Band Structures of Bilayer Graphene Superlattices

We formulate a low energy effective Hamiltonian to study superlattices in bilayer graphene (BLG) using a minimal model which supports quadratic band touching points. We show that a one dimensional (1D) periodic modulation of the chemical potential or the electric field perpendicular to the layers leads to the generation of zero-energy anisotropic massless Dirac fermions and finite energy Dirac points with tunable velocities. The electric field superlattice maps onto a coupled chain model comprised of ``topological'' edge modes. 2D superlattice modulations are shown to lead to gaps on the mini-Brillouin zone boundary but do not, for certain symmetries, gap out the quadratic band touching point. Such potential variations, induced by impurities and rippling in biased BLG, could lead to subgap modes which are argued to be relevant to understanding transport measurements.

Finite Signal-set Capacity of Two-user Gaussian Multiple Access Channel

The capacity region of a two-user Gaussian Multiple Access Channel (GMAC) with complex finite input alphabets and continuous output alphabet is studied. When both the users are equipped with the same code alphabet, it is shown that, rotation of one of the user’s alphabets by an appropriate angle can make the new pair of alphabets not only uniquely decodable, but will result in enlargement of the capacity region. For this set-up, we identify the primary problem to be finding appropriate angle(s) of rotation between the alphabets such that the capacity region is maximally enlarged. It is shown that the angle of rotation which provides maximum enlargement of the capacity region also minimizes the union bound on the probability of error of the sumalphabet and vice-verse. The optimum angle(s) of rotation varies with the SNR. Through simulations, optimal angle(s) of rotation that gives maximum enlargement of the capacity region of GMAC with some well known alphabets such as M-QAM and M-PSK for some M are presented for several values of SNR. It is shown that for large number of points in the alphabets, capacity gains due to rotations progressively reduce. As the number of points N tends to infinity, our results match the results in the literature wherein the capacity region of the Gaussian code alphabet doesn’t change with rotation for any SNR.

Doubly spectral stochastic finite element method (DSSFEM) for random field problems

Uncertainties in complex dynamic systems play an important role in the prediction of a dynamic response in the mid- and high-frequency ranges. For distributed parameter systems, parametric uncertainties can be represented by random fields leading to stochastic partial differential equations. Over the past two decades, the spectral stochastic finite-element method has been developed to discretize the random fields and solve such problems. On the other hand, for deterministic distributed parameter linear dynamic systems, the spectral finite-element method has been developed to efficiently solve the problem in the frequency domain. In spite of the fact that both approaches use spectral decomposition (one for the random fields and the other for the dynamic displacement fields), very little overlap between them has been reported in literature. In this paper, these two spectral techniques are unified with the aim that the unified approach would outperform any of the spectral methods considered on their own. An exponential autocorrelation function for the random fields, a frequency-dependent stochastic element stiffness, and mass matrices are derived for the axial and bending vibration of rods. Closed-form exact expressions are derived by using the Karhunen-Loève expansion. Numerical examples are given to illustrate the unified spectral approach.