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The Fermi surface of a conventional two-dimensional electron gas is equivalent to a circle, up to smooth deformations that preserve the orientation of the equi-energy contour. Here we show that a Weyl semimetal confined to a thin film with an in-plane magnetization and broken spatial inversion symmetry can have a topologically distinct Fermi surface that is twisted into a figure-8—opposite orientations are coupled at a crossing which is protected up to an exponentially small gap. The twisted spectral response to a perpendicular magnetic field B is distinct from that of a deformed Fermi circle, because the two lobes of a figure-8 cyclotron orbit give opposite contributions to the Aharonov–Bohm phase. The magnetic edge channels come in two counterpropagating types, a wide channel of width and a narrow channel of width (with the magnetic length and β the momentum separation of the Weyl points). Only one of the two is transmitted into a metallic contact, providing unique magnetotransport signatures.
Export citation and abstract. The Fermi surface of degenerate electrons separates filled states inside from empty states outside, thereby governing the electronic transport properties near equilibrium.
In a two-dimensional electron gas (2DEG) the Fermi surface is a closed equi-energy contour in the momentum plane. It is a circle for free electrons, with deformations from the lattice potential such as the trigonal warping of graphene or the hexagonal warping on the surface of a topological insulator.
These are all smooth deformations which do not change the orientation of the Fermi surface: the turning number is 1, meaning that the tangent vector makes one full rotation as we pass along the equi-energy contour. The turning number. Figure 1. Three oriented contours (black curves) with turning number ν = 0, 1, 2. Aa2 unlimited.
The red segments show the uncrossing deformation that removes a self-intersection without changing the total turning number. Download figure: The turning number is preserved by any smooth deformation of the contour. This includes so-called 'uncrossing' deformations : as illustrated in figure, uncrossing breaks up a self-intersecting contour Γ into a collection of nearly touching oriented contours Γ i, with turning numbers ν i. The total turning number is invariant against uncrossing deformations, which is another result due to Gauss. All familiar 2D electron gases belong to the universality class. Here we show that a thin-film Weyl semimetal with an in-plane magnetization and broken spatial inversion symmetry can have ν = 0: if the Fermi level lies in between the two Weyl points the circular Fermi surface is twisted into a figure-8 with zero total curvature. The self-intersection introduced when the Fermi level passes through a Weyl point, to ensure that remains odd, is a crossing of Fermi arcs on the top and bottom surfaces of the thin film (width W).
These have a penetration depth ξ 0 into the thin film that can be much less than the Fermi wavelength of the bulk states, so that we can be in the 2D regime of a single occupied subband without appreciable overlap of the surface states –. The effect of a nonzero surface state overlap is to open up an exponentially small gap in the figure-8, as in figure (a). In a perpendicular magnetic field B the signed area enclosed by the Fermi surface is quantized in units of, with the magnetic length. A figure-8 Fermi surface of linear dimension k F has a signed area much smaller than, because the upper and lower loops have opposite orientation. We find that this twisted Fermi surface produces edge states of width —much wider than the usual narrow quantum Hall edge states of width l m. The wide and the narrow edge states are counterpropagating: if the wide channel moves parallel to, the narrow channel moves antiparallel.
An applied voltage selectively populates one of the two types of edge states, resulting in a conductance of e 2/ h instead of 2 e 2/ h—even though there are two conducting edges. The outline of the paper is as follows. In the next section we formulate the problem, on the basis of a two-band model Hamiltonian , , and calculate the band structure in a slab geometry. The way in which the Fermi arcs reconnect with the bulk Weyl cones is described exactly by a simple transcendental equation (Weiss equation). The Fermi surface in the thin-film regime is calculated in section, to show the topological transition from turning number 1 to turning number 0 when the Fermi level passes through a Weyl point. In section we calculate the edge states in a perpendicular magnetic field, by semiclassical analytics and comparison with a numerical solution.
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The implications of the two types of counterpropagating edge channels for electrical conduction are investigated in section. We conclude with an overview of possible experimental signatures of the twisted Fermi surface. 2. Weyl semimetal confined to a slab. The Pauli matrices are σ α, with σ 0 the 2 × 2 unit matrix, acting on a hybrid of spin and orbital degrees of freedom.
