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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 ...

We report conductance measurements of ballistic one-dimensional (1D) wires defined in GaAs/AlGaAs heterostructures in an in-plane magnetic field, B. When the Zeeman energy is equal to the 1D subband energy spacing, the spin-split subband N upward arrow intersects (N+1) downward arrow, where N is the index of the spin-degenerate 1D subband. At the crossing of N=1 upward arrow and N=2 downward arrow subbands, there is a spontaneous splitting giving rise to an additional conductance structure evolving from the 1.5(2e(2)/h) plateau. With further increase in B, the structure develops into a plateau and lowers to 2e(2)/h. With increasing temperature and magnetic field the structure shows characteristics of the 0.7 structure. Our results suggest that at low densities a spontaneous spin splitting occurs whenever two 1D subbands of opposite spins cross.

We demonstrate that a source-drain bias creates a fully spin-polarized current as the 0.25(2e2∕h) plateau in quantum wires even in zero magnetic field. When a source-drain bias lifts the momentum degeneracy, the dc measurements show that it is possible to achieve a unidirectional ferromagnetic order and this ordered spin array is destroyed once transport in both directions commences. The spin polarization of currents, between full spin polarization and partial spin polarization (or spin degeneracy), is thus simply controlled by source-drain bias and split-gate voltage, something of considerable value for spintronics.

We have measured the nonequilibrium current noise in a ballistic one-dimensional wire which exhibits an additional conductance plateau at 0.7x2e(2)/h. The Fano factor shows a clear reduction on the 0.7 structure, and eventually vanishes upon applying a strong parallel magnetic field. These results provide experimental evidence that the 0.7 structure is associated with two conduction channels that have different transmission probabilities.