Multiphysics 4

Numerical modelling uses a Drude-Lorentz model with 3 oscillators for the electric permittivity of gold^{38 (link)} and a constant permittivity of 4.0 for silicon nitride. The optical properties of the metasurface were simulated for normal incidence illumination by a plane wave considering a single unit cell with periodic boundary conditions using finite element method modelling (COMSOL Multiphysics 4.4) in three dimensions.

Particle behavior during EK injection and separation was observed with a ZEISS Axiovert 40 CFL inverted microscope (Carl Zeiss Microscopy, Thornwood, NY, USA). Voltages were applied by employing a high voltage supply (Model HVS6000D, LabSmith, Livermore, CA, USA). The voltage sequencer was controlled with the Sequence software provided by the manufacturer. COMSOL Multiphysics^® 4.4 was also used to simulate the electric field distribution across the channel and particle velocities. A description of the mathematical model used in COMSOL and Table S1, which lists the numerical values used with the COMSOL model, are included in the Supplementary Materials.

MIM measurements were implemented using an Attocube-based scanner stack in a Janis He4 cryostat with 9 T superconducting magnet15 (link). Topography was taken with a quartz tuning fork sensor to which the MIM probe is attached; MIM images were taken separately with the tip ∼30 nm above mesa surface in constant-height mode. The efficient capacitive coupling at microwave frequency enables sub-surface sensing without inducing interband transitions. The tip-sample interaction is in the near-field limit, so the spatial resolution is determined by the tip diameter (∼150 nm) rather than the microwave wavelength. Imaginary and real parts of the complex tip-sample admittance were recorded during scanning to produce the MIM-Im and -Re channel images. Only σ_xx contributes to the screening of the microwave electric field; σ_xy is not probed here. A small bias is applied to the tip to compensate the work function difference. The relationship between QW conductivity and tip-sample admittance was obtained using COMSOL Multiphysics 4.4 (details see Supplementary Note 2).

The simulations of the dispersion relations, wave transmission, effective material parameters and eigenstates were performed by the commercial finite element software ABAQUS 6.11–1. The simulations of the wave imaging and propagation through the hyperlens based on the optimized HEMMs were conducted by COMSOL Multiphysics 4.4. The optimization procedures were implemented on a Linux cluster with Intel Xeon X5650 Core @ 2.66 GHz.

The eigenmodes of the disordered photonic lattices and the optical beam propagation are numerically investigated by solving the 3D Maxwell equations using the finite element method, as implemented in COMSOL Multiphysics 4.4. The mode solver of COMSOL is used to find the eigenmodes, with the simulation domain being surrounded by scattering boundary conditions (SBCs). The frequency domain solver of COMSOL is used to simulate the optical beam propagation. A TM-polarized Gaussian beam with an x-component of electric filed, E_x(x) = exp(−x²/(3λ)²), is used as the profile of the optical beam at the input facet of the optical superlattice. Appropriate SBCs were used to emulate open boundaries.

Phage response was observed and recorded as videos with a Leica DMi8 inverted microscope (Wetzlar, Germany). Direct current (DC) electric potentials were applied with a high voltage supply (Model HVS6000D, LabSmith, Livermore, CA, USA). COMSOL Multiphysics® 4.4 was used to predict the magnitude of the trapping value (Tv, Equation (3)). Each experiment started with a clean channel to which a 5–10 μL sample of the corresponding labeled virus was added, followed by the application of DC electric potentials. For the purpose of this study, a “sufficient” trapping voltage was determined as the required voltage to obtain a visually observable band or cluster of trapped viral particles.

We used the wave optics module in COMSOL Multiphysics 4.4. From SEM images, we extracted the particle geometry by masking the particles greater than 5 nm (using Gwyddion⁴⁴ ). The masks were contoured and saved as .dxf files (by Python 2.7 and OpenCV) to import into COMSOL. We used dielectric functions for gold,^{45 (link)} silicon,^{46 (link)} and silicon dioxide.⁴⁷ For 2D “bird-view” calculations the gold particles were embedded in an effective medium (82.5% air and 17.5% silicon). The excitation field is implemented as an incoming wave in the model plane. For cross-section simulations we used an excitation port and continuous periodic boundary conditions. A maximum mesh size of 1 nm and 5 nm were used for the boundaries and domains, respectively. The mean field enhancement was calculated by the area mean value of |E/E₀|⁴ over the whole nanostructure surface where |E₀| is the mean electric field surrounding the particle ensemble.