$\pi$-bonds in graphene and surface of topological insulator $Bi_2Se_3$
G. J. Shu$^1$
Chemical bond analysis for the topological insulator (TI) Bi$_2$Se$_3$ following valence bond theory and VSEPR rule is proposed and compared to that of graphene. The existence of $\pi$-bond trimers on surface Se-layer is revealed to be responsible for the unusual 2D surface conduction found in the prototype $Z_2$ topological insulator Bi$_2$Se$_3$, which shows great similarity to the ideal ``surface-only" 2D material graphene with surface band in Dirac cone shape. The $\pi$-bond trimers on the surface of TI form a dynamic $\pi$-bond conjugated system of implicit chirality. Local $\pi$-bond energy exchange in cross-bridge model is proposed responsible for the unique 2D surface conduction for topological insulator Bi$_2$Se$_3$, and the similar low dimensional conduction mechanism can also be identified in graphene and conductive polymers. Preliminary supporting evidence for the existence of $\pi$-bonds is provided by the commonly assigned $\pi$ plasmon for the spectral feature near 7 eV in graphite and Bi$_2$Se$_3$ using electron energy-loss spectroscopy (EELS).
GJ-TI_physics_to_the_chemists_plus_EELS-DJ_corrected-APL-070513.tex
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%%%%%%%%%%%%%%%%% END OF PREAMBLE %%%%%%%%%%%%%%%% - for NJP submission and rejected

%Dear Editors:  Please find the attached manuscript titled "Surface conduction and $\pi$-bonds in graphene and topological insulator $Bi_2Se_3$" which I am submitting to the New J. Physics as a regular article.  Topological insulator has been one important topic in condensed matter physics for the past few years, however, the understanding of its physical properties is hindered by the involved mathematical analogy using quantum spin Hall effect and band topology through band structure analysis in reciprocal space.  We believe it is urgent to bridge the gap using a more intuitive language in real space to attract more researchers beyond the theorists and the experimentalists using ARPES as a tool, especially to expand the research base through material design for the development of next generation topological insulators.   This article uses the language of chemical bonds in real space crystal structure to interpret the key physical properties of a topological insulator Bi2Se3, i.e., the band insulator with an unique surface conduction mechanism which is similar to that of graphene.  

%We believe a real space view from the chemical bond perspective should enhance and be complimentary to the understanding and material design of TI.  In particular, we find that there is a great deal of similarity between the ideal 2D material of ``surface-only" graphene and Bi$_2$Se$_3$ surface in terms of the Dirac cone band picture and the existence of $\pi$-bonds. 

%The authors have been working in the field of topological insulators with a focus on materials synthesis and crystal growth for years.  We have co-authored several important papers related to this subject, including the Bi2Se3 crystal growth and defect studies (Huang et al PRB 86-081104, 2012), surface diffraction of Bi2Se3 (Zhu et al., PRL 108-185501, 2012), and the latest topological crystalline insulators (Xu et al, Nat. Comm. DOI: 10.1038/ncomms2191, 2012).  

%Although this article does not fall into the common categories of physics of theoretical calculation, or data generated from  experiments, we believe the proposed pi-bond interpretation to the existing theoretical and experimental results is fresh and would inspire most researchers in the field.  In particular, we did a close comparative analysis to that of graphene and benzene as a strong supporting evidence in a deductive manner at this stage.  The influence and the finding of pi-bond in the topological insulators could be a general phenomenon, which has been demonstrated by the common existence of pi-bond and Dirac cone in both the ideal 2DEG of graphene and the surface of Bi2Se3.  We have included more comparative analysis with graphene and the way of presentation fits the interdisciplinary requirement of NJP very well.  Please do not treat this model interpretation as physics of "speculations without supporting evidence", or as solid state chemistry only, we believe the implication of current finding is general with great significance for the research of topological insulator.     

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Article under review for New Journal of PhysicsSurface conduction and \pi-bonds in graphene and topological insulator Bi$_2$Se$_3$ - Dr Guo-Jiun Shu et alID: NJP/455948/PAP
%BOARD MEMBER'S REPORT============================
%This work draws analogies between conduction on a two-dimensional carbon sheet of graphene and on the two-dimensional surface of a three-dimensional topological insulator. The presentation is entirely qualitative, without equations or calculations, and therefore has very much a preliminary character.  This is one reason I do not consider this suitable for publication in NJP.  I also find the analogy quite unconvincing. In graphene the conduction is not-topological in origin, and indeed the graphene sheet becomes insulating when inversion symmetry is broken (for example by a substrate or hydrogenation). In a topological insulator, in contrast, topology protects the metallic surface state, and a substrate or inversion symmetry breaking has no effect.  As NJP is an open access journal which captures a broad readership of specialists and non-specialists we place a large emphasis on work being of sufficient importance and accessibility to be of interest to our readers.  In this case, I do not believe that NJP is the most appropriate outlet for the work and that it would be in the authors interest to resubmit to a more specialist journal.

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%************************************************************************

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (edited by DJ on 032513) - for honeycomb submission and rejected
%Dear Editors: 

%Attached please find the manuscript titled "�-bond and topological character of �stuffed graphite�-type ternary compounds," which I am submitting to Nature Materials as a Letter. Topological insulator has been one important topic in condensed matter physics for the past few years; however, the understanding of its physical properties is hindered by the involved mathematical analogy, including quantum spin Hall effect and band topology in reciprocal space. We believe that it is urgent to bridge the gap using a more intuitive language in real space to attract more researchers beyond the theorists and experimentalists in the field of condensed matter physics. In fact, a similar view has just been raised by R. J. Cava in his latest introduction letter, which appeared in the News and Views section of Nature Materials in March 2013. It is time for the chemists and material scientists to join efforts.

%To expand the research base through material design for the development of next-generation topological insulators, the language used on topological insulators must be more inclusive. This article uses the language of chemical bond in real-space crystal structure to interpret the key physical properties of a special class of topological insulator that has a unique "stuffed graphite"-type structure. Pi-bond emerges naturally from both graphene and this class of compounds, and can be used directly in classifying whether the compound is topologically trivial or not, in contrast to the involved theoretical calculations.

