1) Angles between Planes in Si6 points total Like the plane distance of parallel planes in the Si crystal, the angle between planes is important for microfabrication, as especially anisotropic etch processes can result in microstructures that have the same angles present in their geometry. Please choose one set of planes from each of the 3 main plane families (100,110,111) and calculate the angles between all possible combinations of these planes from the different families (so the angle between one plane from 100 and one from 110 and so on for all 3 possible combinations).
2) Changing dopant concentration in the Czochralski process 20 points total
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 ofimpurities incorporated into the Si crystal in the Czochralski process. Thus, the resultingingot will have an varying impurity concentration along its length. Here, we will derive aformula to obtain an estimation for the impurity concentration of the ingot dependingon the volume fraction of the crystal at the moment the corresponding ingot sectioncrystallized. As a first simplification assume that the density of the molten Si and the crystalline Siare identical. Therefore, the total volume (and thus the volumeV0of the melt at thestart of the process) is simply the sum of the volume of the meltVland the volume of thecrystalVsat any given time throughout the process. Please also note that the impurityconcentration is the number of impurities per unit volume. The parameters used are the initial VolumeV0and impurity numberI0of the melt. Thevariables are the momentary volume of the crystalVsand impurity number of the meltIlat a given process state. The process state is defined by the fraction of solidified materialVs/V0.1
a) Differential equation5 pointsTo get the differential equation for the problem start with the following consideration: thecrystal volume starts at 0m3(no crystal) with obviously no impurities. The momentaryconcentration of impuritiesCsadded to the solid is then equal to the amount of impuritiesthat are added per volume change. As the amount of impurities added to the crystal isremoved from the melt this can be described in differentials asCs=−dIldVs. To obtain the differential equation, please use this equation together with the definitionof the segregation coefficient, the definition of the impurity concentration of the liquidand the total volume to obtain the differential equation with the two differentials od on the two sides of the equation. (hint: make sure youhave no ’hidden variables’ left)
b) Integration5 pointsNow solve the differential equation by integration and simplify. The result is a formulathat gives you the momentary impurity number of the liquid based onVs/V0. (hint: oneside is an integration from the initial impurity number of the liquid to the momentarynumber, the other side from the initial crystal volume to the momentary volume).
c) Final formula3 pointsUsing the same differential equation for impurity concentration of the crystal as aboveplease derive a formula that gives the (averaged) impurity concentrationCs(Vs/V0) ofthe crystal as a function ofVs/V0.
d) Check your formula3 pointsIntegrate the impurity concentration of the crystal over theVs/V0from 0 to 1 (so overthe whole process) to verify that the formula is correct and you end up with the initialimpurity concentration (no impurities can get lost).
e) Plot/Draw the results3 pointsUse the resulting formula to plot/draw a graph of the momentary impurity concentrationof the crystal as a function ofVs/V0. Use steps at least 10 data points. Assume andinitial impurity concentration ofC0= 10121/m3and boron as the only impurity.2
f ) Stop condition1 pointAt which point of this process would you need to stop it if you wanted to have a maximumimpurity concentrationCmaxs= 10121/m3in any part of the ingot? (Same assumptionsas in the previous task)