Use the drc package which was created to solve your problem exactly: sigmoid regression; its acronym stands for Dose Response Curve. It is a well-established package backed up by scientific publications.

Use the R drc package which was created to solve your problem exactly: sigmoid regression; its acronym stands for Dose Response Curve. It is a well-established package backed up by scientific publications.

The package includes many forms of sigmoid curves as detailed in the documentation (drc package documentation with examples).