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Visualization Eigenstate for Nanodevices with High Symmetry D. J. Mason1,2 1Department 2The of Physics, Harvard University, Cambridge, MA Molecular Foundry, Lawrence Berkeley National Laboratory Motivation & Method User Interface Static Visualizations Graphene devices at Stanford and Columbia [1-3] Recent graphene experiments (above) increasingly use arbitrary geometries of nanoscale-tomacroscopic size. Anticipating future devices requires a thorough understanding of eigenstate contributions to their electron transmission, which can be measured macroscopically, and local density of states (LDOS), which can be measured directly with a scanning tunneling microscope. LDOS and Transmission G = [( E i )1 H] 1 LDOS (left) and eigenenergy decomposition (right) for one Fermi energy The derivation of the LDOS is handled in an external paper, while the formulation for the eigenenergy decomposition is described under the formula in the middle column. In truth, the decomposition is a function of both a generic energy (the x-axis) and the Fermi energy (E0). The user adjusts the energy slider to view a different cut of the function through the generic energy at a particular Fermi energy. The key ingredient to all calculations is the Hamiltonian which completely describes the system. We invert the Hamiltonian to obtain the Green s function matrix, whose diagonal and boundary block (red) are to calculate the LDOS and electronic transmission respectively. The algorithm employed here utilizes features of the Hamiltonian to obtain these results efficiently and automatically, and is currently being submitted for publication. User interface for eigenstate explorer We created a sophisticated user-interface to examine various features of the DOS and transmission. In the instance above, the user is in DOS-mode exploring an eigenstate whose density is plotted in red on the left and whose contribution to the total DOS is plotted in blue on the right. The user can switch between eigenstates using the left-and-right arrows, or by dragging the energy slider. Algorithms determine the best normalization for circle size in the spatial plot. Eigenstates Equation for visualizing DOS eigenenergy decomposition The DOS for a confined system is a discrete set of delta-functions. These are nearly impossible to visualize since the delta functions may overlap. Since the DOS at a particular Fermi energy has different weights, demarked C(E), this weighting further complicates the visualization. We instead convolve the delta functions with a Lorentzian of width and sum the result. The useris able to select the ideal width via the Gamma slider on the right, while the energy slider on the left determines the Fermi energy (above, E0). The LDOS and eigenenergy decomposition for one Fermi energy iare depicted in the upper-right. Eigenspaces Full DOS decomposition eigenenergy (left) and transmission self-similarity matrix (right) Clicking the read out button plots the full three-dimensional information of the eigenenergy decommposition as a single image for comparison. This can be useful for a quick summary of the information but poor for the detailed analysis the user-interface is designed for. The read out button also provides the user the transmission self-similarity matrix which is computed using the above formula. For each point in the above image, the eigenenergies associated with its x- and y-positions are compared for a similar transmission profile. White areas indicate eigenstates which contribute strongly and in tandem, giving an indication of underlying symmetries. In the above example, a few eigenstates at .75eV resonate strongly with a wide spectrum of other eigenenergies. Electron density for eigenstates of a hexagonal device By decomposing the Hamiltonian into its eigenstates, that is, the columns of the Q matrix, we can explore which energies the electrons are scattering off to produce the computed results. The above images plot the density for three eigenstates each of which Is associated with an eigenenergy. These states are important to understand since they illuminate how the geometry of the device enhances or disrupts electron transmission. Future Work 60% Acknowledgements Work at the Molecular Foundry was supported by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, of the U.S. Department of Energy under DE-AC02-05CH11231. Additional support provided by U.S. Department of Energy Computer Science Graduate Research Fellowship under grant number DE-FG02-97ER25308. [1] N. Stander et al. Transport measurements across a tunable potential barrier in graphene. PRL 98 (236803) 2007 [2] K. Bolotin et al. Measurement of scattering rate and minimum conductivity in graphene. PRL 99 (246803) 2007 [3] J. P. Small et al. Landau-level splitting graphene in high magnetic fields. PRL 96 (136806) 2006 10% 25% High-conductance eigenspace expressed at different rotations Devices with rotation or reflection symmetry have pairs of eigenstates, collectively called an eigenspace, that correspond to each eigenenergy. For a two-dimensional device like the ones examined in these visualizations, the user can rotate within an eigenspace in real-time using the lower slider, allowing her to see the full range of possible electron densities contributing to a single energy scattering. 5% Proposed layout for heterogenous small multiples The algorithms used to plot theDOS eigenenergy decompositon (see Variations on ThreeDimensional Data) are scientifically useful but non-intuitive. Future work will plot the eigenstates associated with the DOS and indicate their contribution by scaling their spatial plots accordingly.

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