The momentum varies over the Brillouin zone of a simple cubic lattice (lattice constant, and we also set ). The two Weyl points are at the momenta, and at energies, displaced along the k z-axis by the magnetization and displaced along the energy axis by the strain λ. Time-reversal symmetry and spatial inversion symmetry are broken by β and λ, respectively. We take a slab geometry, unbounded in the y– z plane and confined in the x-direction between x = 0 and x = W. The magnetization along z is therefore in the plane of the slab. We impose the infinite-mass boundary condition on the wave function ψ. Figure 2. Dispersion relation E( k y, k z) for k y = 0.01 as a function of k z, of a thick Weyl semimetal slab (width W = 40), calculated from equations and for β = 1.5, λ = 0.1, t x = t y = t z = t' = 1.
The diagram at the top shows the geometry with the trajectory of an electron in a Fermi arc state spiraling along the surface with velocity in the direction of the magnetization. The two branches of the Fermi arc visible in the dispersion relation correspond to states on the top and bottom surface of the slab (assumed to be of infinite extent in this calculation). For this thick slab the range of Fermi energies in which only a single 2D subband is occupied is very narrow (between the red dotted lines). For thinner slabs a larger energy range is available. Download figure: 2.3. Weyl cones and Fermi arcs. The ± sign distinguishes the Fermi arcs on opposite surfaces (− at x = 0 and + at x = W).
The trajectory of an electron in a Fermi arc state moves chirally along the surface (see top inset in figure ), spiraling in the direction of the magnetization with velocity. The surface Fermi arc reconnects with the bulk Weyl cone near k z = ± β. This 'Fermi level plumbing' is described quantitatively by the Weiss equation , as q switches from imaginary to real at a critical for which γ = 1. The penetration length of the surface state into the bulk is plotted in figure, as a function of k z for k y = 0. Its minimal value near the center of the Brillouin zone is. Figure 4. Fermi surfaces of the thin-film Weyl semimetal with a single occupied subband ( W = 15), calculated from equations and for β = 1.5, t x = t y = t z = t' = 1 at different values of λ and E F.
The turning number ν = 0 in the top row, while ν = 1 in the bottom row. The figure-8 in the top row has a narrowly avoided crossing with a gap δ k z = 3 × 10 −5 (not visible on the scale of the figure).
The color of the contour indicates whether the state is localized on the top surface (red), on the bottom surface (blue), or extended through the bulk (black). Download figure: As discussed in the introduction, the turning number ν is a topological invariant of the equi-energy contour. We see from figure that the Fermi surface is twisted into a figure-8 with ν = 0 when the Fermi level lies between the Weyl points, while for larger Fermi energies the Fermi surface has ν = 1. Because the turning number and the number of self-intersections must have opposite parity, the topological transition when E F passes through a Weyl point must introduce a crossing in the Fermi surface. The crossing of the equi-energy contour for small E F is possible since the intersecting states are spatially separated on the top and bottom surfaces of the slab. For a finite ratio W/ ξ 0 of slab width and penetration length the crossing is narrowly avoided because of the exponentially small overlap of the states at opposite surfaces.
From the Weiss equation we calculate that the δ k z gap in the figure-8 is given. To make contact with some of the older literature –, we note that the figure-8 Fermi surface of a Weyl semimetal is essentially different from the figure-8 equi-energy contour of a conventional metal with a saddle point in the Fermi surface. In that case the figure-8 requires fine tuning of the energy to the saddle point, while here the figure-8 persists over a range of energies between two Weyl points. Moreover, the orientation of the two lobes of the figure-8 is the same in the case of a saddle point, while here it is opposite. 4. Quantum Hall edge channels. With the magnetic length and a B-independent offset. Depending on the clockwise or anti-clockwise orientation of the contour, the enclosed area is negative or positive.
Note that the signed area enclosed by the figure-8 Fermi surface of figure (a) equals zero. The phase shift γ = 0 in a bulk Weyl semimetal, when the equi-energy contour encloses a gapless Weyl point –. For the thin film the numerical data indicates γ = 1/2. If the thin film is confined to the strip, with, the spectrum within the strip remains dispersionless, but at the boundaries y = 0 and y = W y propagating states appear.
