Imaging Fractal Conductance Fluctuations and Scarred Wave Functions in a Quantum Billiard

Abstract

We present scanning-probe images and magnetic-field plots which reveal fractal conductance fluctuations in a quantum billiard. The quantum billiard is drawn and tuned using erasable electrostatic lithography, where the scanning probe draws patterns of surface charge in the same environment used for measurements. A periodicity in magnetic field, which is observed in both the images and plots, suggests the presence of classical orbits. Subsequent high-pass filtered highresolution images resemble the predicted probability density of scarred wave functions, which describe the classical orbits. , are observed in disordered systems due to multiple-path interference as electrons scatter from random impurities [1]. A quantum billiard is a large quantum dot where electron trajectories are ballistic, meaning scattering occurs predominantly at the billiard boundary. If the electron phase coherence length is longer than the billiard dimensions, then conductance fluctuations can also be observed in quantum billiards where electrons scatter off the billiard boundary instead of impurities [2][3][4][5]. A soft-walled quantum billiard is a classically mixed system, with regions of regular and chaotic behavior, characterized by the presence of fractal magnetoconductance fluctuations [4,6,7]. The system is chaotic in the sense that a small change, in the magnetic field for example, strongly modifies conductance on an arbitrarily fine scale. Quantum billiards often exhibit Aharonov-Bohm like [1] periodic conductance fluctuations, which are understood to be the signature of stable closed-loop orbits with well defined areas whose quantum states are preferentially excited due to collimation from the leads [2]. The amplitude of the associated wave functions, which are known as scarred wave functions, are concentrated along the underlying classical trajectories and are found through simulation to also exist periodically in magnetic field [8][9][10]. In this letter we provide a further link between experiment and simulation by presenting high resolution scanning probe images of fractal conductance fluctuations which reveal structure remarkably similar to that seen in theoretical images of scarred wave functions [8]. Figure 1 illustrates the billiard construction. A 2D electron system (2DES) with electron mobility 6 10 5 × cm 2 V -1 s -1 and density 11 10 1 . 3 × cm -2 forms at a GaAs/AlGaAs heterojunction 97 nm beneath the surface. The billiard is defined from the 2DES using erasable electrostatic lithography (EEL) where a conductive scanning probe draws spots of negative charge on the GaAs surface to locally deplete 2DES electrons [11]. Uniquely, EEL uses the same low-temperature high-vacuum environment as used for measurement, so device geometry can be modified during the experiment. A row of EEL spots, separated by 100 nm, creates a linear barrier in the 2DES which defines the quantum billiard walls. The lithographic dimension of the billiard is 2 by 3.5 m, but EEL line width and lateral depletion decrease the 2DES billiard dimension ...

Local density of states in mesoscopic samples from scanning gate microscopy

“…The gate acts as a local electrostatic (repulsive or attractive) potential on the electronic system and allows to obtain two-dimensional (2D) conductance (or resistance) images of the scanned area as a function of the tip position. At the present time, SGM or an alternative technique called scanning capacitance microscopy (SCM) have been adopted to investigate the physics of quantum points contacts, 2,3,4,5,6 quantum dots, 7,8 carbon nanotubes, 9 open billiards 10 and edge states in the integer quantum Hall regime. 11,12,13,14 SGM on InAs nanowires has evidenced the presence of multiple quantum dots inside the structure corresponding to circular Coulomb blockade peaks in the conductance plots.…”

Imaging Electron Wave Functions Inside Open Quantum Rings

“…For example, in the vicinity of a QPC [2], coherent electron flow is imaged due to multiple reflections and interferences of electrons bouncing between the QPC and the tip-induced depleted region. In comparison, the situation seems more complex when the tip scans directly over an open mesoscopic billiard [6]: the tip perturbation extends over the whole system of interest, so that all semi-classical trajectories are modified. The mechanisms that link conductance maps to the properties of unperturbed electrons still need to be clarified.…”

“…SGM consists in mapping the conductance of the system as the polarized tip, acting as a flying nano-gate, scans at a constant distance above the 2DEG. SGM gave many valuable insights into the physics of quantum point contacts (QPCs) [2], Coulombblockaded quantum dots [3], magnetic focusing [4], carbon nanotubes [5], open billiards [6] and 2DEGs in the quantum Hall regime [7].…”

Quantum chaotic tunneling in graphene systems with electron-electron interactions

“…In realistic quantum devices, nonhyperbolic dynamics with mixed phase space [33][34][35][36][37][38][39][40][41][42][43] can be expected to arise typically. Here we investigate the tunneling dynamics in classically nonhyperbolic systems in the presence of electronelectron interactions.…”

“…In closed chaotic Hamiltonian systems, the basic phenomena that have been and continue to be studied include energy level-spacing statistics [3][4][5] and quantum scarring . In open Hamiltonian systems, quantum chaotic scattering [33][34][35][36][37][38][39][40][41][42][43] has been investigated extensively. Quite recently, due to the significant development of graphene physics [44][45][46][47][48][49][50], relativistic quantum manifestations of classical chaos have become an interesting area of study [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65].…”