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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Spin-orbit coupling and broken spin degeneracy in multilayer graphene
AU - McCann, Edward
AU - Koshino, Mikito
N1 - © 2010 The American Physical Society
PY - 2010/6/15
Y1 - 2010/6/15
N2 - Since the lattices of ABA-stacked graphene multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial-inversion symmetry, their electronic bands must be spin degenerate in the presence of time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when symmetry is not broken by external fields, the only spin-orbit coupling present at low energy near the corner of the Brillouin zone is the Kane-Mele term, that opens a bulk energy gap but does not break the spin degeneracy of the energy bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)]. However, spin splitting is allowed in multilayers with an odd number of layers because their lattices do not satisfy spatial-inversion symmetry. We show that, in trilayer graphene, in addition to the Kane-Mele term, there is a second type of intrinsic spin-orbit coupling present at low energy near the corner of the Brillouin zone. It introduces a Zeeman-type spin splitting of the energy bands at each valley with an opposite sign of the effective magnetic field in the two valleys. We estimate the magnitude of the effective field to be ~2 T.
AB - Since the lattices of ABA-stacked graphene multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial-inversion symmetry, their electronic bands must be spin degenerate in the presence of time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when symmetry is not broken by external fields, the only spin-orbit coupling present at low energy near the corner of the Brillouin zone is the Kane-Mele term, that opens a bulk energy gap but does not break the spin degeneracy of the energy bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)]. However, spin splitting is allowed in multilayers with an odd number of layers because their lattices do not satisfy spatial-inversion symmetry. We show that, in trilayer graphene, in addition to the Kane-Mele term, there is a second type of intrinsic spin-orbit coupling present at low energy near the corner of the Brillouin zone. It introduces a Zeeman-type spin splitting of the energy bands at each valley with an opposite sign of the effective magnetic field in the two valleys. We estimate the magnitude of the effective field to be ~2 T.
KW - graphene
U2 - 10.1103/PhysRevB.81.241409
DO - 10.1103/PhysRevB.81.241409
M3 - Journal article
VL - 81
JO - Physical review B
JF - Physical review B
SN - 1550-235X
IS - 24
M1 - 241409
ER -