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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 DOS calculated by SCBA.(a) For ni = 0.5 and different impurity strengths. The insight shows the energy gap Δg as a function of the impurity strength λ for ni = 0.5. (b) For λ = 0.3 and different impurity concentrations. The insight shows the energy gap Δg as a function of the impurity concentration ni for λ = 0.3.
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f3: The DOS calculated by SCBA.(a) For ni = 0.5 and different impurity strengths. The insight shows the energy gap Δg as a function of the impurity strength λ for ni = 0.5. (b) For λ = 0.3 and different impurity concentrations. The insight shows the energy gap Δg as a function of the impurity concentration ni for λ = 0.3.

Mentions: With the formulation of the conductivity tensor developed in the preceding section, we are in a position to perform a numerical calculation of the conductivity spectrums, i.e. the diagonal and the Hall conductivities as functions of the Fermi energy. And then a comparison between the results from Kubo and Boltzmann theories can be made, based on which the validity of the semi-classical Boltzmann transport theory to describe the conductivity tensor of graphene arising from SOC impurities can be clarified. However, before doing this, we would like to present numerical results about the DOS spectrums. From the approximate result of DOS to the weak scattering limit eq. (21), we know that the SOC impurity can open a band gap which is proportional to the product of the scatterer strength and concentration. The numerical results of DOS spectrums are shown in Fig. 3 for different strengths and concentrations of the SOC impurities. We can clearly see that a gap occurs around the Dirac point (zero-energy point) for all the cases. And at relatively weak strength and concentration of the SOC impurities, the approximate expression of DOS to the weak scattering limit, i.e. eq. (21), can give satisfactory results, well agreeing with the numerical results within SCBA. In the insights of Fig. 3, we give the dependence of the band gap on the strength and concentration of the SOC impurities. As given by eq. (21), the gap is 2niλ in the weak scattering limit. The numerical results shown in the insights of Fig. 3 indicate that such a simple relation holds true only in the case of relatively weak scattering. However, some previous works reported that the simple linear dependence of the band gap on the product of the SOC scatterer strength and concentration, i.e. Δg = 2niλ, is still a good approximation even when the system is far away from the weak scattering limit132236. The deviation of the band gap from the linear relation as shown in the insights of Fig. 3 is due to that the SCBA employed in this work excludes some high-order scattering processes. To illustrate this, we can go one step further to calculate the DOS under t-matrix approximation instead of SCBA. Within t-matrix approximation, we add up all the terms represented by the Feynman diagrams shown in Fig. 1(c) to obtain the self-energy self-consistently. The results are shown in Fig. 4(a). We can see that when the impurity strength is relatively weak (λ < 0.4), the t-matrix approximation and the SCBA almost give the same result. The band gap as a function of the SOC impurity strength obtained by t-matrix approximation is closer to the linear rule than that from SCBA, as seen in the insight of Fig. 4(a). Another issue we would like to study is the effect of usual scalar scatterers on the topologically nontrivial gap. To derive the SCBA self-energy in this case, we need to evaluate the Feynman diagrams shown in Fig. 1(d). We assume that the strength of the scalar scatterers has a zero value on average. Thus, we characterize the disorder by the root mean square strength . The concentration of the scalar scatterers is denoted by ns. The DOS results are shown in Fig. 4(b). It can be readily seen that the band gap opened by the SOC scatterers survive a weak scalar disorder. However, a strong enough scalar disorder will close the gap, and hence destroy the QSH state. The derivation of the self-energy within t-matrix approximation and the SCBA self-energy in the presence of usual scalar scatterers are given in the Methods section.


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 DOS calculated by SCBA.(a) For ni = 0.5 and different impurity strengths. The insight shows the energy gap Δg as a function of the impurity strength λ for ni = 0.5. (b) For λ = 0.3 and different impurity concentrations. The insight shows the energy gap Δg as a function of the impurity concentration ni for λ = 0.3.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3: The DOS calculated by SCBA.(a) For ni = 0.5 and different impurity strengths. The insight shows the energy gap Δg as a function of the impurity strength λ for ni = 0.5. (b) For λ = 0.3 and different impurity concentrations. The insight shows the energy gap Δg as a function of the impurity concentration ni for λ = 0.3.
Mentions: With the formulation of the conductivity tensor developed in the preceding section, we are in a position to perform a numerical calculation of the conductivity spectrums, i.e. the diagonal and the Hall conductivities as functions of the Fermi energy. And then a comparison between the results from Kubo and Boltzmann theories can be made, based on which the validity of the semi-classical Boltzmann transport theory to describe the conductivity tensor of graphene arising from SOC impurities can be clarified. However, before doing this, we would like to present numerical results about the DOS spectrums. From the approximate result of DOS to the weak scattering limit eq. (21), we know that the SOC impurity can open a band gap which is proportional to the product of the scatterer strength and concentration. The numerical results of DOS spectrums are shown in Fig. 3 for different strengths and concentrations of the SOC impurities. We can clearly see that a gap occurs around the Dirac point (zero-energy point) for all the cases. And at relatively weak strength and concentration of the SOC impurities, the approximate expression of DOS to the weak scattering limit, i.e. eq. (21), can give satisfactory results, well agreeing with the numerical results within SCBA. In the insights of Fig. 3, we give the dependence of the band gap on the strength and concentration of the SOC impurities. As given by eq. (21), the gap is 2niλ in the weak scattering limit. The numerical results shown in the insights of Fig. 3 indicate that such a simple relation holds true only in the case of relatively weak scattering. However, some previous works reported that the simple linear dependence of the band gap on the product of the SOC scatterer strength and concentration, i.e. Δg = 2niλ, is still a good approximation even when the system is far away from the weak scattering limit132236. The deviation of the band gap from the linear relation as shown in the insights of Fig. 3 is due to that the SCBA employed in this work excludes some high-order scattering processes. To illustrate this, we can go one step further to calculate the DOS under t-matrix approximation instead of SCBA. Within t-matrix approximation, we add up all the terms represented by the Feynman diagrams shown in Fig. 1(c) to obtain the self-energy self-consistently. The results are shown in Fig. 4(a). We can see that when the impurity strength is relatively weak (λ < 0.4), the t-matrix approximation and the SCBA almost give the same result. The band gap as a function of the SOC impurity strength obtained by t-matrix approximation is closer to the linear rule than that from SCBA, as seen in the insight of Fig. 4(a). Another issue we would like to study is the effect of usual scalar scatterers on the topologically nontrivial gap. To derive the SCBA self-energy in this case, we need to evaluate the Feynman diagrams shown in Fig. 1(d). We assume that the strength of the scalar scatterers has a zero value on average. Thus, we characterize the disorder by the root mean square strength . The concentration of the scalar scatterers is denoted by ns. The DOS results are shown in Fig. 4(b). It can be readily seen that the band gap opened by the SOC scatterers survive a weak scalar disorder. However, a strong enough scalar disorder will close the gap, and hence destroy the QSH state. The derivation of the self-energy within t-matrix approximation and the SCBA self-energy in the presence of usual scalar scatterers are given in the Methods section.

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.