%The authors have been working in the field of topological insulators with a focus on materials synthesis and crystal growth for years. We have co-authored several important papers related to this subject, including the Bi2Se3 crystal growth and defect studies (Huang et al., PRB 86-081104, 2012), surface diffraction of Bi2Se3 (Zhu et al., PRL 108-185501, 2012), and the latest topological crystalline insulators on Pb1-xSnxTe (Xu et al., Nat. Comm. DOI: 10.1038/ncomms2191, 2012). 

%While this letter presents a totally different view on the common methods applied in the field of topological insulator research, especially in the condensed matter physics, and it seems too obvious to be ignored, some resistance from the traditional view may exist. We believe Professors R. J. Cava of Princeton U., H. Takagi of U. Tokyo, K. Noveselov, A. K. Geim and J. B. Goodenough of U. Texas should be considered for this highly interdisciplinary work.

%Sincerely yours, Fang-Cheng Chou 
%Professor and Research Fellow, Center for Condensed Matter Sciences, National Taiwan University 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Nature Materials-Letter: 
%1)This format begins with an introductory paragraph (not abstract) of 150 words maximum, summarizing the background, rationale, main results and implications. This paragraph should be referenced, as in Nature style, and should be considered part of the main text, so that any subsequent introductory material avoids too much repetition of the introductory paragraph.
%2) The text is limited to 2000 words, excluding the introductory paragraph, Methods, references and figure legends. References are limited to 30. Letters should have no more than 3�5 display items (figures and/or tables). Letters include received/accepted dates and may be accompanied by supplementary information.
%3) Cover Letter: Authors should provide a cover letter that includes the affiliation and contact information for the corresponding author. Authors should briefly discuss the work's importance and explain why the work is considered appropriate for the diverse readership of Nature Materials.
%4) The manuscript text file should include the following parts, in order: a title page with author affiliations and contact information (the corresponding author should be identified with an asterisk); the sections required for each content type (see information for Letters and Articles) then References, Acknowledgements (optional), Author Contributions, Competing Financial Interests statement, Figure Legends, and Tables.
%5) For the inclusion of graphics, we recommend graphicx.sty. Please use numerical references only for citations.
%6) Authors submitting LaTeX files may use any of the standard class files such as article.cls, revtex.cls or amsart.cls. Non-standard fonts should be avoided; please use the default Computer Modern fonts. 

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%Earlier cover letter to the Nat. Mat. in honeycomb paper+++++++++++++++
%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%Topological insulator has been one important topic in condensed matter physics for the past few years; however, the understanding of its physical properties is hindered by the involved mathematical analogy, including quantum spin Hall effect and band topology in reciprocal space. We believe that it is urgent to bridge the gap using a more intuitive language in real space to attract more researchers beyond the theorists and experimentalists in the field of condensed matter physics. In fact, a similar view has just been raised by R. J. Cava in his latest introduction letter, which appeared in the News and Views section of Nature Materials in March 2013. It is time for the chemists and material scientists to join efforts.  To expand the research base through material design for the development of next-generation topological insulators, the language used on topological insulators must be more inclusive. This article uses the language of chemical bond in real-space crystal structure to interpret the key physical properties of a special class of topological insulator that has a unique "stuffed graphite"-type structure. Pi-bond emerges naturally from both graphene and this class of compounds, and can be used directly in classifying whether the compound is topologically trivial or not, in contrast to the involved theoretical calculations.


%======ubmit to APL in July 2013 ==================
%============================================
%Dear Editors:  Please find the attached manuscript titled "$\pi$-bonds in graphene and surface of topological insulator $Bi_2Se_3$" which I am submitting to the Applied Physics Letters as a regular article.   The main contribution of this work is that the unique physical properties of graphene and topological insulator Bi2Se3 can be explained with the proposed chemical bond model satisfactorily.  In particular, it is found that the open pi-bond conjugated system is the common link responsible for the unique low-d transport mechanism for conductive polymer, graphene, and topological insulator. 

%Topological insulator has been one important topic in condensed matter physics for the past few years; however, the understanding of its physical properties is hindered by the involved mathematical analogy, including quantum spin Hall effect and band topology in reciprocal space. We believe that it is urgent to bridge the gap using a more intuitive language in real space to attract more researchers beyond the theorists and experimentalists in the field of condensed matter physics. In fact, a similar view has just been raised by R. J. Cava in his latest introduction letter that appeared in the News and Views section of Nature Materials in March 2013. It is time for the chemists and material scientists to join efforts.  To expand the research base through material design for the development of next-generation topological insulators, the language used on topological insulators must be more inclusive.  We believe that viewing topological insulators with a correct chemical bond model could  lead to the understanding of surface electronic state naturally.

%The authors have been involved in the development of many topological insulators in the past, including Bi2Se3 (PRB 86-081104(R) 2012) and Pb1-xSnxTe (Nat. Comm. 3-1192 2012).  We believe this work should satisfy the high publication standard required by APL.

%Suggested referees: especially on EELS: P. E. Batson, C. Colliex 




\begin{document}

%\preprint{}

\title{$\pi$-bonds in graphene and surface of topological insulator $Bi_2Se_3$}

\author{G. J. Shu$^1$}
\author{S. C. Liou$^1$}
\author{F. C. Chou$^{1,2}$}
\email{fcchou@ntu.edu.tw}
\affiliation{
$^1$Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan}
\affiliation{
$^2$National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan}
%\affiliation{
%$^3$Center for Emerging Material and Advanced Devices, National Taiwan University, Taipei 10617, Taiwan}

