Phonon structure in dispersion curves and density of states of massive Dirac fermions

Zhou Li and J. P. Carbotte

Phys. Rev. B 88, 045417 – Published 9 July 2013

Abstract

Dirac fermions exist in many solid state systems including graphene, silicene, and other two-dimensional membranes that belong to group VI dichalcogenides, as well as on the surface of some insulators where such states are protected by topology. Coupling of those fermions to phonons introduces new structures in their dispersion curves and, in the case of massive Dirac fermions, can shift and modify the gap. We show how these changes are present in the angular-resolved photoemission spectroscopy of the dressed charge carrier dispersion curves and scanning tunneling microscopy measurements of their density of states. In particular, we focus on the region around the band gap. In this region, the charge carrier spectral density no longer consists of a dominant quasiparticle peak and a smaller incoherent phonon related background. The quasiparticle picture has broken down, and this leads to important modification in both dispersion curves and density of states.

Received 8 March 2013

DOI:https://doi.org/10.1103/PhysRevB.88.045417

Authors & Affiliations

Zhou Li^1,* and J. P. Carbotte^1,2,†

¹Department of Physics, McMaster University, Hamilton, Ontario, Canada,L8S 4M1
²Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8

^*lizhou@mcmaster.ca
^†carbotte@mcmaster.ca

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Vol. 88, Iss. 4 — 15 July 2013

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Images

Figure 1
The real (solid) and imaginary (dashed) parts of the self-energy of a massive Dirac fermion as a function of energy $ω$ . The top frame of the top panel gives the mass renormalization $Σ^{Z} (ω)$ , while the lower frame is for the quasiparticle renormalization $Σ^{I} (ω)$ . The middle panel is the charge carrier density of states DOS as a function of $ω$ . It compares the bare band case (solid line) with the dressed case (dashed line). The bare band chemical potential was set at $μ = 30 meV$ and the bare band gap extends from $-$ 55 to $- 15$ meV (solid line). Coupling to a phonon at $ω_{E} = 7.5$ meV introduces structure at $- μ - Δ / 2 - ω_{E}$ , $- μ + Δ / 2 - ω_{E}$ , $- ω_{E}$ , and $ω_{E}$ . The gap for the interacting case is shifted with respect to its bare value and its magnitude is effectively reduced to about 25 meV. In the bottom panel, we show a schematic of the band structure of ungapped (left) and gapped (right) Dirac fermions. The blue dot in the middle of the Brillouin zone is the $Γ$ point.Reuse & Permissions
Figure 2
The Dirac fermion spectral density $A (k, ω)$ vs $ω$ in electron volts for various values of momentum as labeled. Each curve is restricted to the region below 600 for ease of viewing. The chemical potential is 25 meV and both gap and quasiparticle self-energies $Σ^{Z} (ω)$ and $Σ^{I} (ω)$ are included. The vertical dotted lines mark important energies 7.5, $-$ 7.5(blue), $-$ 12.5, and $-$ 52.5 meV (red).Reuse & Permissions
Figure 3
Color plots of the dressed Dirac fermion dispersion curves $ω = E (k)$ (left frame) as a function of momentum k compared with the bare case (right frame). In this last case, a small constant residual scattering rate of $Γ = 0.1 meV$ was included. The chemical potential is set at 25 meV and the bare gap $Δ$ is 40 meV. For ease of comparison between left and right frames, we have used for the bare case a value of chemical potential shifted by the value of $Re Σ^{I} (ω)$ at $ω = 0$ . (b) is same as (a) but the electron-phonon coupling has been halved to show the progression from the bare to the dressed case.Reuse & Permissions
Figure 4
The dressed Dirac fermion density of states as a function of $ω$ for the case of a single-layer MoS $_{2}$ with spin-polarized bands. In the top frame of the top panel, the chemical potential has been chosen to be equal to $-$ 0.845 eV, which falls below the top of the spin-up and above the top of the spin-down valence bands. The chemical potential is shown by the vertical black dashed line. The phonon structures are at $- Δ / 2 - λ - ω_{E}$ , $μ - ω_{E}$ , $μ + ω_{E}$ , and $- Δ / 2 + λ + ω_{E}$ . In the lower frame of the top panel, $μ = - 0.995$ eV and so falls below the top of both spin-up and -down bands as shown. The phonon structures are at $μ - ω_{E}$ , $μ + ω_{E}$ , $- Δ / 2 - λ + ω_{E}$ , and $- Δ / 2 + λ + ω_{E}$ . The dotted red curves are for the spin-up band and the solid black for the sum of up and down. The bottom panel is a schematic of the band structure of MoS $_{2}$ . The spin splitting of the conduction band is small and not visible in the schematic. The electron-phonon mass renormalization was set at $λ_{ep} = 0.1$ .Reuse & Permissions
Figure 5
The dressed density of state for the massive Dirac fermions of single-layer MoS $_{2}$ . The chemical potential is $μ = - 0.995$ eV and falls below the top of both spin-up (left top frame) and -down (left bottom frame) valence bands as illustrated in the inset of Fig. 4. In each frame, the solid curves are the bare band results shown for comparison with red dashed curve for which we have included only the quasiparticle self-energy $Σ^{I} (ω)$ , and the blue dotted curves, which involves both $Σ^{I} (ω)$ and $Σ^{Z} (ω)$ . The bare case has been shifted by the constant $Re Σ^{I} (ω)$ at $ω$ =0. The right frames show the corresponding spectral density $A (k = 0, ω)$ vs $ω$ .Reuse & Permissions
Figure 6
The density of state vs $ω$ for massive Dirac fermions described by the Hamiltonian (1) (with the $- 1$ in the last term left out) and coupling to phonons described by Eq. (2). The parameters used are much smaller than those for MoS $_{2}$ . They are illustrative of silicene with $Δ / 2 = 20$ meV, $ω_{E} = 7.5$ meV, and $λ / 2 = 10$ meV. Both conduction and valence bands are shown. Solid curves are for spin down and dotted for spin up (green is for the conduction band and blue for the valence band). The bare band edges are shown as heavy vertical red lines and an arrow indicates how they are shifted by interactions. The phonon structures are identified in terms of $Δ$ , $λ$ , and $ω_{E}$ . In the lower frame of the top panel, the phonon energy has been shifted to 16.5 meV. The bottom panel is a schematic of the bands in silicene. The spin splitting is the same size in both valence and conduction bands. The electron-phonon mass renormalization is $λ_{ep} = 0.3$ for $ω_{E} = 7.5$ meV and $λ_{ep} = 0.45$ for $ω_{E} = 16.5$ meV.Reuse & Permissions

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