  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.
