Visualizing the chiral anomaly in Dirac and Weyl semimetals with photoemission spectroscopy

Jan Behrends, Adolfo G. Grushin, Teemu Ojanen, and Jens H. Bardarson

Phys. Rev. B 93, 075114 – Published 8 February 2016

Abstract

Quantum anomalies are the breaking of a classical symmetry by quantum fluctuations. They dictate how physical systems of diverse nature, ranging from fundamental particles to crystalline materials, respond topologically to external perturbations, insensitive to local details. The anomaly paradigm was triggered by the discovery of the chiral anomaly that contributes to the decay of pions into photons and influences the motion of superfluid vortices in ${}^{3}{He}$ -A. In the solid state, it also fundamentally affects the properties of topological Weyl and Dirac semimetals, recently realized experimentally. In this work we propose that the most identifying consequence of the chiral anomaly, the charge density imbalance between fermions of different chirality induced by nonorthogonal electric and magnetic fields, can be directly observed in these materials with the existing technology of photoemission spectroscopy. With angle resolution, the chiral anomaly is identified by a characteristic note-shaped pattern of the emission spectra, originating from the imbalanced occupation of the bulk states and a previously unreported momentum dependent energy shift of the surface state Fermi arcs. We further demonstrate that the chiral anomaly likewise leaves an imprint in angle averaged emission spectra, facilitating its experimental detection. Thereby, our work provides essential theoretical input to foster the direct visualization of the chiral anomaly in condensed matter, in contrast to transport properties, such as negative magnetoresistance, which can also be obtained in the absence of a chiral anomaly.

Received 23 September 2015
Revised 20 January 2016

DOI:https://doi.org/10.1103/PhysRevB.93.075114

Physics Subject Headings (PhySH)

Dirac fermions Electronic structure Weyl fermions

Angle-resolved photoemission spectroscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jan Behrends¹, Adolfo G. Grushin¹, Teemu Ojanen², and Jens H. Bardarson¹

¹Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany
²Low Temperature Laboratory, Department of Applied Physics, Aalto University, FI-00076 AALTO, Finland

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Vol. 93, Iss. 7 — 15 February 2016

