0C_0C0​ is the initial impurity concentration in the melt prior to solidification,

• fSf_SfS​ is the fraction of material that has solidified, and

• keffk_{text{eff}}keff​ is the effective segregation coefficient, defined as the ratio of the impurity concentration in the solid CSC_SCS​ to that in the melt CLC_LCL​.

The effective segregation coefficient keffk_{text{eff}}keff​ depends on the equilibrium segregation coefficient k0k_0k0​ (for example, $k0=0.35$ for phosphorus), the impurity diffusion coefficient DDD in the melt, the crystal growth rate vvv, and the thickness of the solute boundary layer δdeltaδ at the solid–liquid interface.