Since most impurities have a segregation coefficient smaller than 1 they enrich in the melt. However, the concentration of impurities in the melt directly influences the amount of impurities incorporated into the Si crystal in the Czochralski process. Thus, the resulting ingot will have a varying impurity concentration along its length. Here, we will derive a formula to obtain an estimation for the impurity concentration of the ingot depending on the volume fraction of the crystal at the moment the corresponding ingot section crystallized. As a first simplification assume that the density of the molten Si and the crystalline Si are identical. Therefore, the total volume (and thus the volume V0 of the melt at the start of the process) is simply the sum of the volume of the melt Vl and the volume of the crystal Vs at any given time throughout the process. Please also note that the impurity concentration is the number of impurities per unit volume. The parameters used are the initial Volume V0 and impurity number I0 of the melt. The variables are the momentary volume of the crystal Vs and the impurity number of the melt Il at a given process state. The process state is defined by the fraction of solidified material Vs/V0. a) To get the differential equation for the problem start with the following consideration: the crystal volume starts at 0m3 (no crystal) with obviously no impurities. The momentary concentration of impurities Cs added to the solid is then equal to the amount of impurities that are added per volume change. As the amount of impurities added to the crystal is removed from the melt this can be described in differentials as Cs = − dIl dVs . To obtain the differential equation, please use this equation together with the definition of the segregation coefficient, the definition of the impurity concentration of the liquid, and the total volume to obtain the differential equation with the two differentials od the variables dI_L and dv_s separated on the two sides of the equation. (hint: make sure you have no ’hidden variables’ left)
b) Now solve the differential equation by integration and simplify. The result is a formula that gives you the momentary impurity number of the liquid based on Vs/V0. (hint: one side is an integration from the initial impurity number of the liquid to the momentary number, the other side from the initial crystal volume to the momentary volume).
c) Using the same differential equation for impurity concentration of the crystal as above please derive a formula that gives the (averaged) impurity concentration Cs(Vs/V0) of the crystal as a function of Vs/V0.
d) Integrate the impurity concentration of the crystal over the Vs/V0 from 0 to 1 (so over the whole process) to verify that the formula is correct and you end up with the initial impurity concentration (no impurities can get lost).
e) Use the resulting formula to plot/draw a graph of the momentary impurity concentration of the crystal as a function of Vs/V0. Use steps at least 10 data points. Assume and initial impurity concentration of C_o = 10^12 (1/m3)and boron as the only impurity.