There are 1br ($1995) and 2br ($2995) apartments available in the complex I live in the Barron Park area, see below. The first two months of rent are currently free with a one-year lease. The location is very convenient - an easy, 15-20 minute bike ride to campus, and the apartments are quite spacious for the area!

1br: https://sfbay.craigslist.org/pen/apa/d/palo-alto-months-free-rentcharming/7286955920.html
2br: https://sfbay.craigslist.org/pen/apa/d/palo-alto-2br-ba-beauitfulnewly/7293012799.html

Contact the property manager through Craigslist if you're interested, or directly at pcj914@gmail.com