József Cserti

Application of the lattice Green's function for calculating the resistance of an infinite network of resistors

The resistance between two arbitrary grid points of several infinite lattice structures of resistors is calculated by using lattice Green’s functions. The resistance for d dimensional hypercubic, rectangular, triangular, and honeycomb lattices of resistors is discussed in detail. Recurrence formulas for the resistance between arbitrary lattice points of the square lattice are given. For large separation between nodes the asymptotic form of the resistance for a square lattice and the finite limiting value of the resistance for a simple cubic lattice are calculated. The relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian is given. The Green’s function method used in this paper can be applied in a straightforward manner to other types of lattice structures and can be useful didactically for introducing many concepts used in condensed matter physics.

Random walks on directed percolation clusters

An asymptotically exact expression is given for the mean displacement (R(t)) of random walks on directed percolation clusters on lattices in arbitrary dimensions. The critical behaviour of Rinfinity =limt to infinity (R(t)), the mean squared displacement and the relaxation time are discussed near the threshold probability pc=1 in terms of critical exponents.

Unified description of Zitterbewegung for spintronic, graphene, and superconducting systems

We present a unified treatment of Zitterbewegung phenomena for a wide class of systems including spintronic, graphene, and superconducting systems. We derive an explicit expression for the time dependence of the position operator of the quasiparticles which can be decomposed into a mean part and an oscillatory term. The latter corresponds to the Zitterbewegung. To apply our result for different systems, one needs to use only vector algebra instead of the more complicated operator algebra.

Perturbation of infinite networks of resistors

The resistance between arbitrary nodes of an infinite network of resistors is calculated when the network is perturbed by removing one bond from the perfect lattice. A relation is given between the resistance and the lattice Green’s function of the perturbed resistor network. By solving the Dyson equation, the Green’s function and the resistance of the perturbed lattice are expressed in terms of those of the perfect lattice. Numerical results are presented for a square lattice.

Caustics due to a negative refractive index in circular graphene p-n junctions.

We show that the wave functions form caustics in circular graphene p-n junctions which in the framework of geometrical optics can be interpreted with a negative refractive index.

Role of the trigonal warping on the minimal conductivity of bilayer graphene

Using a reformulated Kubo formula we calculate the zero-energy minimal conductivity of bilayer graphene taking into account the small but finite trigonal warping. We find that the conductivity is independent of the strength of the trigonal warping and it is 3 times as large as that without trigonal warping and 6 times larger than that in single layer graphene. Although the trigonal warping of the dispersion relation around the valleys in the Brillouin zone is effective only for low-energy excitations, our result shows that its role cannot be neglected in the zero-energy minimal conductivity.

Trigonal warping and anisotropic band splitting in monolayer graphene due to Rashba spin-orbit coupling

We study the electronic band structure of monolayer graphene when Rashba spin-orbit coupling is present. We show that if the Rashba spin-orbit coupling is stronger than the intrinsic spin-orbit coupling, the low-energy bands undergo trigonal-warping deformation and that for energies smaller than the Lifshitz energy, the Fermi circle breaks up into separate parts. The effect is very similar to what happens in bilayer graphene at low energies. We discuss the possible experimental implications, such as threefold increase in the minimal conductivity for low electron densities, anisotropic, wave-number-dependent spin splitting of the bands, and the spin-polarization structure.

Theory of snake states in graphene

We study the dynamics of the electrons in a non-uniform magnetic field applied perpendicular to a graphene sheet in the low energy limit when the excitation states can be described by a Dirac type Hamiltonian. We show that as compared to the two-dimensional electron gas (2DEG) snake states in graphene exibit peculiar properties related to the underlying dynamics of the Dirac fermions. The current carried by snake states is locally uncompensated even if the Fermi energy lies between the first non-zero energy Landau levels of the conduction and valence bands. The nature of these states is studied by calculating the current density distribution. It is shown that besides the snake states in finite samples surface states also exist.

Minimal longitudinal dc conductivity of perfect bilayer graphene

xx = (J�/2)e 2 /h while from the other formula used in the above mentioned work we findmin xx = (4J/�)e 2 /h, where J = 1 for single layer and J = 2 for bilayer graphene. The two universal values are different although they are numerically close to each other. Our two results are in the same order of magnitude as that of experiments and for single layer case one of our result agrees many earlier theoretical predictions. However, for bilayer graphene only two studies are known with predictions for the minimal conductivity different from our calculated values. Similarly to the single layer case, the physical origin of the minimal conductivity in bilayer graphene is also rooted back to the intrinsic disorder induced by the Zitterbewegung which is related to the trembling motion of the electron.

Uniform tiling with electrical resistors

The electric resistance between two arbitrary nodes on any infinite lattice structure of resistors that is a periodic tiling of space is obtained. Our general approach is based on the lattice Greens function of the Laplacian matrix associated with the network. We present several non-trivial examples to show how efficient our method is. Deriving explicit resistance formulas it is shown that the Kagome, diced and decorated lattice can be mapped to the triangular and square lattice of resistors. Our work can be extended to the random walk problem or to electron dynamics in condensed matter physics.