Magneto-optical conductivity in graphene including electron-phonon coupling

Adam Pound, J. P. Carbotte, and E. J. Nicol

Phys. Rev. B 85, 125422 – Published 19 March 2012

Abstract

We show how coupling to an Einstein phonon $ω_{E}$ affects the absorption peaks seen in the optical conductivity of graphene under a magnetic field $B$ . The energies and widths of the various lines are shifted, and additional peaks arise in the spectrum. Some of these peaks are Holstein sidebands, resulting from the transfer of spectral weight in each Landau level (LL) into phonon-assisted peaks in the spectral function. Other additional absorption peaks result from transitions involving split LLs, which occur when a LL falls sufficiently close to a peak in the self-energy. We establish the selection rules for the additional transitions and characterize the additional absorption peaks. For finite chemical potential, spectral weight is asymmetrically distributed about the Dirac point; we discuss how this causes an asymmetry in the transitions due to left- and right-handed circularly polarized light and therefore oscillatory behavior in the imaginary part of the off-diagonal Hall conductivity. We also find that the semiclassical cyclotron resonance region is renormalized by an effective-mass factor but is not directly affected by the additional transitions. Last, we discuss how the additional transitions can manifest in broadened, rather than split, absorption peaks due to large scattering rates seen in experiment.

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Received 18 October 2011

DOI:https://doi.org/10.1103/PhysRevB.85.125422

Authors & Affiliations

Adam Pound¹, J. P. Carbotte^2,3, and E. J. Nicol⁴

¹School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom
²Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada, L8S 4M1
³The Canadian Institute for Advanced Research, Toronto, Ontario, Canada, M5G 1Z8
⁴Department of Physics, University of Guelph, Guelph, Ontario, Canada, N1G 2W1

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Vol. 85, Iss. 12 — 15 March 2012

