Last update 27-9-2000 by fkc
This Ph.D thesis encompasses a global numerical simulation of the needle-eye oat zone process, used to grow silicon single crystals. The numerical models includes coupled electromagnetic and free surface models and a global heat transfer model, with moving boundaries. An axisymmetric uidow model, including centrifugal, buoyancy, thermocapillary and electromagnetic forces, is used to determine flow field, after the phase boundaries have been determined, by the heat transfer model. A finite element model for calculating dopant transport, using the calculated unsteady flow field, has been developed within this project. This model has furthermore been expanded to two equations coupled by a non-zero right hand side, for simulating transport of point defects in the crystal during growth.
Free surface shapes and induced electric surface current are calculated for t wo different 4" congurations and a 0.8" conguration. The heat transfer calculations of the same three congurations, yields the global temperature field, from which temperature gradients are determined. The heat transfer model is furthermore expanded to study convective cooling of the crystal from natural convection in the pressurized surrounding gas, for one of the 4" congurations. The depth of the lower phase boundaries of both 4" congurations have with good agreement to the calculations been measured, to provide experimental verication of the heat transfer model.
Calculations of melt convection within the foating zone is done for the two 4" congurations, for four dierent setups. The flow is unsteady, but laminar and is seen to be repressed by increased rates of rotation. Gas doping is simulated by prescribing a flux of dopant through the free surface, resulting in unsteady dopant concentration fields, with distinct concentration boundary layers at the lower phase boundaries. The dopant concentrations, at the lower phase boundaries, are used to determine radial resistivity profies, which with fair agreement are compared to measurements.
Simulations of defect transport are conducted for both of the 4", as well as the 0.8" conguration, for two different values of the recombination factor. The calculation of the 0.8" crystal is compared to DLTS measurements, revealing good agreement for one of the recombination factors, which however does not fit the Voronkov theory. Both factors are used in the simulation of the two 4" congurations, which both have so high growth parameters, that the vacancy domination is relatively unaffected by the recombination.
Resumé (in Danish)
Denne Ph.D afhandling indeholder en global numerisk simulering af needle-eye foat zone processen, til dyrkning af silicium enkrystaller. De numeriske modeller inkluderer en koblet elektromagnetisk og fri væskeoverflade model, samt en global varmetransmissions model, med bevægelige rande. En aksesymmetrisk strømningsmodel indeholdende, centrifugal, opdrifts, termocapillare og elektromagnetiske krfter, anvendes til at bestemme strømningsfeltet, efter at fasegrænserne er bestemt af varmetransmissions modellen. En finite element model til at beregne transport af doterstof, udfra det beregnede instationære strømningsfelt, er udviklet i dette projekt. Denne model er ydermere udvidet til to ligninger koblet af højresiden, for at simulere transport af punkt defekter i krystallen under dyrkning.
Formen af frie væskeoverflader og inducerede elektriske overflade strømme er beregnet for to forskellige 4" konfigurationer og en 0.8" konfiguration. Varmetransmissions beregninger af de samme tre konfigurationer giver et globalt temperaturfelt, hvorfra temperaturgradienter bestemmes. Varmetransmissions modellen er ydermere udvidet, for at studere konvektiv køling af krystallen forsaget af naturlig konvektion i dyrkningskammerets tryksatte atmosfære, for en af 4" konfigurationerne. Dybden af den nedre fasegrænse for begge 4" konfigurationer er med god overensstemmelse målt, for at give eksperimentel verifikation af varmetransmissions modellen.
Beregninger af smelte strøming i den ydende zone er udført for de to 4" konfigurationer, i fire forskellige opsætninger. Strømningen er instationær, men laminar og ses undertrykt af øget rotation. Gasdotering simuleres ved at foreskrive en flux af doterstof gennem den frie overflade, resulterende i instationære doterstof koncentrationsfordelinger, med tydelige koncentrations grænselag ved ned de nedre fasegrænser. Doterstof koncentrationerne ved de nedre fasegrænser, anvendes til at bestemme radialle resistivitetsprofiler, som med rimelig overenstemmelse er sammenlignet med målinger.
Simuleringer af defekt transport er gennemført for begge 4" samt 0.8" konfigurationen, for to forskellige værdier af rekombinationsfaktoren. Beregningen for 0.8" krystallen er sammenlignet med DLTS målinger, med god overensstemmelse for en af rekombinations faktorene, denne beregning passer dog ikke med Voronkov teorien. Begge faktore anvendes i simuleringen af de to 4" konfigurationer, som begge har så høje dyrkningsparametre, at vacancy domineringen er relativt uberørt af rekombinationen.