Unbalanced edge modes and topological phase transition in gated trilayer   graphene
  Gapless edge modes hosted by chirally-stacked trilayer graphene display unique features when a bulk gap is opened by applying an interlayer potential difference. We show that trilayer graphene with half-integer valley Hall conductivity leads to unbalanced edge modes at opposite zigzag boundaries, resulting in a natural valley current polarizer. This unusual characteristic is preserved in the presence of Rashba spin-orbit coupling that turns a gated trilayer graphene into a ${Z}_2$ topological insulator with an odd number of helical edge mode pairs. 
Xiao Li, Zhenhua Qiao, Jeil Jung, Qian Niu
cond-mat.mes-hall cond-mat.mtrl-sci