Observation of Gapped Dirac Cones in a Two-Dimensional Su-Schrieffer -Heeger Lattice Daiyu Geng 12 Hui Zhou 12 Shaosheng Yue 12 Zhenyu Sun 12 Peng Cheng 12 Lan

2025-04-26 1 0 727.85KB 16 页 10玖币

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Observation of Gapped Dirac Cones in a Two-Dimensional

Su-Schrieffer-Heeger Lattice

Daiyu Geng,1,2,# Hui Zhou,1,2,# Shaosheng Yue,1,2 Zhenyu Sun,1,2 Peng Cheng,1,2 Lan

Chen,1,2,3 Sheng Meng,1,2,3,4* Kehui Wu,1,2,3,4* Baojie Feng,1,2,4*

1Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China

2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

3Songshan Lake Materials Laboratory, Dongguan, Guangdong, 523808, China

4Interdisciplinary Institute of Light-Element Quantum Materials and Research Center for Light-

Element Advanced Materials, Peking University, Beijing, 100871, China

#These authors contributed equally to this work.

*Corresponding author. E-mail: khwu@iphy.ac.cn; smeng@iphy.ac.cn; bjfeng@iphy.ac.cn.

Abstract

The Su-Schrieffer-Heeger (SSH) model in a two-dimensional rectangular lattice

features gapless or gapped Dirac cones with topological edge states along specific

peripheries. While such a simple model has been recently realized in

photonic/acoustic lattices and electric circuits, its material realization in condensed

matter systems is still lacking. Here, we study the atomic and electronic structure of a

rectangular Si lattice on Ag(001) by angle-resolved photoemission spectroscopy and

theoretical calculations. We demonstrate that the Si lattice hosts gapped Dirac cones

at the Brillouin zone corners. Our tight-binding analysis reveals that the Dirac bands

can be described by a 2D SSH model with anisotropic polarizations. The gap of the

Dirac cone is driven by alternative hopping amplitudes in one direction and staggered

potential energies in the other one and hosts topological edge states. Our results

establish an ideal platform to explore the rich physical properties of the 2D SSH

model.

 
1. Introduction 
The discovery of graphene has inspired enormous research interest in searching 
for topological materials with Dirac cones [1-5]. In a Dirac material, the valence and 
conduction  bands  touch  at  discrete  points  in  the  momentum  space,  and  the  band 
degeneracies are protected by symmetries, such as time-reversal and mirror reflection. 
The  topological  band  structures  of  Dirac  materials  can  give  rise  to  various  exotic 
properties, including the half-integer quantum Hall effect [6], Klein tunneling [7], and 
extremely  large  magnetoresistance  [8].  Compared  to  three-dimensional  Dirac 
materials,  two-dimensional  (2D)  Dirac  materials  are  relatively  rare  because  of  the 
scarcity  of  realizable  2D  materials  [9].  In  addition,  most  of  the  experimentally 
realized 2D Dirac materials are hexagonally symmetric, such as graphene [10,11] and 
silicene [12,13]. In rectangular or square lattices, Dirac states are quite rare [14-16], 
despite the recent prediction of candidate materials including 6,6,12-graphyne [17,18], 
t1(t2)-SiC [19], and g-SiC3 [20,21]. On the other hand, breaking certain symmetries 
can gap out the Dirac point, giving rise to topological (crystalline) insulating states 
with conducting edge channels.  
A  prototypical  model  to  realize  topological  states  is  the  Su-Schrieffer-Heeger 
(SSH) model, which was initially put forward to describe spinless electrons hopping 
in a one-dimensional (1D) dimerized lattice [22]. Recently, the 1D SSH model has 
been extended to 2D square lattices [23,24]. In the 2D SSH model, a gapless Dirac 
cone exists at the M point when electron hoppings are homogeneous, as shown in Fig. 
1(a) and 1(b). However, suppose the hopping amplitudes alternate in one direction 
while keeping constant in the other one. In that case, the application of a staggered 
potential in the other direction can gap out the Dirac cone, as shown in Fig. 1(c) and 
1(d), with topological edge states along the specific peripheries. Experimentally, the 
2D  SSH  model  has  been  realized  in  several  systems,  including  photonic/acoustic 
lattices [26-28] and electric circuits [29]. However, in condensed matter systems, the 
material realization of 2D SSH lattice is still lacking.  
In this work,  we demonstrate  that  Dirac  electronic states can be realized in  a 
rectangular Si lattice, and the topological properties of this system can be described 

