Editing Zone melting (section)

==Process details==
The principle is that the segregation coefficient ''k'' (the ratio at equilibrium of an impurity in the solid phase to that in the liquid phase) is usually less than one. Therefore, at the solid/liquid boundary, the impurity atoms will diffuse to the liquid region. Thus, by passing a crystal [[boule (crystal)|boule]] through a thin section of furnace very slowly, such that only a small region of the boule is molten at any time, the impurities will be segregated at the end of the crystal. Because of the lack of impurities in the leftover regions which solidify, the boule can grow as a perfect [[single crystal]] if a [[seed crystal]] is placed at the base to initiate a chosen direction of crystal growth. When high purity is required, such as in semiconductor industry, the impure end of the boule is cut off, and the refining is repeated.{{Citation needed|date=February 2021}}

In zone refining, solutes are segregated at one end of the ingot in order to purify the remainder, or to concentrate the impurities. In '''zone leveling''', the objective is to distribute solute evenly throughout the purified material, which may be sought in the form of a single [[crystal]]. For example, in the preparation of a transistor or [[diode]] [[semiconductor]], an ingot of germanium is first purified by zone refining. Then a small amount of [[antimony]] is placed in the molten zone, which is passed through the pure germanium. With the proper choice of rate of heating and other variables, the antimony can be spread evenly through the germanium. This technique is also used for the preparation of [[silicon]] for use in [[integrated circuit]]s ("chips").{{Citation needed|date=February 2021}}

===Heaters===
A variety of heaters can be used for zone melting, with their most important characteristic being the ability to form short molten zones that move slowly and uniformly through the ingot. [[Induction coil]]s, ring-wound [[resistance heater]]s, or gas flames are common methods. Another method is to pass an electric current directly through the ingot while it is in a [[magnetic field]], with the resulting [[magnetomotive force]] carefully set to be just equal to the weight in order to hold the liquid suspended. Optical heaters using high-powered [[halogen lamp|halogen]] or [[xenon arc lamp|xenon lamps]] are used extensively in research facilities particularly for the production of insulators, but their use in industry is limited by the relatively low power of the lamps, which limits the size of crystals produced by this method. Zone melting can be done as a [[batch production|batch process]], or it can be done continuously, with fresh impure material being continually added at one end and purer material being removed from the other, with impure zone melt being removed at whatever rate is dictated by the impurity of the feed stock.{{Citation needed|date=February 2021}}

''Indirect-heating floating zone'' methods use an induction-heated tungsten ring to heat the ingot radiatively, and are useful when the ingot is of a high-resistivity semiconductor on which classical induction heating is ineffective.{{Citation needed|date=February 2021}}

===Mathematical expression of impurity concentration===
When the liquid zone moves by a distance <math>dx</math>, the number of impurities in the liquid change.  Impurities are incorporated in the melting liquid and freezing solid.<ref>[[James D. Plummer]], Michael D. Deal, and Peter B. Griffin (2000) ''Silicon VLSI Technology'', Prentice Hall, page 129</ref>{{Clarify|reason=Provide a geometric interpretation|date=May 2022}}

:<math>k_O</math>: segregation coefficient
:<math>L</math>: zone length
:<math>C_O</math>: initial uniform impurity concentration of the solidified rod
:<math>C_L</math>: concentration of impurities in the liquid melt per length
:<math>I</math>: number of impurities in the liquid
:<math>I_O</math>: number of impurities in zone when first formed at bottom
:<math>C_S</math>: concentration of impurities in the solid rod

The number of impurities in the liquid changes in accordance with the expression below during the movement <math>dx</math> of the molten zone
:<math>dI = (C_O - k_O C_L) \, dx\;</math>
:<math>C_L = I/L\;</math>
:<math>\int_0^x dx = \int_{I_O}^I \frac{dI}{C_O - \frac{k_O I}{L}}</math>
:<math>I_O = C_O L\;</math>
:<math>C_S = k_O I / L\;</math>
:<math>C_S (x) = C_O \left ( 1 - (1 - k_O) e^{- \frac{k_O x}{L} } \right )</math>