^{1,2,a)}, Nicholas M. Boffi

^{1}, Mark A. Ratner

^{1}and Tamar Seideman

^{1}

### Abstract

We computationally investigate the decay of surface effects in one-, two-, and three-dimensional materials using two-band tight-binding models. These general models facilitate a direct comparison between materials of differing dimensionality, which reveals that material dimensionality (not material-specific chemistry/physics) is the primary factor controlling the decay of surface effects. Our results corroborate more sophisticated, material-specific studies, finding that surface effects decay after ∼10, ∼25, and ≳ 100 layers in three-dimensional, two-dimensional, and one-dimensional materials, respectively. Physically, higher-dimensional materials screen surface effects more efficiently, as theoretically described by integration over each layer's Brillouin zone. Finally, we discuss several implications of these results.

We thank Scott Thornton and Bobby Sumpter for helpful conversations. M.G.R. performed this research as a Department of Energy (DoE) Computational Science Graduate Fellow (Grant No. DE-FG02-97ER25308) while at Northwestern University and as a Eugene P. Wigner Fellow at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC, for the U.S. DoE under Contract No. DE-AC05-00OR22725. M.A.R. and T.S. acknowledge support from the National Science Foundation (Grant Nos. CHE-1012207/001 and CHE-1058896) and from the NSF's MRSEC program (DMR-1121262) at the Materials Research Center of Northwestern University. Figures 2–6 were prepared with the LevelScheme package.^{71}

I. INTRODUCTION

II. THEORETICAL METHODS

III. ONE-DIMENSIONAL MATERIALS

A. Linear 1D model

B. Ladder 1D model

IV. TWO-DIMENSIONAL MATERIALS

V. THREE-DIMENSIONAL MATERIALS

VI. DIMENSIONALITY AND THE DECAY OF SURFACE EFFECTS

VII. CONCLUSIONS

### Key Topics

- Surface states
- 25.0
- Materials properties
- 18.0
- Bulk materials
- 16.0
- Surface photoemission
- 14.0
- Materials modification
- 8.0

##### B82B1/00

## Figures

Schematic representations of the tight-binding models. (a) A 1D material (linear model). (b) A 2D material. (c) A 3D material. (d) A 1D material (ladder model). In all cases there are two alternating types of sites, one with site energy α (blue circles) and the other with −α (green circles). The linear 1D, 2D, and 3D models alternate between two couplings β_{1} (black lines) and β_{2} (red lines). The ladder 1D model has three couplings: a “rung” coupling β_{1} (black lines) and two alternating “rail” couplings β_{2} (red) and β_{3} (orange).

Schematic representations of the tight-binding models. (a) A 1D material (linear model). (b) A 2D material. (c) A 3D material. (d) A 1D material (ladder model). In all cases there are two alternating types of sites, one with site energy α (blue circles) and the other with −α (green circles). The linear 1D, 2D, and 3D models alternate between two couplings β_{1} (black lines) and β_{2} (red lines). The ladder 1D model has three couplings: a “rung” coupling β_{1} (black lines) and two alternating “rail” couplings β_{2} (red) and β_{3} (orange).

LDOSs for various layers (as labelled) of the linear 1D model. Red: α = 0.3 eV, β_{1} = −0.8 eV, and β_{2} = −1.3 eV. Blue: α = 0.3 eV, β_{1} = −1.3 eV, and β_{2} = −0.8 eV. When formation of the surface breaks the stronger coupling (|β_{2}| > |β_{1}|, red lines), a surface state appears in the odd layers as a broadened δ function at *E* = α. This surface state decays with increasing layer number while oscillations around the bulk limit appear in the bands. In agreement with previous studies, these oscillations do not quickly dampen and the LDOSs do not converge to the bulk.

LDOSs for various layers (as labelled) of the linear 1D model. Red: α = 0.3 eV, β_{1} = −0.8 eV, and β_{2} = −1.3 eV. Blue: α = 0.3 eV, β_{1} = −1.3 eV, and β_{2} = −0.8 eV. When formation of the surface breaks the stronger coupling (|β_{2}| > |β_{1}|, red lines), a surface state appears in the odd layers as a broadened δ function at *E* = α. This surface state decays with increasing layer number while oscillations around the bulk limit appear in the bands. In agreement with previous studies, these oscillations do not quickly dampen and the LDOSs do not converge to the bulk.

