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Interfaces between buckling phases in Silicene: Ab initio density functional theory

calculations

Matheus P. Lima,1, ∗ A. Fazzio,1, † and Antonio J. R. da Silva1, 2, ‡

1Instituto de Fısica, Universidade de Sao Paulo, CP 66318, 05315-970, Sao Paulo, SP, Brazil.2Laboratorio Nacional de Luz Sıncrotron, CP 6192, 13083-970, Campinas, SP, Brazil.

The buckled structure of silicene leads to the possibility of new kinds of line defects that separateregions with reversed buckled phases. In the present work we show that these new grain boundarieshave very low formation energies, one order of magnitude smaller than grain boundaries in graphene.These defects are stable along different orientations, and they can all be differentiated by STMimages. All these defects present local dimerization between the Si atoms, with the formation ofπ-bonds. As a result, these defects are preferential adsorption sites when compared to the pristineregion. Thus, the combination of low formation energy and higher reactivity of these defects maybe cleverly used to design new nano-structures embedded in silicene.

PACS numbers: 61.72.-y,61.46.-w,31.15.E-

I. INTRODUCTION

The first in-lab observation of silicene1 has attractedthe attention of the community that studies new materi-als due to their similarities with graphene2. For this rea-son, efforts have been employed to understand and con-trol the growth of silicene3–8. These experimental mea-surements confirmed some theoretical predictions doneseveral years before9,10. Recent works have suggestedthat silicene is a potential candidate for applications innano-electronics11. However, having in mind device de-velopment and integration, it is fundamental to have adeep understanding of the influences that defects, doping,interactions with substrates, external fields and magneticmoments have on the properties of this material in orderto fully exploit the possibilities of silicene in nanotech-nology. Currently, these tasks represent great challengesand the necessary knowledge is still in its infancy.

When compared to graphene, silicene has a new andimportant ingredient: the presence of a buckled struc-ture. The understanding of the effects caused by thisextra feature is crucial to explore the full potential ofthis material. Few consequences of this buckling pat-tern have already been investigated; among them we canmention: i) the increase of spin-orbit coupling, enhanc-ing the Quantum Spin Hall Effect12; ii) the possibility totune the energy gap and the topological phase by the ap-plication of an external electric field perpendicular to thesheet13–15; and iii) increase of the surface reactivity16.

Due to the buckling, there are, in fact, two equiva-lent, energy degenerate geometric phases (α and β) insilicene. The α phase has a given atom shifted up andits first neighbors shifted down, whereas in the β phasethe shifts are reversed. A new possibility is thus the co-existence of both the α and β phases in the same sample.This situation may be created by different causes, suchas a peculiar growth mechanism or the interaction withthe substrate, since such interactions may pin down par-ticular phases in different regions of the sample.

In this article we investigate these new kinds of line de-

FIG. 1: (Color Online) Fully relaxed geometries for the (a)zz-1 interface in the zigzag direction, (b) zz-2 interface in thezigzag direction, (c) arm-1 interface in the armchair direction,(d) arm-2 interface in the armchair direction. The atoms arecolored according to their out-of-plane dislocation (y). Thegray rectangles emphasize the interface region. Below eachstructure is presented the bond length deviations from thepristine silicene (dbulk − d).

fects created at the interface between the α and β phasesin silicene. These buckling phase interfaces may occuralong different orientations (zigzag, armchair or interme-diated chiral geometries), and have a much lower energythan grain boundaries in graphene. The main character-istic of such defects is the presence of π bonds between theinterface atoms caused by an out-of-plane dislocation re-arrangement. For the zigzag direction, the formation of π

2

bonds is more clear, leading to a slightly lower formationenergy when compared to any other directions. We anal-yse in detail how these interfaces could be observed anddifferentiated in Scanning Tunneling Microscopy (STM)images. Furthermore, we show that the reactivity is en-hanced at these interface regions, exemplified by demon-strating that Au atoms have a lower binding energy whenadsorbed over these buckling phase interfaces in compar-ison with the pristine regions.

