Topological properties and resonant tunneling in graphene with Haldane model and extrinsic Rashba effect

We study the topological phases and resonant tunneling in graphene with the fixed extrinsic Rashba effect. Our study demonstrates that the phase diagram in the space of the Haldane model and ferromagnetic exchange field exhibits a topological phase transition between the Chern numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>4</mml:mn></mml:mrow> </mml:math> . With the staggered potential, the phase diagram presents the richer topological phases that are characterized by the Chern numbers from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn></mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn></mml:mrow> </mml:math> . In particular, the Chern number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:math> corresponds to six distinct helical edge states in zigzag graphene nanoribbon, containing the valley edge state and mixed edge state. Additionally, the Chern numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> correspond to four valley-polarized edge states. In the junction consisting of different chiral edge states, the transmission manifests many resonant peaks with a transmittance of about 1, originating from the backscattering and different loop currents. Our findings offer insights into the edge states with identical Chern numbers and their resonant tunneling.