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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 Hall conductivity of K valley spin-up electrons as a function of the Fermi energy calculated by Kubo formula.(a) For ni = 0.5 and different impurity strengths. (b) For λ = 0.3 and different impurity concentrations.
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f6: The Hall conductivity of K valley spin-up electrons as a function of the Fermi energy calculated by Kubo formula.(a) For ni = 0.5 and different impurity strengths. (b) For λ = 0.3 and different impurity concentrations.

Mentions: In Fig. 6, the Hall conductivity spectrums obtained by Kubo formula are shown for different SOC scatterer strengths and concentrations. From this figure, we can see that for the K valley spin-up electron, the Hall conductivity shows a plateau with a height of e2/2h within the gap, regardless of the scattering strength and concentration. This result indicates that the gap opened by SOC impurities is topologically nontrivial. Considering that the spin-down electron contributes an opposite Hall conductivity due to the time reversal symmetry, the total Hall conductivity vanishes. However, the system is on a QSH state with a quantized spin Hall conductivity . The results shown in Fig. 6 support our previous theoretical prediction that the randomly distributed SOC impurities can drive the graphene into a QSH state22. This quantized spin Hall conductivity has been obtained in ref. 23 where the SOC interaction does not act as a scatterer. Now the interesting thing is that the QSH state emerges even if the SOC is in the scatterer but not in the band. From Fig. 6, we can also see that outside the gap, the Hall conductivity is nonzero, though it decreases rapidly as the Fermi energy goes away from the gap. This indicates that in this region, the system is in a spin Hall regime even though the spin Hall conductivity is not quantized. The nonzero spin Hall conductivities outside the gap are also discussed in ref. 23, in which the intrinsic Hall conductivity agrees with our results. A typical numerical result of σxy from Boltzmann transport theory is shown in Fig. 7. We can see that the Hall conductivity here is completely different from that calculated from Kubo theory. We cannot observe a spin Hall plateau in the gap region and the spin Hall conductivity tends to zero when εF → 0. Although the spin Hall conductivity from Boltzmann theory is not quantized, it has nonzero values especially when the Fermi energy is far away from the Dirac point. This nonzero spin Hall conductivity is the result of skew scattering, which is detailed studied in ref. 21.


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 Hall conductivity of K valley spin-up electrons as a function of the Fermi energy calculated by Kubo formula.(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

f6: The Hall conductivity of K valley spin-up electrons as a function of the Fermi energy calculated by Kubo formula.(a) For ni = 0.5 and different impurity strengths. (b) For λ = 0.3 and different impurity concentrations.
Mentions: In Fig. 6, the Hall conductivity spectrums obtained by Kubo formula are shown for different SOC scatterer strengths and concentrations. From this figure, we can see that for the K valley spin-up electron, the Hall conductivity shows a plateau with a height of e2/2h within the gap, regardless of the scattering strength and concentration. This result indicates that the gap opened by SOC impurities is topologically nontrivial. Considering that the spin-down electron contributes an opposite Hall conductivity due to the time reversal symmetry, the total Hall conductivity vanishes. However, the system is on a QSH state with a quantized spin Hall conductivity . The results shown in Fig. 6 support our previous theoretical prediction that the randomly distributed SOC impurities can drive the graphene into a QSH state22. This quantized spin Hall conductivity has been obtained in ref. 23 where the SOC interaction does not act as a scatterer. Now the interesting thing is that the QSH state emerges even if the SOC is in the scatterer but not in the band. From Fig. 6, we can also see that outside the gap, the Hall conductivity is nonzero, though it decreases rapidly as the Fermi energy goes away from the gap. This indicates that in this region, the system is in a spin Hall regime even though the spin Hall conductivity is not quantized. The nonzero spin Hall conductivities outside the gap are also discussed in ref. 23, in which the intrinsic Hall conductivity agrees with our results. A typical numerical result of σxy from Boltzmann transport theory is shown in Fig. 7. We can see that the Hall conductivity here is completely different from that calculated from Kubo theory. We cannot observe a spin Hall plateau in the gap region and the spin Hall conductivity tends to zero when εF → 0. Although the spin Hall conductivity from Boltzmann theory is not quantized, it has nonzero values especially when the Fermi energy is far away from the Dirac point. This nonzero spin Hall conductivity is the result of skew scattering, which is detailed studied in ref. 21.

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.