Air carrier Captain reported an altitude deviation at the FAF crossing altitude; resulting in ATC issuing a low altitude alert when the flight crossed the FAF lower than the published altitude. The crew climbed and continued the approach to a landing.

Narrative

TIP TOE VISUAL 28L. 240 KIAS past EDDY; instructed to lookout for traffic Aircraft Y for 28R. Requested slower speed; assigned 210 KIAS. Eventually got the traffic in sight and got cleared for the approach. Assigned 180 KIAS. Started our descent and the plane was not slowing down; it was actually accelerating instead. Decided to use speed brakes and extend the landing gear to slow down quicker. As the plane was getting established on the 310 HDG (heading) after CHERA; instructed the FO to sequence the approach from DUYET (FAF ILS 28L). I got fixated on trying to slow down to the assigned speed of 180 KIAS. The plane had switched from PTH (Path) mode to FPA (Flight Path Angle) mode and given the configuration of speed brakes and gear down we reached 1800' sooner than expected; a few miles out from DUYET. ATC said they got a low altitude alert" and instructed us to climb to 2000' and that we were still cleared for the approach. Proceeded to climb to 2000' and intercept the course for 28L. Continued with a normal approach to land.Cause: Task Saturation. Speed management.ATC did not assign a slower airspeed until requested. Instead they were instructing us to find traffic for parallel runway. After we requested slower speed we got assigned 210 KIAS; and then when we had traffic in sight we were cleared for the approach and then assigned a further slower speed (180 KIAS). Decided to use speed brakes and landing gear to slow down to assigned speed quickly; but if we'd been at a slower speed to begin with it would not have been necessary to extend the landing gear at that point. We successfully slowed down to the assigned speed but got to our altitude sooner than expected. Fixation on the airspeed caused a failure to recognize that we were below the vertical profile; and a shallower FPA would be more appropriate to reintercept the path.Suggestions: Request slower speed sooner; we recognized that we were close and they had not assigned a slower speed so we requested it. We could also just slow down on our own and advise ATC that we are slowing down and not let them put us a position where it may be more difficult to slow down and configured."