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Graphene antidot lattices (GALs) are two-dimensional (2D) monolayers with periodically placed holes in otherwise pristine graphene. We investigate the electronic properties of symmetric and asymmetric GAL structures having hexagonal holes, and show that anisotropic 2D GALs can display a dimensional crossover such that effectively one-dimensional (1D) electronic structures can be realized in two-dimensions around the charge neutrality point. We investigate the transport and thermoelectric properties of these 2D GALs by using non-equilibrium Green function (NEGF) method. Dimensional crossover manifests itself as transmission plateaus, a characteristic feature of 1D systems, and enhancement of thermoelectric efficiency, where thermoelectric figure of merit, $zT$, can be as high as 0.9 at room temperature. We also study the transport properties in the presence of Anderson disorder and find that mean free paths of effectively 1D electrons of anisotropic configuration are much longer than their isotropic counterparts. We further argue that dimensional crossover due to broken symmetry and enhancement of thermoelectric efficiency can be nanostructuring strategy virtually for all 2D materials.

Using first-principles density functional theory calculations, we investigate a family of stable two-dimensional crystals with chemical formula $A_2B_2$, where $A$ and $B$ belong to groups IV and V, respectively ($A$ = C, Si, Ge, Sn, Pb; $B$ = N, P, As, Sb, Bi). Two structural symmetries of hexagonal lattices $P\bar{6}m2$ and $P\bar{3}m1$ are shown to be dynamically stable, named as $\alpha$- and $\beta$-phases correspondingly. Both phases have similar cohesive energies, and the $\alpha$-phase is found to be energetically favorable for structures except CP, CAs, CSb and CBi, for which the $\beta$-phase is favored. The effects of spin-orbit coupling and Hartree-Fock corrections to exchange-correlation are included to elucidate the electronic structures. All structures are semiconductors except CBi and PbN, which have metallic character. SiBi, GeBi and SnBi have direct band gaps, whereas the remaining semiconductor structures have indirect band gaps. All structures have quartic dispersion in their valence bands, some of which make the valence band maximum and resemble a Mexican hat shape. SnAs and PbAs have purely quartic valence band edges, i.e. $E{\sim}{-}\alpha k^4$, a property reported for the first time. The predicted materials are candidates for a variety of applications. Owing to their wide band gaps, CP, SiN, SiP, SiAs, GeN, GeP can find their applications in optoelectronics. The relative band positions qualify a number of the structures as suitable for water splitting, where CN and SiAs are favorable at all pH values. Structures with quartic band edges are expected to be efficient for thermoelectric applications.