Dimensional Crossover and Enhanced Thermoelectric Efficiency Due to Broken Symmetry in Graphene Antidot Lattices

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 the nonequilibrium Green function 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.

Figure 1

Structural parameters of GALs are shown in (a). There are two antidots per unit cell, which has side lengths $L_x$ and $L_y$. Antidots are hexagonal and equal in size ($r$). The center of an antidot is chosen at the origin, while the second is at ($\mathrm{\Delta }x$,$\mathrm{\Delta }y$). The studied geometries ($\alpha$, $\beta$, $\gamma$) are shown below.

Figure 2

Band structures and transmission functions of $\alpha$ (blue), $\beta$ (red), $\gamma$ (green) respectively. Solid (dash-dotted) curves represent the asymmetric (symmetric) structures. Zero of the energy corresponds to the charge neutrality point.

Figure 3

Toy model with a square lattice in the first nearest neighbor approximation having longitudinal and transverse hopping parameters $t_x$ and $t_y$. The electronic bands and corresponding transmission spectra are plotted for different values of $\eta =t_y/t_x$, namely $\eta =1$, 0.2 and 0 (orange, magenta and gray, respectively).

Figure 4

Thermoelectric properties. The columns represent $\alpha$, $\beta$ and $\gamma$ geometries, respectively, while the rows stand for the Seebeck coefficient ($S$), power factor ($P$), and TE efficient ($zT$). Values for symmetric (asymmeteric) structures are plotted using dot-dashed (solid) curves. Band gap edges are represented with vertical lines (dot-dashed) for asymmetric (symmetric) GALs.

Figure 5

Three variations of $\alpha$ structure with their band structures, transmission functions and power factors are given (lower half). Blue, red and black lines correspond to the first (${\alpha }_S$), the second and the third (${\alpha }_A$) structure respectively.

Figure 6

Width normalized phonon thermal conductance values as functions of temperature. Asymmetry changes phononic heat conduction for $\alpha$ and $\gamma$ whereas it remains almost unchanged for $\beta$.

Figure 7

Mean free paths for ${\alpha }_A$ and ${\alpha }_S$ structures. Electronic density of states is shown in the inset. Red curves represent the symmetric structure, whereas the blue curves are for the asymmetric ones.