Igor Tsukerman Research

Nanotechnology: Computer Simulation of Nanoscale Systems

	Design of plasmon-enhanced optical tips with molecular-scale resolution.
	Simulation of magnetically driven assembly.
	Molecular dynamics and long-range interactions.
	Colloids: the Poisson-Boltzmann and DLVO models.
	Photonics: wave propagation in photonic crystals, plasmon resonances.

Project: NSF-NIRT (Nanoscale Interdisciplinary Research Teams).

"Magnetically Driven Assembly of Heterogeneous Nanosystems".

Assembling systems on the nanoscale is the greatest challenge of nanotechnology today.

I am working jointly with Prof. Gary Friedman's group at the ECE Department of Drexel on MAGnetically Driven Assembly (MAGDA) proposed by Prof. Friedman.

MAGDA involves charged and magnetized particles in a colloidal solution. The particles can be used either to carry various chemicals or components of nanodevices or, alternatively, to mask (block) parts of the substrate.

Experiments and pictures: Prof. Friedman's group at Drexel University.

Behavior of particles in colloidal solutions is quite complex. Electromagnetic interactions involve a delicate balance of screened Coulomb forces, van der Waals forces, as well as magnetic (in MAGDA) and thermodynamical effects.

Our simulation techniques include a new finite-difference calculus of 'Flexible Local Approximation MEthods' (FLAME). These methods incorporate accurate local approximations of the field directly into the difference scheme and can therefore work nicely even on simple and relatively coarse grids.

"Flexible Local Approximation MEthods"(FLAME): – a new Finite Difference calculus with nanoscale applications.

Geometric and physical features of the problem are represented algebraically, by suitable approximating functions, rather than geometrically, on conforming meshes. In particular, spherical or ellipsoidal particles can be modeled on regular coarse grids with high accuracy.

Solving a photonic waveguide problem: FEM vs. FLAME:

Same accuracy, very different meshes: FEM mesh with 154,000 degrees of freedom vs. the simple Cartesian 50´50 grid for FLAME.

Magnetized particle in external field. FEM vs. FLAME:

Same accuracy, very different meshes: FEM mesh with 125,665 degrees of freedom vs. the simple Cartesian 30´30 grid for FLAME. Magnetized particle (μ = 100) in an external field. Same numerical error (~2.5∙10⁻⁸) for both solutions.

Plasmon resonances: electrostatics at optical frequencies! Design of plasmon-enhanced optical tips for near-field microscopy. (Joint work with Prof. A. Sokolov.)

	Metal nanoparticles can exhibit strong resonances, often in the optical range.
	Electrostatics with a negative real part of the dielectric constant.
	Applications: biosensors, bio-labels, nano-optical devices.

(click on images to enlarge)

Parallel implementation of Generalized FEM.

Read more... (PowerPoint file, 2.1 MB)

Applied Finite Element Analysis and Multigrid Methods

	Adaptive refinement and multigrid methods for electromagnetic applications.
	FEM in unbounded domains.
	A priori error estimates.
	Edge elements.
	Generalized FEM by partition of unity.
	'Spurious modes' in electromagnetic resonance problems.
	Field computation in electric machines.
	Coupled field-circuit problems.
	Geophysical logging for oil exploration.

Read more... (PowerPoint file, 2 MB)

Finite Element Theory: a priori error estimates.

There are many well-known shape conditions for 2D and 3D finite elements (for example, the minimum or the maximum angle conditions for first order tiangular elements). I have proposed two new conditions:

1) A general a priori accuracy criterion via the maximum eigenvalue of the FE stiffness matrix.

2) A minimum singular value condition for first order triangular and tetrahedral elements. (These turn out to be equivalent, up to a small factor, to Jamet's condition, but more easily computable.)

Read more... (same PowerPoint file as above)

Formulations of electromagnetic problems

Read more... (same PowerPoint file as above)

	Nano-optics: plasmon-enhanced optical systems with molecular-scale resolution.
	Long-range electrostatic interactions in heterogeneous media and electrolytes. Methods of Molecular Dynamics.
	Plasmon-enhanced sensors for single molecule detection.
	Metamaterials.
	Assembly of micro- and nanoparticles.
	Wave propagation in photonic crystals.

	“Flexible Local Approximation MEthods” (FLAME) – a new Finite Difference calculus with nanoscale applications. Schemes for wave propagation and scattering: Cartesian grids without 'staircase' effects.
	Applied Finite Element Analysis, Multigrid Methods and Generalized FEM.
	Parallel implementation of Generalized FEM.
	Finite Element Theory: a priori error estimates.
	Formulations of electromagnetic problems.