 
by  the  2D  SSH  model.  The  rectangular  Si  lattice  can  be  synthesized  by  epitaxial 
growth of Si on Ag(001), which was reported as early as 2007 [30]. Previous works 
on  this  system  only  focused  on  the  structural  characterization  using  scanning 
tunneling  microscopy  (STM)  and  surface  X-ray  diffraction,  while  studies  of  its 
electronic structure are still lacking, both experimentally and theoretically.  
Here,  we  systematically  study  the  atomic  and  electronic  structure  of  the 
rectangular  Si  lattice  by  ARPES  measurements,  first-principles  calculations,  and 
tight-binding analysis. Our first-principles calculations show that adsorption of Si on 
Ag(001) will result in the periodic missing of topmost Ag atoms, in contrast to the 
previously  proposed  unreconstructed  Ag(001)  surface  [30,31].  Interestingly,  our 
ARPES measurements proved the existence of a gapped Dirac cone at each M point 
of the  Si  lattice,  which  is  supported  by  our first-principles calculations.  Our  tight-
binding (TB) analysis based on a 2D SSH model reveals that the gap of the Dirac 
cone  is  driven  by  the  alternation  of  bond  lengths  in  one  direction  and  stagger  of 
potential energies in the other one. In addition, our calculations show that topological 
edge  states  exist  along  specific  peripheries,  which  indicates  the  rich  topological 
properties in this system. These results call for further research interest in the exotic 
topological properties of the 2D SSH model in condensed matter physics.  
 
2. Results and Discussion 
2.1 Growth and atomic structure 
When the coverage of Si is less than one monolayer, a 3×3 superstructure with 
respect to the Ag(001) substrate will form [30]. The sharp LEED patterns shown in 
Fig. 2(a) indicate the high quality of the sample. Further deposition of Si will lead to 
the formation of the second layer, which is a more complex phase compared to the 
first layer [30]. Here, we focus on the monolayer phase, i.e., the 3×3 superstructure. 
Figure 2(b) schematically shows the Brillouin zones (BZs) of the Si lattice and the 
Ag(001) substrates together.  
To determine the atomic structure of the Si/Ag(001) system, we performed first-
principles  calculations.  Previously,  Leandri  et  al.  suggested  that  the  3×3 

superstructure is formed by the periodic arrangement of tilted Si dimers on an

unreconstructed Ag(001) surface [30]. However, this structure model is neither stable

nor metastable according to first-principles calculations [31]. He et al. proposed that

Si form hexagons on unreconstructed Ag(001), as shown in Supplementary Fig. 1.

Through an extensive structural search, we discovered a more stable structure, as

shown in Fig. 2(c). Our structure model is composed of Si dimers, analogous to that

proposed by Leandri et al. However, the dimers are not tilted, and the topmost Ag

layer is seriously reconstructed. Each unit cell contains two Si dimers, with one of

them sinking because of the missing of two Ag atoms. The migration of Ag atoms in

the topmost layer has also been found in the Si/Ag(111) and Si/Ag(110) systems [32-

34], which may be a result of interaction between Si and Ag. Figure 2(d) shows a

simulated STM image, which agrees well with previous experimental results [30].

The calculated binding energy of each Si atom (~2.042 eV) is much larger than that of

the hexagonal structure in Supplementary Fig. 1 (~1.039 eV), indicating that our

structure model is more stable. The validity of our structure model is also supported

by the agreement between the calculated band structure and ARPES measurement

results, as discussed in the following.

2.2 ARPES measurements

ARPES measurements were carried out to study the electronic structure of the

Si/Ag(001)-3×3 surface. Constant energy contours (CECs) of Si/Ag(001)-3×3 and

pristine Ag(001) are displayed in Fig. 3(a)-(d), Supplementary Fig. 2 and 3. All the

observable bands from the Fermi surface to EB ~0.5 eV are derived from the Ag(001)

substrate. When the binding energy increases to 0.6 eV, two dot-like features emerge

at the M point of Si, as indicated by the black arrows in Fig. 3(a). With increasing

binding energies, each dot becomes a closed pocket [Fig. 3(b)], indicating a hole-like

band centered at the M point. Further increase of the binding energy leads to the touch

and merge of neighboring pockets, which is a signature of a saddle point or van Hove

singularity at the X point of Si.

ARPES intensity maps along Cuts 1-4 [indicated by the red lines in Fig. 3(a)] are

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ObservationofGappedDiracConesinaTwo-DimensionalSu-Schrieffer-HeegerLatticeDaiyuGeng,1,2,#HuiZhou,1,2,#ShaoshengYue,1,2ZhenyuSun,1,2PengCheng,1,2LanChen,1,2,3ShengMeng,1,2,3,4*KehuiWu,1,2,3,4*BaojieFeng,1,2,4*1InstituteofPhysics,ChineseAcademyofSciences,Beijing,100190,China2SchoolofPhysicalSciences,U...

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Observation of Gapped Dirac Cones in a Two-Dimensional Su-Schrieffer -Heeger Lattice Daiyu Geng 12 Hui Zhou 12 Shaosheng Yue 12 Zhenyu Sun 12 Peng Cheng 12 Lan

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