# Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Details and Method of Calculations

#### 2.1. Experimental Details

#### 2.2. Calculation Method

## 3. Results and Discussion

#### 3.1. Symmetry Analysis

**Figure 1.**(

**a**) The first Brillouin zone of a reciprocal space of 2D hexagonal lattices. The different M and K points are indicated (equivalent points are connected by a reciprocal space vector); (

**b**) The unit cell with the symmetry mirrors for $P6mm$, $P31m$, $P3m1$ and $P3$ space groups; (

**c**) The reciprocal vectors in the first (${\overrightarrow{G}}_{n}^{\left(1\right)}$) and second (${\overrightarrow{G}}_{n}^{\left(2\right)}$) rings around Γ, as well as the corresponding Fourier component of the potential (see text).

#### 3.2. The Ag/Cu(111) Reconstruction

#### 3.3. Symmetry and Potential Properties

**Figure 2.**Brillouin zone with the mirror symmetries (top panel), schematic band structures in the nearly free electron model (center panel) and the corresponding direct space surface potential (bottom panel) for the different space groups: (

**a**) $P6mm$; (

**b**) $P31m$; (

**c**) $P3m1$; and (

**d**) $P3$. Comparison between these band structures shows that the ${\sigma}_{h}$ mirrors are necessary to have a gapless Dirac point at K.

#### 3.4. Electronic Properties of Ag/Cu(111)

**Figure 3.**(

**a**) Scanning tunneling microscopy (STM) image of the Ag/Cu(111) surface with the unit cell and the three-fold ${\sigma}_{v}$ mirror. Solid and dotted lines represent the unit and Wigner–Seitz cells, respectively; (

**b**) Second derivative angle-resolved photoemission spectroscopy (ARPES) signal representing the band dispersions measured on Ag/Cu(111) (left) and a K-doped surface (closeup illustrating the three bands close to the K point).

**Figure 4.**(

**a**) Comparison between the experimental and calculated ARPES intensity for Ag/Cu(111) showing that both the energy gap width and the intensity of the folded band are reproduced. The spectra have been divided by the Fermi function in order to evidence the spectral weight in the 3$kT$ energy range above the Fermi energy; (

**b**) The surface potential built from the ${V}_{1}$ and ${V}_{2}$ components; (

**c**) The surface band structure (lines) calculated from the experimentally determined surface potential and experimental dispersions (symbols) up to the Fermi energy; (

**d**) Experimental scanning tunneling spectrum (symbols) and calculated local density of states (lines) evidencing the energy gaps of the band structure.

## 4. Conclusions

## Conflicts of Interest

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**MDPI and ACS Style**

Vasseur, G.; Fagot-Revurat, Y.; Kierren, B.; Sicot, M.; Malterre, D.
Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points. *Symmetry* **2013**, *5*, 344-354.
https://doi.org/10.3390/sym5040344

**AMA Style**

Vasseur G, Fagot-Revurat Y, Kierren B, Sicot M, Malterre D.
Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points. *Symmetry*. 2013; 5(4):344-354.
https://doi.org/10.3390/sym5040344

**Chicago/Turabian Style**

Vasseur, Guillaume, Yannick Fagot-Revurat, Bertrand Kierren, Muriel Sicot, and Daniel Malterre.
2013. "Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points" *Symmetry* 5, no. 4: 344-354.
https://doi.org/10.3390/sym5040344