\date{\today}

\begin{abstract}
Chemical bond analysis for the topological insulator (TI) Bi$_2$Se$_3$ following valence bond theory and VSEPR rule is proposed and compared to that of graphene. The existence of $\pi$-bond trimers on surface Se-layer is revealed to be responsible for the unusual 2D surface conduction found in the prototype $Z_2$ topological insulator Bi$_2$Se$_3$, which shows great similarity to the ideal ``surface-only" 2D material graphene with surface band in Dirac cone shape.  The $\pi$-bond trimers on the surface of TI form a dynamic $\pi$-bond conjugated system of implicit chirality.  Local $\pi$-bond energy exchange in cross-bridge model is proposed responsible for the unique 2D surface conduction for topological insulator Bi$_2$Se$_3$, and the similar low dimensional conduction mechanism can also be identified in graphene and conductive polymers.  Preliminary supporting evidence for the existence of $\pi$-bonds is provided by the commonly assigned $\pi$ plasmon for the spectral feature near 7 eV in graphite and Bi$_2$Se$_3$ using electron energy-loss spectroscopy (EELS).      
\end{abstract}

\pacs{73.20.-r; 73.25.+i; 31.10.+z; 33.15.Fm; 79.20.Uv }
%31.10.+z	Theory of electronic structure, electronic transitions, and chemical binding
%33.15.Fm	Bond strengths, dissociation energies
%73.20.-r	  Electron states at surfaces and interfaces
%73.25.+i	    Surface conductivity and carrier phenomena
%72.20.-i	     Conductivity phenomena in semiconductors and insulators
%72.25.Dc	Spin polarized transport in semiconductors
%72.80.Vp	Electronic transport in graphene
%73.20.At	       Surface states, band structure, electron density of states
%73.22.Pr	Electronic structure of graphene
%75.70.Tj	    Spin-orbit effects (see also 71.70.Ej Spin-orbit coupling, Zeeman and Stark splitting, Jahn-Teller effect)
%79.20.Uv    Electron energy loss spectroscopy 

\maketitle

The finding of topological insulator has been one important breakthrough in condensed matter physics since Kane and Mele first suggested that the Dirac cone band structure of graphene can also be found in this novel class of material.\cite{Kane2005}  Two major classes of TI have been predicted from calculation and verified experimentally by ARPES since; the first type has surface state protected by the time reversal symmetry (TRS) called $Z_2$ topological insulators, such as Bi$_2$Se$_3$,\cite{Hasan2010} and the second type is the topological crystalline insulators with surface state protected by the crystalline symmetry, such as Pb$_{1-x}$Sn$_x$Se.\cite{Fu2007, Dziawa2012}  Simply put, topological insulator is a unique quantum state of matter of narrow band gap with surface-only conduction as a result of strong spin-orbit coupling, and, in particular, the $Z_2$ TI has a surface spin-polarized state protected by the TRS topologically.  In the language of theoretical calculations commonly used in physics, TI is expected to be found in a material system which has opposite parity for the conduction and valence bands, and the occurrence of band inversion is triggered by parameters such as strain or strong spin-orbit coupling (SOC).\cite{Zhang2009, Fu2007}    However, the Hilbert space topology often used by theorists is too abstract for chemists and material scientists for further material design.\cite{Hasan2010, Qi2011}  We believe that a real space view from the chemical bond perspective should enhance and be complimentary to the understanding and material design of TI.  In particular, with the help of the supporting experimental evidence using EELS technique, we find that there is a great deal of similarity between the ideal 2D material of ``surface-only" graphene and Bi$_2$Se$_3$ surface in terms of the Dirac cone band picture and the existence of $\pi$-bonds. 

Molecular Orbital (MO) theory has shown great success in describing condensed matter in the energy-momentum space, i.e, the lowest-unoccupied-molecular-orbital (LUMO) and the highest-occupied-molecular-orbital (HOMO) for individual molecule could be extended to the valence and conduction bands of solids.\cite{textbook}  However, except for the typical metals such as pure IA and IIA elements, insulators of ionic nature, and the wide band gap semiconductors, more ``exceptions" are found in the band theory of solids.  For example, Coulomb repulsion $U$ is introduced to the strongly correlated electron system that has an unfilled band signified to be metallic but turns out to be insulating, and often categorized as Mott insulator.\cite{Mott}  The latest development of topological insulator brings up yet another ``exception" for a narrow band gap semiconductor, i.e., a band insulator for the bulk with a spin polarized conducting surface state, which can hardly be defined as either a zero gap semiconductor or a conductor with zero density of states in the conduction band.

%Graphene as an ideal "surface-only" 2D material shows a band picture which cannot be classified in the traditional band theory, i.e., it is gapless with energy-momentum in linear dispersion relationship as a Dirac cone.\cite{??}  Whether treating graphene as a gapless semiconductor or a metal with zero density of states in the conduction band is controversial, especially following the band theory of solids directly without corrections, such as considering the term of spin-orbit coupling.  These findings challenge a band theory in 2D that strongly suggests the need of a major paradigm shift.  In fact, the direct challenge for graphene is that the traditional conduction band picture ($\pi$*-band) is no longer valid, i.e., when the assumption of discrete energy levels of individual atoms are coupled into condensed matter of infinitesimally small energy gaps as a band, while the z-direction perpendicular to the honeycomb lattice of graphene hits the quantum limit that must be constrained by the Uncertainty Principle.  The commonly assumed $\pi$*-band conduction mechanism used to explain the graphene conduction mechanism becomes fictitious, not only because of the great uncertainty on momentum near each carbon atom, but also the electron in $p_z$ is confined quantum mechanically within $p_z^\pm$ orbitals.