In the quantum Hall effect these are chiral edge channels, moving in opposite directions on opposite edges ,. The electrical conductance of the strip, for a current flowing in the z-direction, equals the number of edge channels N moving in the same direction times the conductance quantum e 2/ h. The classical skipping orbits that form the edge channels in a magnetic field can be directly extracted from the zero-field Fermi surface: the cyclotron motion in momentum space follows the equi-energy contour E( k y, k z) = E F with period 2 π m c/ eB, where. Is the cyclotron effective mass. (The figure-8 has.) Because, the cyclotron motion in real space is obtained from the momentum space orbit by rotation over π/2 and rescaling by a factor. Specular reflection at the edge (with conservation of k z) then gives for the figure-8 Fermi surface the skipping orbits of figure.
Note that these orbits are 2D projections of 3D trajectories in the thin film: the intersections that are visible in the projected orbit correspond to overpassing trajectories on the top and bottom surfaces. (See figure 10(b) of for a wave packet simulation of such a trajectory.).
Figure 5. Classical cyclotron orbits corresponding to the figure-8 Fermi surface of figure (a). Each edge supports counterpropagating skipping orbits.
The corresponding quantum Hall edge channel is narrow if it propagates opposite to the magnetization, while it is wide if it propagates in the direction of the magnetization. The area enclosed by the cyclotron orbits is shaded, the direction of the shading distinguishes positive and negative contributions to the Aharonov–Bohm phase. Download figure: The real-space counterpart of the quantization rule is that the Aharonov–Bohm phase picked up in one period of the cyclotron motion equals 2 π( n + γ). For the skipping orbits this Bohr–Sommerfeld quantization rule still applies if the contour is closed by a segment along the edge, with an additional contribution to γ from reflection at the edge ,. For small n the skipping orbit should enclose a flux of the order of the flux quantum h/ e, which divides the edge channels into two types, designated narrow and wide: the narrow edge channel propagates along the edge in the direction opposite to the magnetization. It is tightly bound to the edge over a distance of order l m, so that the enclosed area of order encloses a flux of order h/ e. The wide edge channel propagates in the direction of the magnetization and extends further from the edge over a distance of order.
It still encloses a small flux of order h/ e because contributions to from the two sides of the crossing point have opposite sign. The gap δk z at the crossing point has no effect on the quantization if, which is satisfied for when.
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Because the exponent wins it is sufficient that to ensure that the figure-8 is effectively unbroken: the field-induced tunneling through the gap then occurs with near-unit probability, so to a good approximation the wave packet propagates in an unbroken figure-8. The presence of counterpropagating edge channels at each edge requires a Fermi energy in between the Weyl points, for a twisted Fermi surface.
When the Fermi surface is a simple contour without self-intersections the edge channels are chiral, propagating in opposite directions on opposite edges as in figure. Figure 8. Probability density for the four edge states labeled in the dispersion of figure. The density is translationally invariant in the z-direction, the color plots show a cross section in the x– y plane (separated in two panels for clarity). Each edge has a counterpropagating pair of edge states, one with v z 0 penetrating more deeply into the bulk (width ). Download figure: In figure we show the Landau levels in an infinite system as a function of the flux Φ through a unit cell. The Landau fan is fitted to. Figure 9. Left panel: sequence of Landau level energies E n( B) as a function of magnetic field; levels at two values of the energy are marked by colored dots.
Right panel: Landau level index n for these two energies as a function of inverse magnetic field. This 'Landau fan' is fitted to equation to obtain the offset γ. The data is calculated numerically from the Weyl semimetal tight-binding model in an unbounded thin film (thickness W = 30), for parameters β = 1.05, λ = 0.1, t x = t y = t z = t' = 1.
Download figure: 5. Magnetoconductance. To determine the magnetotransport through the Weyl semimetal strip we connect it at both ends z = 0 and z = L to a metal reservoir. Following a similar approach used for graphene , it is convenient to take the same model Hamiltonian throughout the system, with the addition of a z-dependent chemical potential term − μ( z) σ 0. (Physically, this potential could be controlled by a gate voltage.) We set μ( z) = 0 in the semimetal region 0 L).
