Hall quantization and optical conductivity evolution with variable Berry phase in the $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ model

Hall quantization and optical conductivity evolution with variable Berry phase in the $α − T_{3}$ model

E. Illes, J. P. Carbotte, and E. J. Nicol

Phys. Rev. B 92, 245410 – Published 7 December 2015

Abstract

The $α − T_{3}$ model is characterized by a variable Berry phase that changes continuously from $π$ to 0. We take advantage of this property to highlight the effects of this underlying geometrical phase on a number of physical quantities. The Hall quantization of the two limiting cases is dramatically different: a relativistic series is associated with a Berry phase of $π$ , and a nonrelativistic series is associated with the other limit. We study the quantization of the Hall plateaus as they continuously evolve from a relativistic to a nonrelativistic regime. Additionally, we describe two physical quantities that retain knowledge of the Berry phase in the absence of a motion-inducing magnetic field. The variable Berry phase of the $α − T_{3}$ model allows us to explicitly describe the Berry phase dependence of the dynamical longitudinal optical conductivity and of the angular scattering probability.

Received 22 September 2015

DOI:https://doi.org/10.1103/PhysRevB.92.245410

Authors & Affiliations

E. Illes^1,2,*, J. P. Carbotte^3,4, and E. J. Nicol^1,2

¹Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1
²Guelph-Waterloo Physics Institute, University of Guelph, Guelph, Ontario, Canada N1G 2W1
³Department of Physics, McMaster University, Hamilton, Ontario, Canada L8S 4M1
⁴The Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8

^*illese@uoguelph.ca

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Vol. 92, Iss. 24 — 15 December 2015

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Figure 1
(a) The lattice of the $α − T_{3}$ model with hopping $t$ between the atoms in the HCL and hopping $α t$ between the atoms at the $B$ and $C$ sites. (b) Schematic diagram of the energy dispersion at a single $K$ point, with cones for the linearly dispersing valence and conduction band, and a dispersionless flat band at zero energy that cuts through the Dirac point.
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Figure 2
(a) The smooth evolution of the square of the Landau level energies as a function of the Berry phase for the cones as calculated from the square of Eq. (7). The $K$ and $K^{'}$ valleys are shown as red dashed and blue solid lines, respectively, for $n = 1, 2, 3$ . (b) The case of $ϕ^{B} = π / 2$ . Here, the square of the LL energies is schematically represented by dots, and the parabolas are a schematic representation of the square of the Dirac cones. In both panels, arrows denote the Berry-phase-dependent separation between the square of the LLs.
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Figure 3
The density of states of the conical bands in the presence of a magnetic field showing the LLs, as calculated from Eq. (10) with a broadening of $Γ = 0.07 γ_{B}$ . (a)–(c) Density of states for the $K$ valley (blue solid line) and the $K^{'}$ valley (red dashed line). (d) The total density of states (summed over both valleys) for $α = 0, 0.5, 1$ ,shown by light blue dashed, magenta solid, and green dot-dashed lines, respectively. In our units the $B = 0$ background is 2 in (a)–(c) and 4 in (d) and is shown as a black dashed line in (d).
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Figure 4
The dc Hall conductivity curves plotting $2 σ_{H}$ as a function of the chemical potential for various $α$ . Here, we have included a factor of 2 to account for spin degeneracy. (a) $α = 0$ (blue solid line) and $α = 0.3$ (red dot-dashed line), (b) $α = 0.4, 0.5, 0.6$ (green dashed, blue solid, and red dot-dashed lines, respectively), and (c) $α = 1$ (blue solid line) and $α = 0.8$ (red dot-dashed line).
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Figure 5
Optical conductivity curves for representative values of $α$ , including the case of graphene ( $α = 0$ ; blue solid line) and the $T_{3}$ lattice ( $α = 1$ ; yellow dot-dashed line). The Berry phase dependence of the conductivity is highlighted for the intermediate value of $α = 0.5$ (red dashed line). For all three curves the Drude, centered at $ω = 0$ , is broadened by $γ = 0.001 μ$ . The inset is a schematic diagram depicting the set of possible interband transitions.
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Figure 6
Angular scattering probability for zero magnetic field as calculated from Eq. (32). Curves are shown for representative $α$ values of $α = 0$ (red solid curve), $α = 0.5$ (blue dashed curve), and $α = 1$ (green dot-dashed curve).
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Physical Review B

covering condensed matter and materials physics

Hall quantization and optical conductivity evolution with variable Berry phase in the $α − T_{3}$ model

E. Illes, J. P. Carbotte, and E. J. Nicol

Phys. Rev. B 92, 245410 – Published 7 December 2015

Abstract

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Physical Review B

covering condensed matter and materials physics

Hall quantization and optical conductivity evolution with variable Berry phase in the α−T3 model

E. Illes, J. P. Carbotte, and E. J. Nicol

Phys. Rev. B 92, 245410 – Published 7 December 2015

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Hall quantization and optical conductivity evolution with variable Berry phase in the $α − T_{3}$ model