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Images

Figure 1
Visualization of the chiral anomaly in Dirac and Weyl semimetals. (a) Low-energy spectrum of a Weyl semimetal film with two bulk Weyl nodes of different chirality separated in momentum space. The gray plane represents the surface state at the film's top surface, occupied up to the equilibrium chemical potential $μ_{eq}$ . Applying external magnetic and electric fields that satisfy $E \cdot B \neq 0$ results in a steady state with left and right cone chemical potentials $μ_{L} \neq μ_{R}$ , linearly interpolated by a tilted Fermi arc. (b) Two constant energy cuts (A and B) through the band structure, with occupied and empty surface states depicted by solid light blue and white dashed lines, respectively. The occupation at these cuts shows a characteristic blue note-shaped pattern depicted in the lower panel. (c) Dirac semimetals host pairs of Weyl cones, each pair with fixed isospin $(↑$ or $↓)$ and both left and right chiralities, that respond to the chiral anomaly in the opposite way. Two edge states with opposite velocities (light red and light blue planes) appear at each boundary of the Dirac semimetal. Scattering processes within and between cones with scattering times $τ_{c}, τ_{v}$ , and $τ_{i}$ are depicted by arrows. (d) The two pairs of Weyl nodes in (c) together comprise a pair of Dirac nodes. At fixed energy cuts (C and D) between $μ_{L}$ and $μ_{R}$ , both bulk nodes are occupied while the surface states are only partially occupied. The total occupation in these planes describes two facing note-shaped patterns, illustrated in the bottom panel.
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Figure 2
ARPES signatures of the chiral anomaly in Weyl semimetals. Numerically computed ARPES spectra for a doped Weyl semimetal film with $L = 2000$ layers in (a) equilibrium and (b) and (c) the chiral anomaly induced steady state. The parameters are such that the equilibrium chemical potential is $μ_{eq} = 0.02 v$ while in nonequilibrium the left and right cones are filled up to $μ_{L} = 0.025 v$ and $μ_{R} = 0.008 v$ , respectively. The lower panels in (a) and (b) show the momentum resolved spectra at a fixed energy, located as schematically shown with gray lines in the upper panels. In equilibrium (a), bulk cones and the surface state are observed, while the chiral anomaly (b) results in the disappearance of one cone and the emerging of the characteristic note-shaped pattern between $μ_{L}$ and $μ_{R}$ . The panel (c) shows the ARPES spectrum in nonequilibrium at a fixed $k_{y} = 0$ , depicted in the top panel. The upper plot in the lower panel shows the total ARPES spectra, demonstrating the conelike structure from the bulk and the flat surface state; the lower plot displays the intensity difference $Δ I = I_{ca} - I_{eq}$ between the out-of-equilibrium and the equilibrium states. The dashed lines mark the chemical potentials $μ_{L}, μ_{R}$ , and $μ_{eq}$ . These plots were obtained for $v = v_{z}, t = 0.5 v, ε = 6 t$ , and $b_{z} = 0.9 v$ with the temperature set to $T = 0.001 v$ , which corresponds to $T \approx 5$ K for a typical Fermi velocity of $v = 0.45$ eV (see Sec. 2b and the Appendix pp1 for details).
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Figure 3
ARPES signatures of the chiral anomaly in Dirac semimetals. Numerically computed ARPES spectra for a doped Dirac semimetal film of $L = 2000$ layers in (a) equilibrium and (b) and (c) the presence of $E \cdot B \neq 0$ . The spectra in (a) and (b) are the momentum resolved spectra at the fixed energies schematically shown at the top as gray lines. The equilibrium chemical potential is set to $μ_{eq} = 0.020 v$ , while the left and right out-of-equilibrium chemical potentials are chosen to be $μ_{L} = 0.025 v$ and $μ_{R} = 0.008 v$ . While there is a slight bulk Dirac cone intensity reduction from the equilibrium to the nonequilibrium situation, a stark qualitative difference is observed in the surface states that leads to the characteristic double note-shaped pattern. In (c) we plot the intensity difference $Δ I = I_{ca} - I_{eq}$ for a fixed momentum $k_{z} = \pm 0.5$ , which reveals the linear dispersion of the edge state. These plots were obtained for $M_{0} = - 0.2 v, M_{1} = - 0.25 v, M_{2} = - 0.75 v$ , and a temperature of $T = 0.001 v$ . With $v = 0.45 eV$ for ${Na}_{3} Bi$ , the temperature corresponds to $T \approx 5 K$ and the induced chemical potential difference to $δ μ \approx 8 meV$ .
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Figure 4
Visualization of the chiral anomaly in ${Na}_{3} Bi$ . (a) Numerically computed equilibrium ARPES spectra for a doped Dirac semimetal film ${Na}_{3} Bi$ with $μ_{eq} = 0$ . The pair of Fermi arcs are well within experimental energy and momentum resolution, confirmed by the recent experiment in Ref. [13]. (b) Chiral anomaly induced ARPES spectra for the same material and an estimate of the chiral chemical potential difference $δ μ \approx 20 meV$ derived in section pp1. The Fermi arcs show evidence of partial occupation within experimental resolution. The upper panels in (a) and (b) show a schematic representation of the bulk band structure with two Dirac nodes connected at higher energies. (c) The lower panel shows two cuts through momentum space at $k_{y} = - 0.1$ and $k_{z} = - 0.85$ in units of the lattice constants and represented schematically by the horizontal and vertical light gray lines in the upper panel. The parameters of the low energy model used to obtain these figures are extracted from first principles in Ref. [27] and take the values $v = 0.45 eV, M_{0} = - 0.087 eV, M_{1} = - 0.11 eV, M_{2} = - 0.35 eV$ and $C_{0} = - 0.064 eV, C_{1} = 0.094 eV, C_{2} = - 0.28 eV$ . All calculations were performed at a temperature of $T = 1 meV = 11.6 K$ and for a film thickness of $L = 1000$ layers.
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Figure 5
Chiral anomaly in PES spectra for Dirac and Weyl semimetals. Numerically computed PES intensities for Weyl and Dirac semimetals for two different temperatures. In equilibrium the intensity $I_{eq}$ presents a single step around $μ_{eq}$ , that is smeared out by increasing temperature and resembles the occupation of the cones. In nonequilibrium the intensity $I_{ca}$ shows a double-step profile with steps at $μ_{R}$ and $μ_{L}$ . Both $μ_{R / L}$ include the temperature dependence of the chiral anomaly [2, 21]. The difference $Δ I = I_{eq} - I_{ca}$ highlights the effect of the chiral anomaly by a characteristic peak-dip structure for low temperatures, resulting from the shift in occupation of the two cones, that evolves into a single peak as temperature is increased. The parameters used to obtain these plots are $μ_{eq} = 0.020 v, μ_{L} = 0.025 v, μ_{R} = 0.008 v$ at $4 K$ and $μ_{L} = 0.024 v, μ_{L} = 0.005 v$ at $15 K$ . The temperatures are chosen to be $T = 8 \times 10^{- 4} v$ and $T = 2.9 \times 10^{- 3} v$ , corresponding to $4 K$ and $15 K$ for a typical value of $v = 0.45 eV$ . The rest of the parameters are chosen as in Figs. 2 and 3 for the upper and lower panel, respectively.
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Figure 6
Proposed magnetic circuit for studying the chiral anomaly and ARPES in a magnetic field, based on ARPES experiments on ferromagnetic materials [65]. The sample is included in a magnetic circuit to minimize stray field effects on photoelectron trajectories. The yellow arrows show the confined magnetization in the ferromagnetic material (e.g., nickel or iron) shaped into the picture-frame geometry. The Weyl or Dirac semimetal sample, a photon of energy $ℏ ω$ , and a photoelectron with energy and momentum $\{E, k\}$ are depicted schematically as a blue rectangle, a purple curved arrow, and a red straight arrow, respectively.
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