Quantum oscillations in Weyl semimetals: A surface theory approach

Jan Borchmann and T. Pereg-Barnea

Phys. Rev. B 96, 125153 – Published 27 September 2017

Abstract

Weyl semimetals are characterized by an even number of bulk Weyl points and surface Fermi arcs connecting their surface projections. It has been shown that when a magnetic field is applied perpendicular to the surface, an unusual kind of quantum oscillations appear. Semiclassically, these oscillations are associated with electron motion in momentum space on the surface Fermi arcs as well as tunneling between the top and bottom surfaces through the bulk chiral Landau level. In this work, we develop an effective surface theory for Weyl semimetals by integrating out the bulk, leaving only the top and bottom infinite surfaces. We use our model to analyze the Landau level spectrum and compare the results with a full three-dimensional calculation as well as the semiclassical theory. We find that the surface theory is a good approximation for the three-dimensional model and agrees with the semiclassical analysis whenever the Weyl points are far from each other. We analytically analyze the case of overlapping Weyl points with short Fermi arcs and find an energy gap which appears as a phase shift in the quantum oscillations, in agreement with our numerics. We also study the case of a type-II Weyl semimetal and show that the absence of the bulk chiral Landau level prevents the formation of surface Landau levels.

Received 24 April 2017

DOI:https://doi.org/10.1103/PhysRevB.96.125153

Physics Subject Headings (PhySH)

Landau levels Surface states

Weyl semimetal

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jan Borchmann and T. Pereg-Barnea^*

Department of Physics and the Centre for Physics of Materials, McGill University, Montreal, Quebec, Canada H3A 2T8

^*Present address: Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel.

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Vol. 96, Iss. 12 — 15 September 2017

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Images

Figure 1
The surface spectral function of our lattice model in a slab geometry [Eq. (8)]. We analytically continue $i ω_{n} \to ω + i η$ and set $η = 0.01$ . (a) The surface spectral function presented as a function of energy in units of $t_{s}$ and the momentum $k_{x}$ with $k_{z} = - π / 2$ . The bright diagonal lines are the surface chiral states where the (negative) positive slope corresponds to the (bottom) top surface. (b)–(d) The surface spectral function at energy $ω = 0$ presented as a function of momentum $(k_{z}, k_{x})$ for varying arc lengths with (b) $m = 0.9 t_{s}$ , (c) $m = 0.5 t_{s}$ , and (d) $m = - 0.5 t_{s}$ .
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Figure 2
The density of states as obtained numerically using the effective surface theory with the magnetic field fixed to $B = Φ_{0} / q a^{2}$ with the flux quantum $Φ_{0} = h c / e$ , the lattice constant $a$ , and with $q = 30$ . (a) Varying the arc length $k_{a} = 2 arccos (m / t_{s})$ by changing the model parameter $m$ . (b) Keeping $m = 0$ and changing the slab thickness $N_{y}$ .
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Figure 3
(a) Energy offset of the zeroth surface Landau level for varying arc lengths. (b) The energy for the $n = 0$ , chiral bulk Landau level as a function of $m$ as calculated numerically from the lattice model (data points) and in the WKB approximation in Eqs. (23) and (24) (lines) for varying magnetic flux $B a^{2} = \frac{Φ_{0}}{q}$ .
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Figure 4
Top: the energy dispersion of the lattice Hamiltonian [Eq. (3)] with the addition of the lattice tilt term in Eq. (25). The horizontal axis is $k_{z}$ , the $k_{x}$ direction is into the page, and $k_{y} = 0$ . The vertical axis is the energy between $- 0.5 t_{s}$ and $0.5 t_{s}$ . Bottom: the surface density of states for $q = 30, N_{y} = 30$ , and $m = 0$ for varying parameter $t_{2}$ , measured in units of $t_{s}$ . The regular part of the spectrum is related to the surface quantum oscillations described semiclassically by Eq. (2). The scale at which the regular spectrum ends is roughly the $n = 1$ bulk Landau level.
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Figure 5
(a) Landau level spectrum of the three-dimensional lattice model with parameters $q = 80$ and $m = 0$ for varying slab widths. (b) The energy difference of the first and zeroth Landau level for $q = 40$ as a function of slab width $N_{y}$ . The 3D lattice calculation (symbols) is compared with the semiclassical expression of Eq. (2) (line). (c) The energy difference of the first and zeroth Landau levels for $q = 60$ and $N_{y} = 60$ as predicted by the semiclassical theory (line) and the full 3D model (symbols). (d) The energy offset of the zeroth Landau level of the full 3D model as a function of arc length for the same parameter values as in (c). The 3D calculation (symbols) is compared with the semiclassical offset expected from Eq. (2) with $γ = \frac{1}{2}$ (dotted line) and with the our WKB-calculated offset [Eq. (23)] (line).
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