%===========================================================
%\begin{figure}
%\includegraphics[width=3.5in]{fig-graphene}
%\caption{\label{fig-graphene} (color online) (a) The 2D honeycomb lattice of graphene, with hybridized $sp^2$ orbitals as $\sigma$-bonds in triangular shape, and the remaining one electronin $p_z$ forms weak $\pi$-bond with one of the three neighboring carbon atoms instantaneously.  (b) The valence bond model of graphene starting from the atomic electronic configuration of C= [He]$2s^22p^2$. }  
%\vspace{-5mm}
%\end{figure}
%============================================================

The unique $sp^2$+$p$ orbital hybridization of graphene allows the honeycomb structure formation at the lowest 2D atomic packing density, and one electron remains in $p_z$-orbital.\cite{graphene}  It is desirable to find the link between the real space chemical bond of $sp^2+p$ and the unique energy band of gapless Dirac cone shape.  While the honeycomb lattice is bridged by the covalent $\sigma$-bonds through $sp^2$ hybridization, the remaining electron in $p_z$ is unpaired with relatively higher energy without the $\sigma$-bond energy gain.  Judging from the metallic nature of graphene, the $p_z$ electrons cannot be localized completely, yet the traditional band picture cannot describe the co-existing bonding and nonbonding states reasonably well. Such a dilemma could be resolved by the unique bond order of 1$\frac{1}{3}$ for graphene, i.e., one $\pi$-bond is shared by three neighboring caron atoms statistically.\cite{Fasolino2007}  In fact, the linear dispersion relationship of energy-momentum suggests the tunneling nature of a massless fermion excitation.\cite{Geim2009, Katsnelson2006}  The conduction mechanism shown in Dirac cone shape band picture must be closely related to the semi-localized $p_z$ electron behavior in graphene.  While the topological insulator Bi$_2$Se$_3$ shows Dirac cone surface band similar to that of graphene, and with the reasonable assumption on the close connection between the dynamics of the unpaired $p_z$ electrons and the Dirac cone surface band picture, it would be important to find out whether there also exist unpaired electrons of similar dynamic nature on the surface of Bi$_2$Se$_3$.

%===========================================================
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{fig-structure}
\end{center}
\caption{\label{fig-structure}(color online) The crystal structure of Bi$_2$Se$_3$ is shown with two quintuple layers. The insets show the actual octahedral environment near Bi, Se2 in the middle, and Se1 at the van der Waals gap. }
\end{figure}
%============================================================

%===========================================================
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{fig-VBmodel}
\end{center}
\caption{\label{fig-VBmodel}(color online) The valence bond model of Bi$_2$Se$_3$ constructed from atomic electronic configurations of Bi ([Xe]5d$^{10}$6s$^2$6p$^3$) and Se ([Ar]4s$^2$4p$^4$) through $sp^3d^2$ hybridization for the octahedral coordination, which is adapted from the VB model for Bi$_2$Te$_3$ proposed by Drabble \textit{et al.}\cite{Drabble1958} }
\end{figure}
%============================================================

Single Dirac cone at the $\bar{\Gamma}$-point has been predicted and experimentally confirmed in the $Z_2$ topological insulator Bi$_2$Se$_3$.\cite{Hasan2010}  The real space view of the surface chemical bond of Bi$_2$Se$_3$ should be examined in order to find out the peculiar non-localization of surface electrons that maybe related to the existence of Dirac cone.  While Bi$_2$e$_3$ is a narrow band gap semiconductor of E$_g$$\sim$0.3 eV, a complete $\sigma$-bond construction following valence bond (VB) theory should be applicable using orbital hybridization, similar to that of the $sp^3$ hybridization for the typical semiconductor silicone with E$_g$$\sim$1 eV.   The crystal structure of Bi$_2$Se$_3$ can be described as layers of quintuple unit with van der Waals interlayer coupling, as shown in Fig.~\ref{fig-structure}.\cite{Huang2012}  The chemical bonds of Bi$_2$Te$_3$ with identical crystal structure has been under debate since Mooser and Pearson first introduced the concepts of orbital hybridization and resonating valence bonds to the system.\cite{Mooser1958, Drabble1958, Bhide1971}  The focus of the early debate was on what kind of hybrid orbitals for Bi and Te are the most suitable for explaining the observed transport property well.  However, the early proposed hybridization models have not taken the van der Waals and surface chemical bondings into consideration.  The quintuple unit of Bi$_2$Se$_3$ shown in Fig.~\ref{fig-structure} indicates that all of the Bi and Se atoms have octahedral environment, which implies that the $sp^3d^2$ hybrid is required under the valence bond theory and VSEPR rule consideration.\cite{inorganic}  Fig.~\ref{fig-VBmodel} shows the VB model proposed for Bi$_2$Se$_3$ starting from the atomic electronic configuration.  We find that the $sp^3d^2$ hybrid is likely for all Se atoms because of the required six outer shell electrons in n=4; on the other hand, it is required to move one electron from 3d$^{10}$ to satisfy the required $sp^3d^2$ hybrid for the two Bi atoms.  The heavy elements of both Bi and Se suggests that the formed covalent $\sigma$-bonds must be relatively weak considering the large atom size of high principal quantum number n ($>$4), and indeed, the melting point of Bi$_2$Se$_3$ is relatively low near 705$^\circ$C.  As a typical layered compound with van der Waals force in between quintuple layers, it is interesting to note that the Se atoms near the van der Waals gap have three unpaired electrons remaining at the gap interface, and these unpaired electrons will be exposed to the surface when the sample is cleaved open along the van der Waals gap.  

%===========================================================
\begin{figure}
\begin{center}
\includegraphics[width=3in]{fig-trimer}
\end{center}
\caption{\label{fig-trimer}(color online) The $\pi$-bond trimers on surface Se-layer of coordination number six in Bi$_2$Se$_3$ form a $\pi$-bond conjugated system (in dashed lines) with an implicit chirality. }
\end{figure}
%============================================================

