Edge density of bulk states due to relativity

Letter

Edge density of bulk states due to relativity

Matthew D. Horner and Jiannis K. Pachos

Phys. Rev. B 104, L081402 – Published 10 August 2021

Abstract

The boundaries of quantum materials can host a variety of exotic effects such as topologically robust edge states or anyonic quasiparticles. Here, we show that fermionic systems such as graphene that admit a low-energy Dirac description can exhibit counterintuitive relativistic effects at their boundaries. As an example, we consider carbon nanotubes and demonstrate that relativistic bulk spinor states can have nonzero charge density on the boundaries, in contrast to the sinusoidal distribution of nonrelativistic wave functions that are necessarily zero at the boundaries. This unusual property of relativistic spinors is complementary to the linear energy dispersion relation exhibited by Dirac materials and can influence their coupling to leads, transport properties, or their response to external fields.

Received 17 May 2021
Revised 7 July 2021
Accepted 28 July 2021

DOI:https://doi.org/10.1103/PhysRevB.104.L081402

Physics Subject Headings (PhySH)

Density of states Edge states Electronic structure Fermi surface

Graphene Honeycomb lattice Nanotubes

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Matthew D. Horner^* and Jiannis K. Pachos

School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom

^*py13mh@leeds.ac.uk

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Issue

Vol. 104, Iss. 8 — 15 August 2021

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Images

Figure 1
Left: The honeycomb lattice comprising two triangular sublattices $A$ and $B$ . The lattice basis vectors $n_{x} = \frac{1}{2} (1, - \sqrt{3})$ and $n_{y} = (1, 0)$ are depicted with corresponding non-Cartesian coordinates $x$ and $y$ . A nanotube with zigzag boundaries (red lines) is depicted with length $L$ corresponding to $L + 1$ unit cells (dashed ovals) in the $n_{x}$ direction while the $n_{y}$ direction is periodic with circumference $N$ . The boundary condition is given by the top $A$ sites and bottom $B$ sites having zero population. Right: The band structure of a zigzag nanotube of circumference $N = 10$ is displayed, where $n = 3$ (dashed lines) is one of the two minimum gap bands. All bands have a single minimum except $n = N / 2 = 5$ which is completely flat (dotted line).
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Figure 2
Left: The phase shifts $θ \equiv θ_{n, p}$ of Eq. (6b) and the numerically measured edge densities $ρ (0) \equiv ρ_{n, p} (0)$ vs circumference $N$ for the ground state of a system of fixed length $L = 200$ . When $N$ is a multiple of three, i.e., when the system is gapless, the phase shift is $θ = π / 2$ and the edge density is $ρ (0) = 1 / L = 0.005$ , confirming Eq. (10). Right: The phase shifts $θ$ and edge densities $ρ (0)$ vs system length $L$ for the ground state of the gapless systems $N = 30$ and its two neighboring gapped systems $N = 29$ and $N = 31$ . The solid line represents the analytical formulas while the points represent numerics. The edge density for the gapless system $N = 30$ goes as $1 / L$ , while for gapped systems $N = 29$ and $N = 31$ the edge density tends to zero quickly, in agreement with Eq. (9). Inset: The integrated LDOS at the edge $x = 0$ for a nanotube of circumferences $N = 29, 30, 31$ and lengths $L = 25$ (dashed lines) and $L = 100$ (solid lines). The LDOS displays the predicted behavior of maximizing for gapless systems ( $N = 30$ ) and increasing with smaller system size $L$ .
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Figure 3
Left: A comparison of the analytical wave functions $| ψ_{A} |^{2}$ and $| ψ_{B} |^{2}$ of Eq. (6a) and charge densities $ρ$ of Eq. (9) to the numerical simulation (markers) for the gapless system $(N, L) = (30, 200)$ . We present the first three excited states above the Fermi energy $E = 0$ . We observe the wave functions act highly relativistically, with a large boundary support and uniform charge density of $1 / L = 0.005$ . Right: We present the same information for the gapped system $(N, L) = (31, 200)$ . The behavior contrasts highly with the gapless system despite $N$ only being greater by one. The wave functions and densities display a Schrödinger-like profile with a much smaller edge density that tends to zero as $L$ increases as seen in Fig. 2.
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Figure 4
The full single-particle ground state $Ψ_{0, j}$ and the first state above the Fermi energy $Ψ_{j}$ for a system of size $(N, L) = (30, 9)$ . The emergent relativistic physics near the Fermi energy can be seen clearly as a result of aliasing of a high-frequency Schrödinger wave function. Comparing this with the left-hand column of Fig. 3, we see how the sublattice wave functions $ψ_{A}$ and $ψ_{B}$ described by the spinor Eq. (6a) emerge.
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