GaAs-based two-dimensional electron gases (2DEGs) show a wealth of remarkable electronic states [1][2][3] , and serve as the basis for fast transistors, research on electrons in nanostructures 4,5 , and prototypes of quantum-computing schemes 6 . All these uses depend on the extremely low levels of disorder in GaAs 2DEGs, with low-temperature mean free paths ranging from microns to hundreds of microns 7 . Here we study how disorder affects the spatial structure of electron transport by imaging electron flow in three different GaAs/AlGaAs 2DEGs, whose mobilities range over an order of magnitude. As expected, electrons flow along narrow branches that we find remain straight over a distance roughly proportional to the mean free path. We also observe two unanticipated phenomena in high-mobility samples. In our highest-mobility sample we observe an almost complete absence of sharp impurity or defect scattering, indicated by the complete suppression of quantum coherent interference fringes.Also, branched flow through the chaotic potential of a high-mobility sample remains stable to significant changes to the initial conditions of injected electrons.Scanning gate microscopy (SGM) images of electron flow in two-dimensional electron gases (2DEGs) [8][9][10][11][12][13][14][15][16][17][18] provide direct spatial information not available in conventional electrical transport measurements. Our SGM studies show how varying disorder affects electron flow, and enable us to infer information about the disorder potential in our different samples. Achieving a detailed picture of the disorder potential 19, 20 may help to understand why exotic electron organization emerges in some 2DEGs and not others, and to aim for ever weaker disorder or even tailored disorder 21 .By analyzing the differences between images of flow in our samples, we find that the highermobility samples are increasingly dominated by small-angle scattering instead of hard-scattering 22 .2 Finally, we investigate an unusual property of electron flow through such a small-angle scattering disorder potential: though the disorder potential is classically chaotic, branches of flow are stable to significant changes in initial conditions.On each of three 2DEG samples defined in GaAs/AlGaAs heterostructures (Table 1), we use a home-built scanning gate microscope to image the flow of electrons emanating from a split-gate quantum point contact (QPC) 23, 24 at 4.2 K, as schematically shown in Fig. 1d. Using a recently established technique 9-15 , we measure the conductance across the QPC while scanning a sharp conducting tip ∼20 nm above the surface of the sample. We negatively bias the tip to create a depletion region in the 2DEG below. When the tip is above a region of high electron flow from the QPC, it backscatters electrons through the QPC, reducing the measured conductance. By scanning the tip and recording the drop in conductance ∆G for each tip location, we thus image electron flow. Images of flow gathered in this way have been found to accurately reproduce the under...

We describe the construction and performance of a scanning tunneling microscope (STM) capable of taking maps of the tunneling density of states with sub-atomic spatial resolution at dilution refrigerator temperatures and high (14 T) magnetic fields. The fully ultra-high vacuum system features visual access to a two-sample microscope stage at the end of a bottom-loading dilution refrigerator, which facilitates the transfer of in situ prepared tips and samples. The two-sample stage enables location of the best area of the sample under study and extends the experiment lifetime. The successful thermal anchoring of the microscope, described in detail, is confirmed through a base temperature reading of 20 mK, along with a measured electron temperature of 250 mK. Atomically-resolved images, along with complementary vibration measurements, are presented to confirm the effectiveness of the vibration isolation scheme in this instrument. Finally, we demonstrate that the microscope is capable of the same level of performance as typical machines with more modest refrigeration by measuring spectroscopic maps at base temperature both at zero field and in an applied magnetic field.PACS numbers: 07.79.Cz,07.79.Fc Scanning tunneling microscopy (STM), since its development almost 30 years ago, has become a powerful technique in condensed matter physics, providing not only structural information about surfaces, but also spectroscopic measurements of the electronic density of states at the atomic length scale. However, most instruments operate at temperatures above 1 K, limiting access to exotic electronic phases and quantum effects expected at lower temperatures, which are studied as a matter of routine by other techniques. Generally, very little spectroscopic information about the electronic density of states is known at dilution refrigerator temperatures, usually being limited to what can be learned using either point contact spectroscopy or planar tunnel junctions. Moreover, STM can make such measurements on the atomic length scale, allowing it to probe systems, such as single spins and atomic chains, which are not directly accessible any other way.While the integration of STM with a dilution refrigerator can be conceptually reduced to simply attaching the microscope to the end of a mixing chamber in lieu of some other cryogenic refrigerator, the technical requirements for sub-Angstrom positioning of an STM tip above an atomically clean surface are often at odds with those for cooling a sample to milli-Kelvin temperatures. For example, when attaching the microscope to the refrigerator, the former would favor the use of a soft mechanical joint using springs, which would isolate vibrations, while a) Electronic mail: yazdani@princeton.edu the latter would favor the use of a rigid mechanical joint with a metal rod, which would provide a strong thermal contact. Nevertheless, a number of STM instruments have been developed that cool the sample using a dilution refrigerator 1-8 . However, among these, few feature ultra-high vacuum (UHV...

Our measurements of the low frequency ac conductivity in strongly disordered two-dimensional films near the magnetic-field-tuned superconductor-to-insulator transition show a sudden drop in the phase stiffness of superconducting order with either increased temperature or magnetic field. Surprisingly, for two different material systems, the abrupt drop in the superfluid density in a magnetic field has the same universal value as that expected for a Berezinskii-Kosterlitz-Thouless transition in a zero magnetic field. The characteristic temperature at which phase stiffness is suddenly lost can be tuned to zero at a critical magnetic field, following a power-law behavior with a critical exponent consistent with that obtained in previous dc transport studies on the dissipative side of the transition.

The surface critical behaviour of the semi-infinite one-dimensional quantum Ising model in a transverse field is studied in the presence of an aperiodic surface extended modulation. The perturbed couplings are distributed according to a generalized Fredholm sequence, leading to a marginal perturbation and varying surface exponents. The surface magnetic exponents are calculated exactly whereas the expression of the surface energy density exponent is conjectured from a finitesize scaling study. The system displays surface order at the bulk critical point, above a critical value of the modulation amplitude. It may be considered as a discrete realization of the Hilhorst-van Leeuwen model. † Permanent address: Department of Theoretical Physics, University of Szeged, H-6720 Szeged, Hungary ‡ Unité de Recherche Associée au CNRS No 155 cond-mat/9411030