Usually, the unpaired electrons exposed to the cleaved surface could end up stabilizing into different surface states, including passivation by the external molecules, surface phonon-related reconstruction that leads to the band bending, or collective itinerant behavior like surface plasmon.  We believe that the most interesting and critical condition is when the unpaired electrons carry energy that is not too high to be itinerant or too low to be localized, as often seen in the low melting point semimetal materials with narrow band gap - Bi$_2$Se$_3$ and most confirmed topological insulators seem to fall in this category.\cite{TI}  In view of the weak bonding energy of Bi$_2$Se$_3$ and the surface hexagonal close packing of Se atoms in vacuum, we believe that the three unpaired electrons per Se on the exposed surface could lead to a novel surface state that is related to the unique surface property for the topological insulators.  We propose that the three unpaired electrons in virtual octahedron in $sp^3d^2$ hybrid per Se1 could form a critical state as trimer of 120$^\circ$ apart to avoid the Coulomb repulsion, in accord with the VSEPR rule, as shown in Fig.~\ref{fig-trimer}.  Since all Se1 atoms are in 2D triangular lattice of hexagonal close packing on the surface, the three unpaired electrons per surface Se1 must either avoid each other to reduce Coulomb repulsion, or form $\pi$-bond with the three neighboring surface Se1 atoms instantaneously.  It should be noted that when the proposed $\pi$-bond is weak with minimum orbital overlap on the surface, the reduction of Gibbs free energy through entropy gain ($-T\Delta S$) is not negligible anymore, i.e., the $\pi$-bond trimer could form a dynamic conjugated system instead of stabilizing in one of the configurations thermodynamically, as illustrated in Fig.~\ref{fig-trimer}.  In fact, such a $\pi$-bond conjugated system is commonly found in graphene, benzene, superconductor MgB$_2$, and conductive polymers.\cite{Choi2002, ppp, Fasolino2007} 


$\pi$-bond is a unique class of chemical bond formed with instantaneous Coulomb attraction between semi-localized electrons in mostly side-to-side orbitals without actual overlap, in contrast to the $\sigma$-bond which requires electron sharing in the head-to-head overlapped orbital space.\cite{inorganic}  Because of its extremely weak bonding strength and the dynamic conjugated nature which is easily perturbed by thermal or strain energy, it is difficult to describe $\pi$-bond (or $\pi$-band) in energy-momentum space for the ground state description.  In fact, while graphene is in the thickness limit of one molecular thick perpendicular to its 2D honeycomb lattice, it is not appropriate to describe the related physical phenomena in a continuous band picture anymore; instead, a discrete quantum state for electron in $p_z$ orbital with the greatest uncertainty on momentum must exist, according to the Uncertainty Principal.  Quantum tunneling phenomenon of Klein paradox and the Dirac equation for massless fermions have all pointed toward this direction for the description of electrons near the Dirac cone for graphene.\cite{Katsnelson2006}  Following the link between the conjugated $\pi$-bond system and the existence of Dirac cone for graphene, the commonly observed gapless Dirac cone on topological insulator surface suggests that the similar $\pi$-bond conjugated system could also be identified in Bi$_2$Se$_3$.  

%===========================================================
\begin{figure}
\includegraphics[width=3.5in]{fig-EELS}
\caption{\label{fig-EELS}(color online) The STEM-EELS spectra for graphite and Bi$_2$Se$_3$ are compared.  The first two main features are assigned coming from the $\pi$ and $\pi+\sigma$ plasmons as described in the text. } 
\vspace{-5mm}
\end{figure}
%============================================================

Herein, we offer a positive experimental evidence to the existence of $\pi$-bond using electron energy-loss spectroscopy (EELS) through comparative studies of graphite (graphene) and topological insulator Bi$_2$Se$_3$.  EELS can provide the information of both valence and conduction band electronic states effectively.  In particular, the low-loss region (energy loss $<$ 50 eV) in EELS spectra contains useful information on the collective modes of valence electron excitation, e.g., surface and volume plasmons; and the single-particle excitations, e.g., interband transitions and low-lying core-level ionizations.\cite{Raether1980, Egerton2011}  The representative EELS spectra for graphite and Bi$_2$Se$_3$ are compared in Fig.~\ref{fig-EELS}.\cite{Liou2013}   The spectral features near 7 and 27 eV for graphite have been demonstrated as the plasma resonances related to the $\pi$ and $\pi$+$\sigma$ electrons, respectively.\cite{Ichikawa1958, Taft1965}  The effective number of electrons per atom ($n_{eff}(\omega)$) participating in the collective oscillation modes near 7 eV and 27 eV have been estimated to be $n_{eff}$$\sim$1 and 4 electrons, respectively.\cite{Taft1965}  The extracted $n_{eff}$ values strongly suggest that the participating electrons for the assigned $\pi$ plasmon (one $\pi$-bond) and the $\pi$+$\sigma$ plasmon (one $\pi$- plus three $\sigma$-bonds) are coming from the unpaired electron in $p_z$ orbital and the three electrons in the hybridized $sp^2$ orbitals per atom, respectively.   

The EELS spectrum for Bi$_2$Se$_3$ shown in Fig.~\ref{fig-EELS} reveals two main spectral features near 7 and 17 eV, which are in good agreement with the spectra reported in the literature.\cite{Nascimento1999}   The two weak peaks at $\sim$26.4 and 28.4 eV can be identified as the Bi \textit{O}$_{4,5}$-edge excitation from Bi $5d$ electrons, which agrees with the $5d^9$ configuration after the required $sp^3d^2$ hybridization shown in Fig.~\ref{fig-VBmodel}.\cite{Liou2013, Nascimento1999}  We can tentatively assign the two main spectral features near 7 and 17 eV to be volume plasmons first.  Considering there are six $\pi$-bonds and twelve $\sigma$-bonds per quintuple Bi$_2$Se$_3$ formula unit (see Fig.~\ref{fig-VBmodel}), we expect the participating $n_{eff}$ for the $\pi$ and $\pi+\sigma$ plasmons for Bi$_2$Se$_3$ be close to 1.2 (6/5 per atom, estimated from the six $\pi$-bonds per quintuple unit) and 3.6 (18/5 per atom, estimated from the 6$\pi$+12$\sigma$ bonds per quintuple unit), respectively.  The $n_{eff}$ derived from the energy-loss function near 7 and 17 eV are $\sim$1 and $\sim$3 electrons per atom, which can be largely interpreted as coming from the plasma oscillations of the $\pi$ and $\pi + \sigma$ electrons, respectively.\cite{Liou2013}  The reasonable agreement between the $\pi$-bond model prediction and the EELS estimation strongly supports the existence of $\pi$-bond in Bi$_2$Se$_3$. 

