Air carrier pilot reported loss of directional control during taxi to the gate when the aircraft main gear encountered snow on the ramp. Pilot shut down engine and aircraft was towed to the gate.

Narrative

This report concerns a loss of directional control taxiing in to a gate in ZZZ during the recent deep freeze that hit in the middle of the month. Coming in to gate X; it was about 15 minutes before sunset and the ramp was quite cold and frozen (approx OAT of -4F / -20C). I was taxiing in single engine as the ramp; taxiways; and runways had good braking action when we had departed ZZZ 3 hours earlier for our ZZZ1 turn; and seemingly on our taxi back in. It was so cold and no sun when we left that everything was just packed snow. However; the clouds had dispersed at ZZZ during our turn and the sun indeed created some slight melting and black ice; mostly near the lead-in line and gate area. Small amounts of deicing fluid (they deice engine nacelles at the gate in ZZZ) may have been part of the mix as well. Bottom line; it was definitely more slippery coming in but wasn't noticed until on the lead-in line.Coming off the taxi line; I turned left and headed straight for the gate and Marshaller. There was about 2-3 inches of snow/packed snow on the ramp; but also a long ridge of snow about 6-12 inches high running perpendicular to the lead-in line. It was probably caused by a plow clearing the road near the vicinity of the start of the lead-in line. The nose gear went through it fine; but the mains almost came to a complete stop hitting this small ridge at slow taxi speed. I applied a small amount of thrust on the number 1 engine (number 2 was shut down) and got through it and immediately went back to idle. The lead-in line was clearly visible as it had been swept previously. The aircraft continued a few feet on the line and then abruptly started going to the right. I applied left brakes to stop the aircraft and help get it back towards the line to the left; but this seemed to not do anything and so I stopped the aircraft completely. The Marshaller was trying to get me to come back to the lead-in line as if I had suddenly got confused where it was. I slowly applied the tiller and a small amount of thrust to angle back towards it. It was challenging because I was trying to turn slightly into the good engine ; while trying not to add too much thrust creating an asymmetric skid. It seemed to be working initially; but then the aircraft began veering to the right again; even with the thrust back at idle and using left brakes. At this point; I just stopped the aircraft for good; set the brakes; and flashed the Marshaller 3 times to hook up and talk to me. I told him we needed to be towed-in the rest of the way due to a loss of directional control from the very slippery ramp. He coordinated with ramp while we pulled out the Tow-in checklist in the cockpit; and shut down the number 1 engine (APU was running). I told the Flight Attendants and pax what the issue was and to remain seated with their belts fastened. I took a picture of the amount we had slid off the lead-in line as well as the brake temps on the left mains; which were higher than the right side due to a slight crosswind on landing. A common ZZZ occurrence; I include them only because it did seem they were not providing much braking during the taxi-in on the lead-in line. This was unusual because they were not terrifically into the overheat range (300C and above) and seemed to work fine until we were on the lead-in line. Perhaps the small ridge of snow got on them; melted immediately and created some kind of reduction in braking effectiveness. The right side brakes; though not in the overheat range; would still have been hot and they seemed to still be effective at braking after going through the same snow ridge.The Tow-in was uneventful; with the exception that it was extremely slow. The Ramp used the lift-tug already there; and it was having traction problems as well with its smaller wheels. They almost called over one of the older diesel tugs with the large wheels; but we finally did get to the gate; albeit it at a very slight angle. It didn'taffect the Jetway operation or the ability to deplane the passengers.