Probing the surface-to-bulk transition: A closed-form constant-scaling algorithm for computing subsurface Green functions

A closed-form algorithm for computing subsurface Green functions—the blocks of a material’s Green function between the surface and the bulk—is presented, where we assume the system satisfies a common principal-layer approximation. By exploiting the block tridiagonal and nearly block Toeplitz structure of the Hamiltonian and overlap matrices, this method scales independently of the system size (constant scaling), allowing studies of large systems. As a proof-of-concept example, we investigate the decay of surface effects in an armchair graphene nanoribbon, demonstrating the persistence of surface effects hundreds of atomic layers ($~$0.5 $\mu$m) away from a surface. We finally compare the surface-to-bulk transitions of finite and semi-infinite systems, finding that finite systems exhibit amplified surface effects.

Figure 1

The armchair graphene nanoribbon considered throughout this work. The left and right PLs contain 12 atomic layers, while the bulk contains two. For reference, one atomic layer in the bulk PL is highlighted. Finite-sized systems are constructed by inserting the desired number of bulk PLs between the left and right PLs, whereas a semi-infinite system is simply the left PL and an infinite number of bulk PLs.

Figure 2

Relevant system properties for using the effective coupling matrix approximation to model the ACGNR system. (a) The condition number of ${\mathbf{M}}_{10}$, showing that the matrix is numerically singular across the entire spectrum. (b) $L$, the minimum number of PLs recommended for using ${\mathbf{M}}_{10}^{\mathrm{eff}}$, when the condition number of ${\mathbf{M}}_{10}^{\mathrm{eff}}$ is ${\delta }_{\mathrm{SVD}}^{-1}$. Systems with fewer than $\mathrm{ceiling}\hspace{0.5em}(L)$ PLs should use an iterative method with ${\mathbf{M}}_{10}$, whereas ${\mathbf{M}}_{10}^{\mathrm{eff}}$ is a good approximation for larger systems.

Figure 3

Root-mean-square (RMS) absolute error between all elements of surface GFs calculated with the recursive technique (using ${\mathbf{M}}_{10}$) and the modified MTM (${\mathbf{M}}_{10}^{\mathrm{eff}}$). The surface GFs are for systems with (a) two bulk PLs, (b) three bulk PLs, and (c) four bulk PLs. When ${\delta }_{\mathrm{SVD}}$ is too small (gray line), the effective coupling matrix remains numerically singular, incurring error. Increasing ${\delta }_{\mathrm{SVD}}$ improves the effective coupling matrix approximation (black line); however, increasing ${\delta }_{\mathrm{SVD}}$ too much (dashed black line) worsens it by modifying physically important states. While the optimal choice of ${\delta }_{\mathrm{SVD}}$ is most likely dependent on both the system and the numerical precision (e.g. double precision), the errors here are small over a large range of ${\delta }_{\mathrm{SVD}}$ values.

Figure 4

LDOSs for the central atomic layers in finite ACGNRs. The number inset in the bottom-left corner of each plot is the minimum depth (in atomic layers) from either surface. Discrete levels are evident in small systems, and these levels coalesce into bands as the ACGNRs become longer. After $~$100 atomic layers, the bands appear as fluctuations around the bulk limit. These fluctuations dampen as the depth from the surfaces increases, and the $~$2500th atomic layer effectively mimicks the bulk ACGNR. Two surface states persist in the band gap for $~$25 atomic layers.

Figure 5

LDOSs for various atomic layers in a semi-infinite ACGNR. The number inset in the bottom-left corner of each plot is the depth from the surface (in atomic layers). As with the finite ACGNRs, fluctuations are observed around the bulk limit, after $~$10 atomic layers. However, unlike the finite systems, these fluctuations are seemingly milder and the $~$1500th atomic layer resembles the bulk. The sole surface state decays in $~$25 atomic layers.

Figure 6

Deviations of atomic layer $m$ from the bulk, ${\delta }_m$, for $m⩽7500$. At a particular distance from the surface, the atomic layers in semi-infinite systems more closely resemble the bulk than do those from a finite system. Furthermore, for small $m$, ${\delta }_m$ follows a power law, ${\delta }_m\approx a/m^b$. This approximation fails for large $m$ due to broadening, $\eta$. For large or reasonable amounts of broadening [panels (a) and (b), respectively], surface effects seem to decay more rapidly in finite systems; however, the effects decay at the same rate for small $\eta$ [panel (c)]. The best-fit power laws are (a) semi-infinite: $a=3.3$, $b=0.57$, $R^2=0.97$, finite: $a=35$, $b=0.94$, $R^2=0.99$; (b) semi-infinite: $a=2.7$, $b=0.28$, $R^2=0.98$, finite: $a=14$, $b=0.37$, $R^2=0.99$; (c) semi-infinite: $a=2.3$, $b=0.15$, $R^2=0.93$, finite: $a=8.9$, $b=0.13$, $R^2=0.99$.