%===========================================================
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{fig-piBonds}
\end{center}
\caption{\label{fig-piBonds}(color online) The $\pi$-bonds (dashed line) can be identified in (a) benzene C$_6$H$_6$ with closed $\pi$-bond conjugated system, (b) trans-polyacetylene [C$_2$H$_2$]$_n$ with 1D open $\pi$-bond conjugated system, (c) graphene with 2D open $\pi$-bond conjugated system, and (d) Bi$_2$Se$_3$ with 2D $\pi$-bond trimer conjugated system. The red circle represents the dynamic $\pi$-bond conjugated system of inherited chirality.}
\end{figure}
%============================================================


The $\pi$-bond conjugated system has been applied to successfully explain the unique transport behavior of conductive polymer.\cite{ppp}  As shown in Fig.~\ref{fig-piBonds}(b), the 1D conduction for trans-polyacetylene [C$_2$H$_2$]$_n$ can be explained by the local energy transfer through the instantaneous $\pi$-bond bonding/nonbonding of the $\pi$-bond conjugated system.  In contrast, benzene has three $\pi$-bonds within each unit (Fig.~\ref{fig-piBonds}(a)), but the formed conjugated system is closed, which leads to its insulating nature.  It is interesting to note that the 1D bridged $\pi$-bond conjugated system shown in trans-polyacetylene [C$_2$H$_2$]$_n$ can be extended to 2D and form one honeycomb network similar to graphene, as illustrated in Fig.~\ref{fig-piBonds}(c).  We believe that the 2D transport behavior of graphene can be explained using the same open $\pi$-bond conjugated system.  Considering the $\pi$-bond trimer conjugated system found on the Bi$_2$Se$_3$ surface, similar transport mechanism can also apply, as illustrated in Fig.~\ref{fig-piBonds}(d).  As long as all surface Se1 atoms contain three unpaired electrons of energy not too high to be itinerant or too low to be localized, the critical energy equivalent to the $\pi$-bond bonding energy would co-exist with the entropy-dominated dynamic $\pi$-bond conjugated system, which can also explain why TI is so sensitive to strain as found in  the theoretical prediction.\cite{Fu2007}  The $\pi$-bond conjugated trimer could bond and break spontaneously (i.e., with nonzero configurational entropy) and the small critical bonding energy is transferred locally, which would lead to a successful long range 2D electronic energy transfer beyond the traditional concept of electric conduction based on diffusion of itinerant electrons.  We might name such 2D conduction mechanism as a ``cross-bridge" model, because the tri-bridge with inherited chirality of random handedness can always maintain an infinite percolation route in 2D at any instance.  


%===========================================================
%\begin{figure}
%\begin{center}
%\includegraphics[width=3in]{fig-vdW}
%\end{center}
%\caption{\label{fig-vdW}(Color online) The van der Waals attractive force can be found in the interlayer instantaneous electric dipole formation for Bi$_2$Se$_3$. }
%\end{figure}
%============================================================

%The revelation of $\pi$-bond trimers on Bi$_2$Se$_3$ surface under the VB model description can also be applied to explain the origin of the van der Waals force between quintuple layers.  The origin of van der Waals force is mostly related to either permanent or instantaneous electric dipole formation.  The upper and lower Se1 surfaces across the van der Waals gap could have two choices each, and the electric dipoles can be formed once the two $\pi$-bond trimers are out-of-phase to avoid Coulomb repulsion.  Once the $\pi$-bond trimers are locked by electric dipole formation, the ``cross-bridge" model 2D conduction would break down, because the dynamic $\pi$-bond conjugated system is broken and fall into one of the static choices.  It is also interesting to note that the Dirac cone observed by ARPES is gapped for few layer Bi$_2$Se$_3$, until more than five layers of quintuple units to regain its gapless Dirac cone.\cite{He2010}  We believe this phenomenon is related to the locked $\pi$-bond trimer, i.e, when the $\pi$-bond trimers on top-most surface and bottom-most surface are too close, they could also be partially locked to cause Dirac cone gap opening.  Current model can also be applied to interpret why Mn substitution is able to open up the Dirac cone gap of Bi$_2$Se$_3$,\cite{Hsieh2009} must be due to the conservation of chirality is broken by the Mn spin, which leads to the break down of the dynamic $\pi$-bond conjugated system.  These findings once again demonstrates the validity of current proposal on close relationship between the 2D conduction mechanism originated from the dynamic $\pi$-bond conjugated system in real space and the existence of Dirac cone on the surface band of topological insulators.   

%===========================================================
%\begin{figure}
%\begin{center}
%\includegraphics[width=3.5in]{fig-parity}
%\end{center}
%\caption{\label{fig-parity}(Color online) 2D parity symmetry can be identified on Bi2Se3 surface. }
%\end{figure}
%============================================================

%With the help of chemical bond analysis under valence bond theory and VSEPR rule, the disclosure of $\pi$-bond trimer on Bi$_2$Se$_3$ surface can also help to explain the unique TRS protected surface state of the topological insulator.\cite{??}  Theoretically the surface state described by Dirac cone is spin polarized, as experimentally verified by spin-ARPES.\cite{??}  We find that the $\pi$-bond trimer conjugated system inherits the chirality of the surface electrons because of the broken parity symmetry in 2D, as shown in Fig.~\ref{fig-parity}.   Parity operation of the wave function in 2D requires \textbf{P}$\Psi$(x,y)=$\Psi${-x,y} or $\Psi${x,-y}, which could be verified from the choice of a mirror plane that lead to the reconstruction of its mirror image, as illustrated in Fig.~\ref{fig-parity}.  Indeed, we can always find the mirror line for the $\pi$-bond trimer on Bi$_2$Se$_3$ surface, which implies that the parity symmetry breaking and the existence of chirality with the choice of RH or LH handedness.  While Abelian group $Z_2$ has often been used to describe the symmetry protected topological  order of $Z_2$ topological insulators, and $Z_2$ topological invariant is usually calculated from the band structure of the bulk, the disclosed parity in 2D actually forms the Abelian group $Z_2$ due to the relationship of \textbf{P}$^2$=1, which is a more intuitive approach to understand the character of topological insulator from the real space view.   The spin polarization depicted in the reciprocal space can now be understood in real space view through the preferred handedness choice of RH or LH chirality.  In fact, the pseudospin discussed in graphene has all of the attributes of real angular momentum,\cite{Mecklenburg2011} which implies the chiral nature of $\pi$-bond also.  Although statistically it is expected to have a balanced choice of chirality for the $\pi$-bod trimer on topological insulator surface, we believe the subtle imbalance between the two could supply additional energy gain through -\textbf{$\Delta$ M}H in Gibbs free energy, which can be viewed as a unique form of spin-orbit coupling in 2D. 

