Breakdown of the Law of Reflection at a Disordered Graphene Edge

E. Walter, T. Ö. Rosdahl, A. R. Akhmerov, and F. Hassler

Phys. Rev. Lett. 121, 136803 – Published 25 September 2018

Abstract

The law of reflection states that smooth surfaces reflect waves specularly, thereby acting as a mirror. This law is insensitive to disorder as long as its length scale is smaller than the wavelength. Monolayer graphene exhibits a linear dispersion at low energies and consequently a diverging Fermi wavelength. We present proof that for a disordered graphene boundary, resonant scattering off disordered edge modes results in diffusive electron reflection even when the electron wavelength is much longer than the disorder correlation length. Using numerical quantum transport simulations, we demonstrate that this phenomenon can be observed as a nonlocal conductance dip in a magnetic focusing experiment.

Received 20 April 2018
Revised 24 August 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.136803

Physics Subject Headings (PhySH)

Ballistic transport Dirac fermions Electrical conductivity Quantum transport Roughness

Graphene Nanoribbon

S-matrix method in transport Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

E. Walter^1,2,*, T. Ö. Rosdahl^3,†, A. R. Akhmerov³, and F. Hassler¹

¹JARA Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany
²Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-University Munich, 80333 Munich, Germany
³Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 4056, 2600 GA Delft, Netherlands

^*elias.walter@rwth-aachen.de
^†torosdahl@gmail.com

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Vol. 121, Iss. 13 — 28 September 2018

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Images

Figure 1
Sketch of the setup. Electrons injected at the source ( $S$ ) follow cyclotron trajectories due to the perpendicular magnetic field $B = B \hat{z}$ , forming a hot spot at the boundary where most trajectories scatter. If the trajectories specularly reflect at the boundary and the separation $W_{x}$ between the midpoints of the source and the drain ( $D$ ) matches two cyclotron diameters, most trajectories enter the drain, and a focusing peak manifests in the nonlocal conductance. The focusing is evident in the classical cyclotron trajectory of an electron normally incident from $S$ at the Fermi level (solid curves), and in the computed current distribution that is superimposed on the device (flow lines, colored background). A side gate $V_{G}$ controls the average potential at the disordered boundary (dotted line), and allows us to tune between regimes of specular and diffusive reflection (see main text). In the diffusive regime, electrons scatter into random angles as shown schematically with the dashed lines, resulting in a drop in the focusing peak conductance compared to the regime of specular reflection. The graphene sheet is grounded, such that current due to off-resonance trajectories may drain away to the sides (open boundaries).
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Figure 2
(a) Solid lines: $Var (ϕ)$ at the Dirac points ( $E_{F} = 0$ ) as a function of the boundary length $L$ , for a disorder strength $s_{d} = 0.05 t$ obtained from the tight-binding model. Markers: $σ^{2} (φ)$ at finite $E_{F}$ , averaged over all incoming modes and $1 0^{2}$ disorder configurations, as a function of the Fermi wavelength $λ_{F}$ for the same disorder strength, obtained numerically for a semi-infinite graphene sheet with a boundary of length $L = 300 a$ . The values chosen for $λ_{F} = \sqrt{3} π t a / E_{F}$ correspond to $E_{F}$ ranging from $0.2 t$ to $0.03 t$ . (b) Same as (a), as a function of the disorder strength $s_{d}^{2}$ , for a value of $2 π L \approx 27 a$ [ $λ_{F} \approx 27 a$ , $E_{F} = 0.2 t$ ]. The dotted line indicates the value of $s_{d}$ used in (a). For $V_{d} = E_{F}$ the variances of both the scattering phase at $E_{F} = 0$ and the reflection angle at $E_{F} > 0$ increase linearly with $s_{d}$ , independent of the Fermi wavelength, exhibiting the breakdown of the law of reflection. For $| V_{d} - E_{F} | ≳ s_{d}$ , $Var (ϕ)$ [ $σ^{2} (φ)$ ] decays with increasing $L$ [ $λ_{F}$ ] as $1 / L$ [ $1 / λ_{F}$ ] and increases quadratically with the disorder strength [as given by Eq. (6)]. Reflection is thus specular, but becomes diffusive for $| V_{d} - E_{F} | ≲ s_{d}$ . Setting $V_{d}$ closer to $E_{F}$ moves transition between the regimes of specular and diffusive reflection to smaller $s_{d}$ . This is because of the overlap of $E_{F}$ with the disorder-broadened edge band. (c),(d) Momentum-resolved density of states at the disordered zigzag edge of a semi-infinite graphene sheet with a boundary of length $L = 300 a$ . A band of edge states with bandwidth $\propto s_{d} = 0.05 t$ extends between the Dirac cones, residing mostly at energy $V_{d}$ , with $V_{d} = 0.03 t$ in (c) and $V_{d} = 0.2 t$ in (d) [dashed lines].
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Figure 3
(a) Conductance as a function of Fermi energy and magnetic field showing the first 4 magnetic focusing peaks for the device sketched in Fig. 1 in the absence of edge disorder and with $V_{G} = 0$ . Superimposed are the predicted locations of the focusing peaks (dotted lines), $1 \leq p \leq 4$ from left to right across the diagonal. The color scale is linear and ranges from about $4 e^{2} / h$ (dark) to $28 e^{2} / h$ (bright). (b) Conductance around the $p = 2$ focusing peak at $E_{F} = 0.093 eV$ [dashed line in (a)] versus gate voltage. We include disorder with $V_{d} = 0.062$ and $s_{d} = 0.047 eV$ in the first $N = 6$ rows next to the boundary. Reflection at the boundary is specular and the conductance smooth in $V_{G}$ , except for a dip when the disordered edge band overlaps with the Fermi level, and reflection becomes diffusive. (c) Line cut from (b) at $B = 0.256 T$ with the predicted voltage value for the dip marked. Within the dip, the conductance exhibits fluctuations dependent on the particular disorder configuration, that are washed out by disorder averaging in (d). We assume the scaling factor $s = 9$ in the tight-binding model, such that $W_{x} = 1.6$ , $W_{y} = 1$ , and $W_{L} = 0.2 μ m$ .
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