Quantum spin Hall effect
Encyclopedia
The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional, semiconductors with spin-orbit coupling
. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, but, unlike the latter, it does not require the application of a large magnetic field. The quantum spin Hall state does not break any discrete symmetries (such as time-reversal or parity). 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. It has been recently proposed and subsequently experimentally realized in mercury (II) telluride (HgTe) semiconductors.
Overall the Kane-Mele model has a charge-Hall conductance of exactly zero but a spin-Hall conductance of exactly
(in units of 
). 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.
Strictly speaking, the models with spin-orbit coupling do not have a quantized spin Hall conductance
. Those models are more properly referred as topological insulator
which is an example of topologically ordered
states.
invariant who characterizes a state as exhibiting or not exhibiting a quantum spin Hall effect. Further stability studies of the edge liquid through which conduction takes place in the quantum spin Hall state proved, both analytically and numerically that the state is robust to both interactions and extra spin-orbit coupling terms that mix spin-up and spin-down electrons.
Since graphene has extremely weak spin-orbit coupling, it is very unlikely to support a quantum spin Hall state at temperatures achievable with today's technologies. A very realistic theoretical proposal for the existence of the quantum spin Hall state has been put forward by Bernevig, Hughes and Zhang1: (BHZ) in Cadmium Telluride/Mercury Telluride/Cadmium Telluride (CdTe/HgTe/CdTe) quantum wells in which a thin (5-7 nanometers) sheet of HgTe is sandwiched between two sheets of CdTe. Different quantum wells of varying HgTe thickness can be built. When the sheet of HgTe in between the CdTe is thin, the system behaves like an ordinary insulator and does not conduct when the Fermi level resides in the band-gap. When the sheet of HgTe is varied and made thicker (this requires the fabrication of separate quantum wells), an interesting phenomenon happens. Due to the inverted band structure of HgTe, BHZ predicted that, at some critical HgTe thickness, a Lifshitz transition occurs in which the system closes the bulk band gap to become a semi-metal, and then re-opens it to become a quantum spin Hall insulator.
In the gap closing and re-opening process, two edge states are brought out from the bulk and cross the bulk-gap. As such, when the Fermi level resides in the bulk gap, the conduction is dominated by the edge channels that cross the gap. The two-terminal conductance is
in the quantum spin Hall state and zero in the normal insulating state. As the conduction is dominated by the edge channels, the value of the conductance should be insensitive to how wide the sample is. A magnetic field should destroy the quantum spin Hall state by breaking time-reversal invariance and allowing spin-up spin-down electron scattering processes at the edge. All these predictions have been experimentally verified in an experiment performed in the Molenkamp labs at Universitat Würzburg
in Germany
Spin-orbit interaction
In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, but, unlike the latter, it does not require the application of a large magnetic field. The quantum spin Hall state does not break any discrete symmetries (such as time-reversal or parity). 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. It has been recently proposed and subsequently experimentally realized in mercury (II) telluride (HgTe) semiconductors.
Overall the Kane-Mele model has a charge-Hall conductance of exactly zero but a spin-Hall conductance of exactly
(in units of ). 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.Strictly speaking, the models with spin-orbit coupling do not have a quantized spin Hall conductance
. Those models are more properly referred as topological insulatorTopological insulator
A topological insulator is a material that behaves as an insulator in its interior or bulk while permitting the movement of charges on its surface....
which is an example of topologically ordered
Topological order
In physics, topological order is a new kind of order in a quantum state that is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations...
states.
In HgTe quantum wells
Real experimental systems, however, are far from the idealized picture presented above in which spin-up and spin-down electrons are not coupled. A very important achievement was the realization that the quantum spin Hall state is robust to the introduction of spin-up spin-down scattering. In a separate paper, Kane and Mele introduced a topological invariant who characterizes a state as exhibiting or not exhibiting a quantum spin Hall effect. Further stability studies of the edge liquid through which conduction takes place in the quantum spin Hall state proved, both analytically and numerically that the state is robust to both interactions and extra spin-orbit coupling terms that mix spin-up and spin-down electrons.Since graphene has extremely weak spin-orbit coupling, it is very unlikely to support a quantum spin Hall state at temperatures achievable with today's technologies. A very realistic theoretical proposal for the existence of the quantum spin Hall state has been put forward by Bernevig, Hughes and Zhang1: (BHZ) in Cadmium Telluride/Mercury Telluride/Cadmium Telluride (CdTe/HgTe/CdTe) quantum wells in which a thin (5-7 nanometers) sheet of HgTe is sandwiched between two sheets of CdTe. Different quantum wells of varying HgTe thickness can be built. When the sheet of HgTe in between the CdTe is thin, the system behaves like an ordinary insulator and does not conduct when the Fermi level resides in the band-gap. When the sheet of HgTe is varied and made thicker (this requires the fabrication of separate quantum wells), an interesting phenomenon happens. Due to the inverted band structure of HgTe, BHZ predicted that, at some critical HgTe thickness, a Lifshitz transition occurs in which the system closes the bulk band gap to become a semi-metal, and then re-opens it to become a quantum spin Hall insulator.
In the gap closing and re-opening process, two edge states are brought out from the bulk and cross the bulk-gap. As such, when the Fermi level resides in the bulk gap, the conduction is dominated by the edge channels that cross the gap. The two-terminal conductance is
in the quantum spin Hall state and zero in the normal insulating state. As the conduction is dominated by the edge channels, the value of the conductance should be insensitive to how wide the sample is. A magnetic field should destroy the quantum spin Hall state by breaking time-reversal invariance and allowing spin-up spin-down electron scattering processes at the edge. All these predictions have been experimentally verified in an experiment performed in the Molenkamp labs at Universitat WürzburgUniversity of Würzburg
The University of Würzburg is a university in Würzburg, Germany, founded in 1402. The university is a member of the distinguished Coimbra Group.-Name:...
in Germany