Magnetic Breakdown and Klein Tunneling in a Type-II Weyl Semimetal

T. E. O’Brien, M. Diez, and C. W. J. Beenakker

Phys. Rev. Lett. 116, 236401 – Published 8 June 2016

Abstract

The band structure of a type-II Weyl semimetal has pairs of electron and hole pockets that coexist over a range of energies and touch at a topologically protected conical point. We identify signatures of this Weyl point in the magnetic quantum oscillations of the density of states, observable in thermodynamic properties. Tunneling between the electron and hole pockets in a magnetic field is the momentum space counterpart of Klein tunneling at a $p − n$ junction in real space. This magnetic breakdown happens at a characteristic field strength that vanishes when the Fermi level approaches the Weyl point. The topological distinction between connected and disconnected pairs of type-II Weyl cones can be distinguished by the qualitatively different dependence of the quantum oscillations on the direction of the magnetic field.

Received 21 April 2016

DOI:https://doi.org/10.1103/PhysRevLett.116.236401

Physics Subject Headings (PhySH)

Density of states Topological materials

Weyl semimetal

de Haas-van Alphen effect

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

T. E. O’Brien, M. Diez, and C. W. J. Beenakker

Instituut-Lorentz, Universiteit Leiden, P. O. Box 9506, 2300 RA Leiden, The Netherlands

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Vol. 116, Iss. 23 — 10 June 2016

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Images

Figure 1
(a) Fermi surface of a type-II Weyl semimetal, calculated from the model Hamiltonian (1), showing the electron and hole pockets touching at the Weyl point. Equienergy contours in planes perpendicular to the magnetic field $B$ are indicated. The magnetic quantum oscillations have a periodicity in $1 / B$ determined by the contour that encloses an extremal area. (b) Intersection of the Fermi surface with a plane perpendicular to $B$ that passes near the Weyl point. Electron and hole pockets are bounded by a contour $C_{\pm}$ enclosing an area $A_{\pm}$ . The semiclassical orbit of an electron follows the contour in the direction of the arrow. Tunneling between the pockets happens with a probability $T$ that tends to unity when their minimal separation $Δ k \to 0$ . This magnetic breakdown is a manifestation of Klein tunneling in momentum space.
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Figure 2
Energy spectrum at $k_{y} = 0$ of the type-II Weyl semimetal with Hamiltonian (11) (parameters $t = 1$ , $t^{'} = 2$ , $μ = 3$ , $b = 1.2$ , $a_{tilt} = 1.7$ , $ξ = 0.08$ ). The black dotted curves are the exact numerical results, the red dashed lines form the semiclassical Landau fan (15) for tunnel-coupled electron and hole pockets. The individual pockets are responsible for the high-frequency oscillations superimposed on the fan.
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Figure 3
Fourier amplitudes of the magnetic quantum oscillations. The numerical data for the partial density of states $ρ (E, k_{y} = 0)$ (smoothed with a Gaussian of width $Γ = t / 500$ ) are Fourier transformed over the field range $B ≲ 0.005 h / e a_{0}^{2}$ ( $200 < N < 1500$ ). The fundamental frequencies from the electron and hole pockets are indicated by $F_{+}$ and $F_{-}$ , respectively (the first harmonics are also faintly visible). Klein tunneling between the pockets when the Fermi energy approaches the Weyl point ( $E = 0$ ) suppresses these high-frequency oscillations, introducing a new component at the difference frequency $| F_{+} - F_{-} |$ . The colored data points for $F_{\pm}$ are the semiclassical prediction (5) from the extremal areas.
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Figure 4
Energy dependence of the Fourier amplitudes from Fig. 3. The curves are fits to $Ω_{\pm} \sqrt{1 - T}$ and $Ω_{Σ} T$ , with the transmission probability $T (E)$ calculated from Eq. (10) and energy-independent fit parameters $Ω_{\pm}$ , $Ω_{Σ}$ . When two frequency lines in Fig. 3 cross we cannot reliably determine the individual amplitudes—which explains some of the large scatter in the data points.
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Figure 5
Magnetic field axis $L_{\pm} = (θ_{\pm}, ϕ_{\pm})$ for which the extremal contour $C_{\pm}$ at $E = 0$ touches the Weyl point. The dashed curves correspond to the Hamiltonian (11) with the parameters of Fig. 1. For the solid curves we have broken the mirror symmetry by adding the term $V_{0} τ_{0} σ_{0} \sin k_{y}$ with $V_{0} = 0.5$ . The intersection of $L_{+}$ and $L_{-}$ (encircled) is the special axis at which Klein tunneling between electron and hole pockets produces magnetic quantum oscillations with the difference frequency $| F_{+} - F_{-} |$ , suppressing both the electron and hole frequencies $F_{\pm}$ . The intersection is protected by the topology of the Möbius strip (indicated by arrows, which show how the edges at $ϕ = 0, π$ should be glued with a twist).
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Figure 6
Left panels: Dependence on the orientation of the magnetic field of the amplitude of the magnetic quantum oscillations (normalized to unit maximal amplitude), for a fixed Fermi energy $E_{F} = 0$ . Pairs of type-II Weyl points at $E = 0$ with disconnected or connected Fermi surfaces are compared. The right panels show a cross section through the electron and hole pockets. For each curve in the left panels the corresponding orbit is indicated. The calculations, detailed in Appendix C in the Supplemental Material [38], are for the Hamiltonian (11) with parameters $t = 1$ , $μ = 3$ , $b = 1.2$ , $a_{tilt} = 1.7$ for all panels and $t^{'} = 2$ , $ξ = 0.08$ (top panels); $t^{'} = 1.7$ , $ξ = 0.24$ (bottom panels).
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