This corresponds to n-type doping of the reservoir. (For p-type doping we would take μ( z) − E 0.) We distinguish n-type and p-type edge channels in the Weyl semimetal depending on whether they reconnect at large with the upper Weyl cones (n-type) or with the lower Weyl cones (p-type). Referring to the dispersion of figure, the channels L ± at the y = 0 edge are n-type, while the channels R ± at the y = W y edge are p-type. The distinction is important, because only the n-type edge channels can be transmitted into the n-type reservoirs. As indicated in figure, the p-type channels are confined to the semimetal region, without entering into the reservoirs. Figure 10. Undoped Weyl semimetal (chemical potential ) connected to heavily doped metal reservoirs ( for n-type doping). Edge channels in a perpendicular magnetic field are shown in red, with arrows indicating the direction of propagation.
The L ± edge channels are n-type and can enter into the reservoirs, while the R ± edge channels are p-type and remain confined to the semimetal region (dotted lines). The current I flows along the n-type edge in the semimetal, irrespective of the sign of the applied voltage V. Download figure: Upon application of a bias voltage V between the two n-type reservoirs a current I will flow along the n-type edge, with a conductance. Determined by the backscattering probability T y=0 along the edge at y = 0, so G = e 2/ h without impurity scattering—see figure. This is not the usual edge conduction of the quantum Hall effect: as shown in figure, the current flows along the same edge when we change the sign of the voltage bias (switching source and drain), while in the quantum Hall effect the current switches between the edges when V changes sign. The only way to switch the edge here is to change the sign of the magnetic field, so that the n-type edge is at y = W y rather than at y = 0.
Figure 11. Color-scale plot in the y– z plane of the occupation numbers of current-carrying states at the Fermi level, in response to a voltage bias between source and drain. The data is calculated numerically from the tight-binding Hamiltonian in the geometry of figure (parameters β = 1.05, λ = 0.25, t x = t y = t z = t' = 1, W = 10, l m = 4). The chemical potential is μ = 0 in the Weyl semimetal region (between green lines, from z = 0 to z = 60), while μ = 0.75 in the metal reservoirs ( z 60). The current keeps flowing along the same edge when source and drain are switched, carried either by a narrow edge channel (top panel) or by a wide edge channel (bottom panel). The opposite edge is fully decoupled from the reservoirs.
Download figure. We have discussed the unusual magnetic response of a 2DEG with a twisted Fermi surface. The topological transition from turning number ν = 1 (the usual deformed Fermi circle) to turning number ν = 0 (the figure-8 Fermi surface) happens when the Fermi level passes through the Weyl point of a thin-film Weyl semimetal with an in-plane magnetization and broken spatial inversion symmetry.
We discuss several transport properties that could serve as signatures for the topological transition from ν = 1 to ν = 0. In a magnetic field the figure-8 Fermi surface supports counterpropagating edge channels, see figure. At E F = 0, with an equal number of left-movers and right-movers at each edge, the Hall resistance will vanish.
This is the first magnetotransport signature. If we vary the Fermi level and enter the regime of chiral edge channels, we should see the appearance of a voltage difference between the edges in response to a current flowing along the edges. The second signature is the edge-selectivity: although both edges support counterpropagating states, the current flows entirely along one of the two edges, determined by the direction of. This edge-selective current flow might be detected directly, or indirectly by introducing disorder on one edge only and measuring a difference between the conductance G for positive and negative B.
Note that does not violate Onsager reciprocity, since for that we would need to change the sign of both magnetic field and magnetization. A third signature is in the cyclotron resonance condition for the optical conductivity. As explained by Koshino in the context of a type-II Weyl semimetal (which has a figure-8 cyclotron orbit at a specific energy where electron and hole pockets touch ), the resonance frequency is twice as small for an electric field oriented along the long axis of the figure-8, than it is for an electric field oriented along the short axis. In the geometry of figure, the resonance frequency equals for σ yy and 2 eB/ m c for σ zz.
In our analysis we have not included disorder effects. The counterpropagating edge channels can be coupled by disorder, and this would reduce the conductance below the quantized value of G = e 2/ h seen in figure. There is no symmetry to protect this quantization, like there is for the helical edge channels in the quantum spin Hall effect, but there is a spatial separation of wide and narrow edge channels (see figure ), which may provide some robustness against backscattering by disorder. We have focused here on Fermi surfaces with turning number ν = 0 and ν = 1. It would be of interest to compare with other values of ν. A model Hamiltonian for ν = 2, that could be a starting point for such a study, is given in the.
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