Quantum spin Hall effect

The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional, semiconductors that have a quantized spin-Hall conductance and a vanishing charge-Hall conductance. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, and both states can be realized on lattice that does not require the application of a large magnetic field. The quantum spin Hall state does not break charge conservation symmetry and spin-${\displaystyle S_{z}}$ conservation symmetry (in order to have well defined Hall conductances). The first proposal for the existence of a quantum spin Hall state was developed by Kane and Mele who adapted an earlier model for graphene by F. Duncan M. Haldane which exhibits an integer quantum Hall effect. The Kane and Mele model is two copies of the Haldane model such that the spin up electron exhibits a chiral integer quantum Hall Effect while the spin down electron exhibits an anti-chiral integer quantum Hall effect.

Overall the Kane-Mele model has a charge-Hall conductance of exactly zero but a spin-Hall conductance of exactly ${\displaystyle \sigma _{xy}^{spin}=2}$ (in units of ${\displaystyle {\frac {e}{4\pi }}}$). Independently, a quantum spin Hall model was proposed by Bernevig and Zhang in an intricate strain architecture which engineers, due to spin-orbit coupling, a magnetic field pointing upwards for spin-up electrons and a magnetic field pointing downwards for spin-down electrons. The main ingredient is the existence of spin-orbit coupling, which can be understood as a momentum-dependent magnetic field coupling to the spin of the electron.

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