LDOSs for various layers (as labelled) of the ladder 1D model (α = 0.3 eV, β_{1} = −0.8 eV, β_{2} = −1.3 eV, and β_{3} = −0.8 eV). Similar to the linear 1D model, a surface state appears at *E* = α that decays with increasing layer number, disappearing by layer 20. Oscillations in the bands are also evident; however, they slowly dampen and allow convergence to the bulk limit. The ladder 1D model is less effectively 1D than the linear 1D model and although surface effects decay slowly, they decay more quickly than in the linear 1D model.

LDOSs for various layers (as labelled) of the ladder 1D model (α = 0.3 eV, β_{1} = −0.8 eV, β_{2} = −1.3 eV, and β_{3} = −0.8 eV). Similar to the linear 1D model, a surface state appears at *E* = α that decays with increasing layer number, disappearing by layer 20. Oscillations in the bands are also evident; however, they slowly dampen and allow convergence to the bulk limit. The ladder 1D model is less effectively 1D than the linear 1D model and although surface effects decay slowly, they decay more quickly than in the linear 1D model.

LDOSs for various layers (as labelled) of the 2D model (α = 0.3 eV, β_{1} = −0.8 eV, and β_{2} = −1.3 eV). Similar to the 1D models, the surface state decays in ∼10 layers (most easily seen by comparing asymmetry in the valence and conduction bands). The LDOSs again oscillate around the bulk limit with increasing depth; however, they converge to the bulk limit much more quickly than in the 1D materials.

LDOSs for various layers (as labelled) of the 2D model (α = 0.3 eV, β_{1} = −0.8 eV, and β_{2} = −1.3 eV). Similar to the 1D models, the surface state decays in ∼10 layers (most easily seen by comparing asymmetry in the valence and conduction bands). The LDOSs again oscillate around the bulk limit with increasing depth; however, they converge to the bulk limit much more quickly than in the 1D materials.

LDOSs for various layers (as labelled) of the 3D model (α = 0.3 eV, β_{1} = −0.8 eV, and β_{2} = −1.3 eV). Similar to materials with other dimensionalities, the surface state decays in roughly 10 layers. Unlike the other materials, however, the LDOS oscillations significantly dampen in these initial layers and the bulk LDOS is essentially recovered by layer ∼10.

LDOSs for various layers (as labelled) of the 3D model (α = 0.3 eV, β_{1} = −0.8 eV, and β_{2} = −1.3 eV). Similar to materials with other dimensionalities, the surface state decays in roughly 10 layers. Unlike the other materials, however, the LDOS oscillations significantly dampen in these initial layers and the bulk LDOS is essentially recovered by layer ∼10.

Deviations between the bulk and *n*th layer LDOSs (δ_{ n }) vs. layer (*n*). The circles are calculated values and the lines are the best-fit functions *a*/*n* ^{−b } (for *n* sufficiently large). (a) Linear 1D material with a surface state (red lines in Fig. 2 ). (b) Linear 1D material without a surface state (blue lines in Fig. 2 ). (c) Ladder 1D material with parameters α = 0.3 eV, β_{1} = −0.8 eV, β_{2} = −1.3 eV, and β_{3} = −1.3 eV. (d) Ladder 1D material from Fig. 3 . (e) 2D material from Fig. 4 . (f) 3D material from Fig. 5 . The decay parameters *b* are (a) 0.0_{0}, (b) 0.0_{0}, (c) 0.2_{1}, (d) 0.2_{6}, (e) 0.6_{3}, and (f) 1.0_{5}. Surface effects decay more rapidly in materials of higher dimensionality.

Deviations between the bulk and *n*th layer LDOSs (δ_{ n }) vs. layer (*n*). The circles are calculated values and the lines are the best-fit functions *a*/*n* ^{−b } (for *n* sufficiently large). (a) Linear 1D material with a surface state (red lines in Fig. 2 ). (b) Linear 1D material without a surface state (blue lines in Fig. 2 ). (c) Ladder 1D material with parameters α = 0.3 eV, β_{1} = −0.8 eV, β_{2} = −1.3 eV, and β_{3} = −1.3 eV. (d) Ladder 1D material from Fig. 3 . (e) 2D material from Fig. 4 . (f) 3D material from Fig. 5 . The decay parameters *b* are (a) 0.0_{0}, (b) 0.0_{0}, (c) 0.2_{1}, (d) 0.2_{6}, (e) 0.6_{3}, and (f) 1.0_{5}. Surface effects decay more rapidly in materials of higher dimensionality.

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