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Figure 1
Schematic depiction of energy bands or levels in graphene and the possible optical transitions between them. In all cases, blue indicates a filled state and yellow indicates an empty one. Each transition is excited by an incoming photon (indicated by a red arrow), but we show only one photon in each diagram to prevent overcrowding. (a) Bare bands forming two cones meeting at the Dirac point, with a photon exciting an interband transition from the valence band to an unoccupied state in the conduction band. (b) Bands dressed by coupling to a phonon, with widened cones and kinks at plus or minus the phonon frequency (relative to the Fermi energy). The emission of a phonon (indicated by the green arrow) allows additional transitions which can occur at lower energies and can alter the quasiparticle momentum. (c) Bare bands condensed into discrete Landau levels when under an applied magnetic field, with an intraband transition from the level just below the Fermi energy to the one just above, and interband transitions from levels labeled by negative integers $- n$ ( $n$ positive or zero) to levels $n \pm 1$ . (d) Landau levels dressed by coupling to a phonon, introducing split levels and phonon-assisted peaks, allowing additional transitions between them.Reuse & Permissions
Figure 2
Schematic decomposition of the levels and transitions in the final diagram of Fig. 1. Sets of transitions are labeled by a non-negative integer $n$ , corresponding to $σ^{(n)}$ . For $n = 0$ , the set is made up of the transition $0 \to 1$ . In the bare case, the members of each set for $n > 0$ would be the transitions $- (n + 1) \to n$ and $- n \to n + 1$ . In the dressed case, each of the additional levels or phonon-assisted peaks derives from a bare level, and the transitions involving it obey the same selection rules as those involving that associated level. To the left of each set, we show the absorption peaks arising from the transitions. In the bare case, the two involved transitions have the same energy, leading to a single peak; in the dressed case, all the involved transitions generically differ in energy and intensity, leading to a deformed set of peaks.Reuse & Permissions
Figure 3
The spectral function $A_{0} (ω)$ in the bare and dressed cases. While the peaks in the two cases have the same height, the dressed level is noticeably narrower, leading to lost spectral weight. The inset shows that this lost weight has gone into phonon-assisted peaks starting at the phonon frequency $ω_{E} = 200$ meV (and the same below $- ω_{E}$ ). Here $B = 2.4$ T, $μ = 0$ , $A = 500$ meV, and $Γ = 5$ meV.Reuse & Permissions
Figure 4
The spectral function $A_{- 2} (ω)$ (top panel) and the functions $ω - Σ_{1} (ω) + μ$ (middle) and $Γ - Σ_{2} (ω)$ (bottom) that determine it, each shown for two values of broadening $Γ$ . In both cases, $B = 18.2$ T, $μ_{0} = 100$ meV, $ω_{E} = 200$ meV, and $A = 250$ meV. Also shown in the bottom panel is $M_{- 2}$ (horizontal dashed line). Peaks in the top panel are either phonon-assisted (corresponding to peaks in $Γ - Σ_{2}$ ) or quasiparticle levels (corresponding to intersections of $ω - Σ_{1} + μ$ with $M_{- 2}$ ); the number of intersections that occur is seen to depend on $Γ$ . LLs are labeled with $n^{α}$ , where $α = \pm$ , while phonon-assisted peaks are labeled with $n_{p}^{m}$ , where $m$ is the index on the energy $P_{m}$ of the corresponding peak in $Γ - Σ_{2} (ω)$ . The inset in the top panel shows that the peaks in $A_{- 2}$ corresponding to those in $Γ - Σ_{2}$ are very weak when far from the largest peak in $A_{n}$ .Reuse & Permissions
Figure 5
Two spectral functions $A_{0} (ω)$ and $A_{1} (ω)$ (top panel) and the transitions between them, together with the functions $ω - Σ_{1} (ω) + μ$ and $Γ - Σ_{2} (ω)$ (bottom two panels), which determine their form. The Fermi energy, which must lie between the initial and final state, is indicated by the dashed green vertical line. Examples of allowed transitions are indicated by arrows; an example of a disallowed transition is crossed out with an $X$ . Solid black circles mark intersections of $ω - Σ_{1} + μ$ with $M_{0}$ and $M_{1}$ , where quasiparticle peaks occur in $A_{0}$ and $A_{1}$ , respectively. Here $ω_{E} = 50$ meV, $B = 12.2$ T, $Γ = 4$ meV, and $A = 250$ meV.Reuse & Permissions
