Ic and Vce equations of IGBT
You obviously want a physics based model for IGBTs. Most of us are happy with much less. Web searches show that there exists at least Allen Hefner's IGBT physics model. Here's a link to an academic work which applies it:

https://mospace.umsystem.edu/xmlui/bitstream/handle/10355/10539/research.pdf?sequence=3&amp;isAllowed=y

Check, if you can understand it.