OFFPRINT Quantum chaotic scattering in graphene systems

We investigate the transport fluctuations in both non-relativistic quantum dots and graphene quantum dots with both hyperbolic and nonhyperbolic chaotic scattering dynamics in the classical limit. We find that nonhyperbolic dots generate sharper resonances than those in the hyperbolic case. Strikingly, for the graphene dots, the resonances tend to be much sharper. This means that transmission or conductance fluctuations are characteristically greatly enhanced in relativistic as compared to non-relativistic quantum systems. Copyright c © EPLA, 2011 In the last three decades, quantum chaos, an interdisciplinary field focusing on the quantum manifestations of classical chaos, has received a great deal of attention [1]. In fact, the quantization of chaotic Hamiltonian systems and the ensuing quantum signatures of classical chaos are fundamental procedure and process, respectively, in physics, having direct applications in condensed matter physics, atomic physics, nuclear physics, optics, and acoustics. However, most existing works on quantum chaos are concerned with non-relativistic quantummechanical systems described by the Schrödinger equation. Since the quasi-particles of graphene are chiral, massless Dirac fermions [2,3], the fundamental issue of relativistic quantum manifestations of chaos in graphene systems has attracted a great deal of recent attention. Topics that have been studied include level-spacing statistics, transition from regular to chaotic dynamics, relativistic quantum scars, and weak localization, etc. [4,5]. In this letter, we study the fundamental problem of relativistic quantum scattering using graphene chaotic billiards and compare the results with those from non-relativistic quantum-dot systems. In open Hamiltonian systems, there are two kinds of dynamically relevant chaotic scattering processes: hyperbolic and nonhyperbolic. Both are highly relevant experimentally. In hyperbolic scattering, all the periodic orbits are unstable and the particle decay law is exponential. As a result, the magnitude squared of the autocorrelation function of the quantum S-matrix elements is Lorentzian, where the classical escape rate determines its half-width [6]. The Lorentzian form has been observed experimentally [7]. For nonhyperbolic chaotic scattering, there are non-attracting chaotic sets coexisting with Kolmogorov-Arnold-Moser (KAM) tori in the phase space [8], leading to an algebraic particle decay law. In this case, the fine-scale semiclassical quantum fluctuations of the S-matrix elements with energy difference are enhanced as compared to the hyperbolic case [8]. We note that for a classically integrable billiard system, Bardarson et al. [9] solved the Dirac equation and observed sharp resonances in the conductance-fluctuation pattern. To uncover the relativistic quantum manifestations of chaotic scattering, in this letter we investigate the electronic transport properties in open graphene quantum dots (GQDs) with both hyperbolic and nonhyperbolic scattering dynamics in the classical limit. We compare the GQD analysis with the one we carry out for nonrelativistic quantum dot (NRQD) systems. A striking finding is that GQDs generally have sharper conductance fluctuations than NRQDs [10]. Moreover, GQDs tend to stabilize unstable periodic orbits, which support the hyperbolic scattering. As a result, even in the hyperbolic GQDs, pronounced quantum pointer states [14,15]