%===========================================================
%\begin{figure}
%\begin{center}
%\includegraphics[width=3.5in]{fig-LiAuSe}
%\end{center}
%\caption{\label{fig-LiAuSe}(Color online) The crystal structures and valence bond models of (a) LiAuSe and (b) KHgSb. }
%\end{figure}
%============================================================

%We find that the similarity between graphene and topological insulator surface is significant, especially on the existence of $\pi$-bond.  Using the half-Heusler ternary compounds LiAuSe and KHgSb as examples, both can be viewed as honeycomb lattice with IA elements stuffed in between layers.\cite{??}  DFT calculations by Zhang \textit{et al.} suggest that the LiAuSe is topologically nontrivial but KHgSb is not.\cite{Zhang2011}  The crystal structures and the corresponding valence bond models for graphene, LiAuSe, and KHgSb are shown in Fig.~\ref{fig-graphene} and Fig.~\ref{fig-LiAuSe}.  The graphene in honeycomb lattice has been described with $sp^2+p$, where honeycomb lattice is formed by $\sigma$-bonds of $sp^2$ hybrid orbitals of carbon in coordination number of three, and one electron remain in $p_z$ to form $\pi$-bond with one of the three neighboring carbon atoms instantaneously as a conjugated system of three choices.\cite{??}  We find the honeycomb lattice can both form for both LiAuSe and KHgSb, but the IA elements of Li And K faces the choice of electron accept ion depending on the electronegativity (EN) between the two elements within the honeycomb lattice.  Based on the EN values, Li must donate its 2s$^1$ electron to Au in LiAuSe, while K must donate its 4s$^1$ electron to Sb in KHgSb, which leads to the possible $\pi$-bond formation in LiAuSe to be greatly resemble that of graphene, but not in KHgSb.  The implication that LiAuSe to be topological nontrivial strongly implies the close connection between graphene and topological insulator, especially on the common existence of $\pi$-bond.  

In summary, with the help of chemical bond analysis under valence bond theory and VSEPR rule considerations, $sp^3d^2$ hybridization for Bi and Se atoms in Bi$_2$Se$_3$ leads to the revelation of $\pi$-bond trimer on the surface of topological insulator Bi$_2$Se$_3$.  The existence of $\pi$-bond in both graphite and Bi$_2$Se$_3$ has been supported from EELS, which not only helps to explain the unique 2D conduction mechanism in the ``cross-bridge" model , but also helps to explain the unique spin polarization of Dirac cone-shaped surface band for topological insulator.  Current real space description of Bi$_2$Se$_3$ can be valuable to understand the topological insulator beyond the reciprocal space view.  The role of $\pi$-bond on topological insulator surface could be the key to the understanding of both graphene and topological insulators in general.       

\section*{Acknowledgment}
We thank G. Y. Guo, M.-W. Chu, and C. H. Chen for helpful discussions and R. Sankar for the single crystal sample Bi$_2$Se$_3$ used in the EELS studies.  FCC acknowledges support from NSC-Taiwan under project number  NSC 101-2119-M-002-007.  GJS acknowledges support from NSC-Taiwan under project number NSC 100-2112-M-002-001-MY3.

%\section{\label{sec:level1}Notes added\protect\\}

%In addition to the explanations of the published experimental works given above, more ongoing experimental results are emerging to support our proposed model.  For example, the existence of a $\pi$-bond trimer has been supported by the observation of C$_{3v}$ symmetry on the surface of Bi$_2$Se$_3$ from a study of second harmonic generation, and, the STEM-EELS studies have revealed signatures of plasmons with $\pi$- and $\sigma+\pi$-characters of different dispersion behaviors.  More experimental works are necessary to trace phenomena related to the existence and behaviors of $\pi$-bond trimer.     
%= GJ 102612 == on our collaborators on Bi2Se3 with Evidences including
%1) Trimer evidence of C3v is observed by Raman-SHG, not six fold not three fold symmetry.  Also Se loss that breaks trimer symmetry. (by YM Chang)
%2) The three different electronic states from STEM-EELS: pi, sigma+pi, and inter band transition has been observed.(SCLiou)
%3) The d^9 configuration has been suggested by the STEM-EELS in Bi. (SCLiou)
%4) The phosflourence has been observed in Bi2Se3 (Chunwei Chen)

%\bibliography{JabRef database} (if use this then no need to use the direct list below)
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%\bibitem{Hsieh2012} Timothy H. Hsieh, Hsin Lin, Junwei Liu, Wenhui Duan, Arun Bansil, and Liang Fu, Nat. Comm. 3, 982 (2012)
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\end{thebibliography}