Figure 6
(Top) The real part of the diagonal conductivity in the bare (dashed red curve) and dressed (solid black) cases for a small coupling $A = 80$ meV. The other parameters are $B = 18.2$ T, $μ = 0$ , $Γ = 5$ meV, and $ω_{E} = 200$ meV. (Bottom) The level diagrams for the two cases with all significant transitions between them. The thick, dashed green line shows the position of the Fermi energy. The dotted lines indicate phonon-assisted peaks rather than quasiparticle levels.Reuse & Permissions
Figure 7
Transition energies associated with peaks in $Re σ_{x x}^{(0)}$ , $Re σ_{x x}^{(1)}$ , and $Re σ_{x x}^{(2)}$ , as a function of $\sqrt{B}$ , for parameters $μ = 0$ , $Γ = 5$ meV, $ω_{E} = 200$ meV, $A = 250$ meV. Only those transitions that give rise to significant absorption peaks are shown. Solid curves correspond to transitions between levels; dashed curves to transitions between levels and phonon-assisted peaks; and the dot-dashed lines to $(M_{n_{f}} - M_{n_{i}}) / (1 + λ)$ , where $n_{i}$ is the initial level and $n_{f}$ is the final level. The dotted horizontal lines mark $ω_{E}$ and $2 ω_{E}$ .Reuse & Permissions
Figure 8
The $n = 2$ piece of the diagonal conductivity (solid black curve) as a function of frequency at $B = 5 . 5^{2}$ T, with parameters as in Fig. 7. Each peak is labeled with the transitions giving rise to it. The dashed red curve is the bare conductivity shifted down by a factor of $1 / (1 + λ)$ . Blue arrows point to the two absorption peaks corresponding to the solid circles on the $n = 2$ curves in Fig. 7.Reuse & Permissions
Figure 9
The weights of all significant peaks in $A_{0} (ω)$ , $A_{1} (ω)$ , $A_{2} (ω)$ , and $A_{3} (ω)$ , with parameters as in Fig. 7. The top panel shows weights of peaks in $Re σ_{x x}^{(0)}$ ; the bottom panel those involved in $Re σ_{x x}^{(2)}$ . Open symbols correspond to phonon-assisted peaks, while solid ones correspond to quasiparticle levels. The dotted black line in both cases represents $1 / (1 + λ)$ . Other lines connecting symbols are to guide the eye.Reuse & Permissions
Figure 10
Effect of increasing chemical potential on the conductivity. The three values of $μ_{0}$ are chosen such that the Fermi energy lies between the zeroth and first LL, the first and second, and the second and third, respectively. The remaining parameters are $B = 27.4$ T, $Γ = 5$ meV, $ω_{E} = 200$ meV, and $A = 150$ meV.Reuse & Permissions
Figure 11
The level diagrams corresponding to Fig. 10. As $μ_{0}$ is increased, the Fermi energy (indicated by the thick, dashed, green line) moves up through the levels, causing the lowest-energy, intraband transition to decrease in energy as the level spacing at the Fermi energy decreases, and progressively eliminating interband transitions as more levels in the upper band become occupied.Reuse & Permissions
Figure 12
Partial optical sums for $μ_{0} = 97$ meV (dashed blue curves) and $μ_{0} = 230$ meV (solid red curves), with the same parameters as in Fig. 10. For comparison, the $B = 0$ case with $μ_{0} = 97$ meV (dotted black curves) is also shown. Despite the splitting of levels in the dressed case, the optical weight follows the same pattern as in the bare case.Reuse & Permissions
Figure 13
Bare (dashed red curves) and dressed (solid black curves) optical conductivity as a function of frequency for right- and left-handed circularly polarized light, with parameters $B = 18.2$ T, $μ_{0} = 100$ meV, $Γ = 5$ meV, $ω_{E} = 200$ meV, and $A = 250$ meV. The bottom panel shows the imaginary part of the Hall conductivity, equal to one-half the difference of $Re σ_{+}$ and $Re σ_{-}$ . Additional features not seen in the bare case appear in this difference, due to the renormalized LLs being asymmetrically arranged about $E_{0}$ .Reuse & Permissions
Figure 14
Level diagrams corresponding to the conductivities shown in Fig. 13. In the dressed case, level spacings are not symmetric about the zeroth level; most significantly, the $n = - 2$ and $- 3$ LLs are split, while the $n = 2$ and 3 are not. This asymmetry causes peaks in $Re σ_{+}$ and $Re σ_{-}$ to appear at differing energies, as seen in Fig. 13.Reuse & Permissions
Figure 15
