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Conductivity tensor of graphene dominated by spin-orbit coupling scatterers: A comparison between the results from Kubo and Boltzmann transport theories.

Liu Z, Jiang L, Zheng Y - Sci Rep (2016)

Bottom Line: By performing numerical calculations, we find that the Kubo quantum transport result of the diagonal conductivity within the self-consistent Born approximation exhibits an insulating gap around the Dirac point.In contrast, the semi-classical Boltzmann theory fails to predict such a topological insulating phase.The Boltzmann diagonal conductivity is nonzero even in the insulating gap, in which the Boltzmann spin Hall conductivity does not exhibit any quantized plateau.

View Article: PubMed Central - PubMed

Affiliation: Key Laboratory of Physics and Technology for Advanced Batteries, College of Physics, Jilin University, Ministry of Education, Changchun 130012, China.

ABSTRACT
The diagonal and Hall conductivities of graphene arising from the spin-orbit coupling impurity scattering are theoretically studied. Based on the continuous model, i.e. the massless Dirac equation, we derive analytical expressions of the conductivity tensor from both the Kubo and Boltzmann transport theories. By performing numerical calculations, we find that the Kubo quantum transport result of the diagonal conductivity within the self-consistent Born approximation exhibits an insulating gap around the Dirac point. And in this gap a well-defined quantized spin Hall plateau occurs. This indicates the realization of the quantum spin Hall state of graphene driven by the spin-orbit coupling impurities. In contrast, the semi-classical Boltzmann theory fails to predict such a topological insulating phase. The Boltzmann diagonal conductivity is nonzero even in the insulating gap, in which the Boltzmann spin Hall conductivity does not exhibit any quantized plateau.

No MeSH data available.


The diagonal conductivity of K valley spin-up electrons as a function of the Fermi energy calculated by both the Kubo formula and the Boltzmann equation.(a) For ni = 0.5 and different impurity strengths. (b) For λ = 0.3 and different impurity concentrations.
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f5: The diagonal conductivity of K valley spin-up electrons as a function of the Fermi energy calculated by both the Kubo formula and the Boltzmann equation.(a) For ni = 0.5 and different impurity strengths. (b) For λ = 0.3 and different impurity concentrations.

Mentions: Now, we turn to discuss the numerical results about the conductivity tensor. The diagonal conductivity spectrums are shown in Fig. 5 for different SOC scatterer strengths and concentrations. From this figure, we can see that the result from Boltzmann theory is almost a nonzero constant even in the gap. It comes to the conclusion that the semi-classical Boltzmann theory fails to describe the electronic transport properties of graphene arising from the SOC scattering not only in the vicinity of the Dirac point, but also around the band edges. Notice that when the band gap is large, the band edges are far away from the Dirac point. As shown in Fig. 5(a,b), σxx from Kubo formula vanishes in the SOC band gap. It increases gradually and tends to the Boltzmann result as the Fermi energy goes away from the gap. This result tells us that the Boltzmann theory is only valid in the short electronic wavelength regions where the quantum interference effect becomes weak. Before ending the discussion about the diagonal conductivity spectrum, we would like to point out that the electronic contributions to the diagonal conductivity from other subband spaces, e.g. the spin-down electrons or K′ valley electrons, are exactly the same as shown in Fig. 5.


Conductivity tensor of graphene dominated by spin-orbit coupling scatterers: A comparison between the results from Kubo and Boltzmann transport theories.

Liu Z, Jiang L, Zheng Y - Sci Rep (2016)

The diagonal conductivity of K valley spin-up electrons as a function of the Fermi energy calculated by both the Kubo formula and the Boltzmann equation.(a) For ni = 0.5 and different impurity strengths. (b) For λ = 0.3 and different impurity concentrations.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4815021&req=5

f5: The diagonal conductivity of K valley spin-up electrons as a function of the Fermi energy calculated by both the Kubo formula and the Boltzmann equation.(a) For ni = 0.5 and different impurity strengths. (b) For λ = 0.3 and different impurity concentrations.
Mentions: Now, we turn to discuss the numerical results about the conductivity tensor. The diagonal conductivity spectrums are shown in Fig. 5 for different SOC scatterer strengths and concentrations. From this figure, we can see that the result from Boltzmann theory is almost a nonzero constant even in the gap. It comes to the conclusion that the semi-classical Boltzmann theory fails to describe the electronic transport properties of graphene arising from the SOC scattering not only in the vicinity of the Dirac point, but also around the band edges. Notice that when the band gap is large, the band edges are far away from the Dirac point. As shown in Fig. 5(a,b), σxx from Kubo formula vanishes in the SOC band gap. It increases gradually and tends to the Boltzmann result as the Fermi energy goes away from the gap. This result tells us that the Boltzmann theory is only valid in the short electronic wavelength regions where the quantum interference effect becomes weak. Before ending the discussion about the diagonal conductivity spectrum, we would like to point out that the electronic contributions to the diagonal conductivity from other subband spaces, e.g. the spin-down electrons or K′ valley electrons, are exactly the same as shown in Fig. 5.

Bottom Line: By performing numerical calculations, we find that the Kubo quantum transport result of the diagonal conductivity within the self-consistent Born approximation exhibits an insulating gap around the Dirac point.In contrast, the semi-classical Boltzmann theory fails to predict such a topological insulating phase.The Boltzmann diagonal conductivity is nonzero even in the insulating gap, in which the Boltzmann spin Hall conductivity does not exhibit any quantized plateau.

View Article: PubMed Central - PubMed

Affiliation: Key Laboratory of Physics and Technology for Advanced Batteries, College of Physics, Jilin University, Ministry of Education, Changchun 130012, China.

ABSTRACT
The diagonal and Hall conductivities of graphene arising from the spin-orbit coupling impurity scattering are theoretically studied. Based on the continuous model, i.e. the massless Dirac equation, we derive analytical expressions of the conductivity tensor from both the Kubo and Boltzmann transport theories. By performing numerical calculations, we find that the Kubo quantum transport result of the diagonal conductivity within the self-consistent Born approximation exhibits an insulating gap around the Dirac point. And in this gap a well-defined quantized spin Hall plateau occurs. This indicates the realization of the quantum spin Hall state of graphene driven by the spin-orbit coupling impurities. In contrast, the semi-classical Boltzmann theory fails to predict such a topological insulating phase. The Boltzmann diagonal conductivity is nonzero even in the insulating gap, in which the Boltzmann spin Hall conductivity does not exhibit any quantized plateau.

No MeSH data available.