\end{document}



1307.2078.extracted_bib
\begin{thebibliography}{99}

\bibitem{Kane2005} C. L. Kane and E. J. Mele, Phys. Rev. Lett. \textbf{95}, 226801 (2005).
\bibitem{Hasan2010} M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. \textbf{82}, 3045 (2010).
\bibitem{Fu2007} L. Fu and C. L. Kane, Phys. Rev. B \textbf{76}, 045302 (2007).
\bibitem{Dziawa2012} P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A. Szczerbakow, M. Szot, E. ?usakowska, T. Balasubramanian, B. M. Wojek, M. H. Berntsen, O. Tjernberg, and T. Story, Nat Mater \textbf{11}, 1 (2012).
\bibitem{Zhang2009} H. Zhang, C. X. Liu, X. Qi, X. Dai, Z. Fang, and S. C. Zhang, Nat. Phys. \textbf{5}, 438 (2009).
\bibitem{Qi2011} X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. \textbf{83}, 1057 (2011).
\bibitem{textbook} J. D. Lee, Concise Inorganic Chemistry, 5th edition, Blackwell Science, Ltd.
\bibitem{Mott} P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. \textbf{78}, 17 (2006).
\bibitem{graphene} C. K. Chua and M. Pumera, Chemical Society Reviews, \textbf{42}, 3222 (2013).
\bibitem{Fasolino2007} A. Fasolino, J. H. Los, and M. I. Katsnelson, Nat. Mater. \textbf{6}, 858 (2007). %multiple bonds graphene
\bibitem{Geim2009} A. K. Geim, Science \textbf{324}, 1530-1534 (2009).
\bibitem{Katsnelson2006} M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nature Physics \textbf{2}, 620 (2006).
\bibitem{Huang2012} F.-T. Huang, M.-W. Chu, H. H. Kung, W. L. Lee, R. Sankar, S.-C. Liou, K. K. Wu, Y. K. Kuo, and F. C. Chou, Phys. Rev. B, \textbf{86}, 081104 (2012).
\bibitem{Mooser1958} E. Mooser and W. B. Pearson, Journal of Physics and Chemistry of Solids \textbf{7}, 65 (1958), and E. Mooser and W. B. Pearson,  Phys. Rev. \textbf{101}, 1608 (1956).
\bibitem{Drabble1958} J. Drabble and C. Goodman, J. Phys. Chem. Sol. \textbf{5}, 142 (1958).
\bibitem{Bhide1971} V. G. Bhide and B. A. Patki, J. Phys. Chem. Sol. \textbf{32}, 1565 (1971).
\bibitem{inorganic} G. L. Miessier and D. A. Tarr, Inorganic Chemistry (4th Edition),  Prentice Hall 2010. 
\bibitem{Choi2002} H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G. Louie, Nature \textbf{418}, 758 (2002).%MgB2
\bibitem{ppp} J. Roncali, Chemical Reviews \textbf{92}, 711 (1992).
\bibitem{TI} Y. Ando, J. Phys. Soc. Jpn., in press, (2013).  %TI review 
\bibitem{Raether1980} H. Raether, Springer Tracts in Modern Physics (volume 88): Excitation of Plasmons and Interband Transitions by Electrons (Springer-Verlag, Berlin, Heidelberg, Germany, 1980).
\bibitem{Egerton2011} R. F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3th edition (Springer New York Dordrecht Heidelberg London, 2011).
\bibitem{Liou2013} S. C. Liou, M. -W. Chu, R. Sankar, F. -T. Huang, G. -J. Shu, F. C. Chou, and C. H. Chen, Phys. Rev. B., \textbf{87}, 085126 (2013).
\bibitem{Ichikawa1958} Y. H. Ichikawa. Phys. Rev. \textbf{109}, 653 (1958).
\bibitem{Taft1965} E. A. Taft and H. R. Philipp, Phys. Rev. \textbf{138}, A197 (1965).
\bibitem{Nascimento1999} V. B. Nascimento, V. E. De Carvalho, R. Paniago, E. A. Soares, L. O. Ladeira, and H. D. Pfannes, Journal of Electron Spectroscopy and Related Phenomena \textbf{104}, 99 (1999).

%\bibitem{He2010} K. He, Y. Zhang, C. Chang, C. Song, L. Wang, X. Chen, J. Jia, Z. Fang, X. Dai, and W. Shan, Nat Phys \textbf{6}, 584 (2010).
%\bibitem{Hsieh2009} D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature \textbf{460}, 1101 (2009).
%\bibitem{Novoselov2005} K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
%\bibitem{Neto2009} A. Neto, F. Guinea, N. Peres, and K. S. Novoselov, Rev. Mod. Phys. 81, 109-162 (2009).
%\bibitem{Bostwick2006} A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, Nat. Phys. 3, 36 (2006).
%\bibitem{Geim2009} A. K. Geim, Science 324, 1530-1534 (2009).
%\bibitem{Drabble1958} J. Drabble and C. Goodman, J. Phys. Chem. Sol. 5, 142 (1958).
%\bibitem{Bhide1971} V. G. Bhide and B. A. Patki, J. Phys. Chem. Sol. 32, 1565 (1971).
%\bibitem{textbook} L. G. Jr. Wade, Organic Chemistry (7th Edition), Prentice Hall 2009.
%\bibitem{Nakajima1963} S. Nakajima, J. Phys. Chem. Sol. 24 (1963) 479-485
%\bibitem{Wallace1947} P. R. Wallace, Phys. Rev., 71, 622 (1947).
%\bibitem{Redaelli2001} A. Redaelli, M. Soncini, F. M. Montevecchi, J. Biomechanics, 34, 1607 (2001).
%\bibitem{Hsieh2009} D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature 460, 1101 (2009).
%\bibitem{Bryce1991} M. R. Bryce, Chem. Soc. Rev. 20, 355 (1991).
%\bibitem{Wu1957} C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, R. P. Hudson, Physical Review 105, 1413 (1957).
%\bibitem{Dai2008} X. Dai, T. L. Hughes, X. L. Qi, Z. Fang, and S. C. Zhang, Phys. Rev. B 77, 125319 (2008).
%\bibitem{Hsieh2008} D. Hsieh, D. Qian, L. Wray, Y. Q. Xia, Y. S. Hor, and R. J. Cava, Nature 452, 970 (2008).
%\bibitem{Hsieh2012} Timothy H. Hsieh, Hsin Lin, Junwei Liu, Wenhui Duan, Arun Bansil, and Liang Fu, Nat. Comm. 3, 982 (2012)
%\bibitem{Tanaka2012} Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T. Takahashi, K. Segawa, and Y. Ando, Nat. Phys.  8, 1 (2012). 
%\bibitem{Lenoir1998} B. Lenoir, A. Dauscher, M. Cassart, Y. I. Ravich, and H. Scherrer, J. Phys. Chem. Sol. 59, 129 (1998).

\end{thebibliography}