Optical conductivity corresponding to CR for three values of chemical potential: $μ_{0} = 202.5$ meV (top two curves), $μ_{0} = 105$ meV (middle two curves), and $μ_{0} = 55$ meV (bottom two curves). The remaining parameters are $B = 1.5$ T, $Γ = 2.5$ meV, $ω_{E} = 200$ meV, and $A = 250$ meV. Dashed curves are bare, and solid are dressed. The inset shows the level diagram and transitions giving rise to the absorption peaks, with the thick, dashed, green curve marking the location of the Fermi level for the three values of chemical potential. At the lower right, we see the transition from semiclassical to quantum behavior as $μ$ decreases and more peaks (due to $- 3 \to 2$ and $- 2 \to 3$ transitions) become visible at reasonably low energies.Reuse & Permissions
Figure 16
The CR frequency $ω_{c r}$ as a function of $B$ for the same parameter values as in Fig. 17. In both the bare and dressed cases, $ω_{c r}$ closely tracks the expected linear approximation, represented by the dotted red curve in the bare case and the dashed black curve in the dressed case.Reuse & Permissions
Figure 17
Bare (thin curves) and dressed (thick curves) conductivity corresponding to CR for three values of magnetic field. The CR frequency increases with increasing $B$ , as the level spacing at the Fermi energy increases. The remaining parameters are $μ_{0} = 310$ meV, $Γ = 1.0$ meV, $ω_{E} = 200$ meV, $A = 250$ meV.Reuse & Permissions
Figure 18
Effect of finite broadening on the CR frequency for $B = 2.4$ T, $μ_{0} = 310$ meV, $ω_{E} = 200$ meV, and $A = 250$ meV. The solid red curve is for $Γ = 1.0$ meV; the dashed blue curve for $Γ = 2.5$ meV. Black arrows point to the peak position in each case, which shifts away from the expected value $B v_{F}^{2} / [μ_{0} (1 + λ)]$ (marked by the vertical dotted line) as $Γ$ increases. The inset shows the levels and transition between them giving rise to the CR, with the widths of the levels causing noticeable portions of the initial (or final) state to fall above (or below) the Fermi energy.Reuse & Permissions
Figure 19
Effects of large broadening on the spectrum. The top panel shows the diagonal conductivity for $Γ = 40$ and 5 meV; the inset shows how the separate contributions to $Re σ_{x x}$ overlap for the $Γ = 40$ meV case. The bottom two panels show a spectral function, $A_{- 3}$ , involving a split peak, along with the function $ω - Σ_{1} + μ$ that determines the nature of the peaks. Here results are shown for both a large impurity scattering and a broadened phonon; either effect, if sufficiently large, can annul the splitting of levels. In all cases, $B = 18.2$ T, $μ_{0} = 100$ meV, $ω_{E} = 200$ meV, $A = 250$ meV. The broadened phonon distribution is a truncated Lorentzian with parameters $A^{'} = 555$ meV, $δ = 15$ meV, and $δ_{0} = 30$ meV.Reuse & Permissions
Figure 20
Effect of varying magnetic field on the conductivity for $Γ = 10$ meV, $μ_{0} = 15$ meV, $ω_{E} = 150$ meV, $A = 150$ meV, and $v_{F} = 1.27 \times 10^{6}$ m/s. The top panel shows the diagonal conductivity as a function of photon frequency for three different values of $B$ ; to allow easy comparison with the bottom panel, the legend indicates the values of $\sqrt{B}$ for which each curve was calculated. The bottom panel shows the half-widths of the first two peaks in the conductivity as a function of $\sqrt{B}$ . Circles indicate the half-width of the first peak, which arises from the transition $T_{0}^{1}$ , while squares indicate that of the second peak, which arises from the two transitions $T_{- 2}^{1}$ and $T_{- 1}^{2}$ .Reuse & Permissions
Figure 21
Splitting of levels such that inverted transitions are possible for $B = 27.4$ T, $μ_{0} = 230$ meV, $Γ = 1.0$ meV, $ω_{E} = 160$ meV, and $A = 300$ meV. The top panel shows $A_{2} (ω)$ and $A_{3} (ω)$ , with the $3^{-}$ peak in $A_{3}$ falling below the Fermi energy and the $2^{0}$ peak (among others) falling above the Fermi energy, making a transition $3^{-} \to 2^{0}$ possible. The bottom panel displays $ω - Σ_{1} + μ$ , showing how this set of peaks arise. The vertical dotted line marks the Fermi energy, and the dotted green line represents the $B = 0$ behavior about which $ω - Σ_{1} + μ$ oscillates. Points C, D, and E mark points relevant to deriving the constraints on when an inverted transition is possible, as discussed in the text.Reuse & Permissions

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