II. COMPUTATIONAL DETAILS

Our results were obtained with ab-initio density func-tional theory calculations. We used the SIESTA code17

within the Local Density Approximation18 (LDA) for theexchange correlation functional. Two dimensional (2D)periodic boundary conditions were employed in all cal-culations, with a grid of 50 × 50 k-points in the unitcell, a Double-ζ polarized basis, and a mesh cut-off of300 Ry to define the grid in the real space. A vacuumof 20 A is sufficient to avoid undesirable interactionsbetween the periodic images of silicene sheets. For thepristine system we obtain a lattice constant of 3.854 A,bond lengths of 2.281 A, and out-of-plane dislocationsof ±0.25 A. These parameters are consistent with earlierresults found in the literature.9,19,20. We also calculateda barrier of 35 meV/atom to revert the buckling phasein the whole system.The simulations of linear defects are done with two

complementary interfaces to reach the 2D periodicity inthe sheet plane. To generate fully relaxed geometries wefirst take a supercell with the pristine geometry. Sub-sequently, we revert the out-of-plane dislocation only ina certain region where we wish to observe the invertedphase. Subsequently, we perform a fully conjugate gra-dient relaxation using a force criterion of 0.015 eV/A toquench the forces in the whole system. As expected, themajor modifications will occur only close to the phaseinterface.

III. RESULTS

A. fully relaxed geometries

The fully relaxed geometries (which remain planar) forthe interfaces considered in this work are shown in Fig.1.21 We considered four kinds of interfaces, being twoalong the zigzag direction, and two along the armchair di-rection. In (a) we show the zz-1 interface along the zigzagdirection, which has pairs of neighboring atoms with thesame out-of-plane dislocation (y). In (b) we show the zz-2interface along the zigzag direction. In this case, there isa line of zigzagged atoms with y = 0. In (c) it is depictedthe arm-1 interface, which occurs along the armchair di-rection. Here, the interface atoms form a line with dimer-ized out-of-plane dislocations, which means two atoms

FIG. 2: (Color Online) DOS and LDOS for the α-β inter-faces: (a) zz-1 (b) zz-2, (c) arm-1 and (d) arm-2. For (a) and(b) the LDOS were calculated around the peaks localized at−0.814 eV and −0.918 eV below the Fermi energy, respec-tively. An energy window of ±0.025 eV was adopted. For(c) and (d) the LDOS were calculated integrating the stateswithin the energy range −0.9 eV ≤ E − Ef ≤ −0.4 eV .

with positive out-of-plane dislocations followed by twoatoms with negative out-of-plane dislocations. And, in(d) it is presented the arm-2 interface along the arm-chair direction. There are pairs of neighboring atomswith y = 0 at this interface. Below each structure wepresent the bond length variation from the pristine dis-tance. Although the interfacial atoms have a bond lengtharound 0.03A shorter than the pristine atoms, the inter-atomic distances are quickly reestablished as one movesaway from the line defect. Such analysis allows to con-clude that the width of these line defects is not more than7A.

B. Electronic Structure

Aiming to understand the influences of such line de-fects in the electronic structure of silicene, we calculatedthe Density of States (DOS) within an energy window of±1 eV around the Fermi energy. We present the resultsin Fig. 2. For the zz-1 interface, shown in (a), the DOS isvery similar to the pristine case, with exception of a sharppeak localized at −0.815 below the Fermi energy (Ef ).

3

The Local Density of States (LDOS) calculated withinan energy window of ±0.025 eV centered at this peak isshown below its DOS. This LDOS reveals that this peakis primarily composed by states with a π-bond signaturelocalized at the interface atoms. The narrow shape ofthis resonance peak indicates that the zz-1 defect levelshave a small dispersion as well as a small coupling to therest of the system. In Fig. 2 (b) we present the results forthe zz-2 interface. Again, only a sharp peak is present inthe DOS, however, its energy is lower (−0.918 eV ), andits amplitude is higher. Similarly to the zz-1 case, theLDOS centered at this peak allows us to conclude thatthese states have a π bond signature, and a small cou-pling to the rest of the system. Therefore, the creationof both kinds of zigzag defects leads to the formation ofπ states localized at the interface atoms.To better illuminate the understanding about the in-

fluence of such linear defects in the electronic structureof silicene, we modeled the system with an effective firstnearest-neighbour tight-binding (TB) Hamiltonian, givenby:

H =∑

<i,j>

−tc†icj +∑

i

Uic†i ci + h.c. (1)

c†i (ci) creates (annihilates) an electron at the site i, t isthe transfer integral, and Ui is the on-site energy.Considering this model, the main effect caused by the

buckling phase inversion are changes in the on-site en-ergy Ui only for the interfacial atoms due to the local re-hybridization occurring at them22. However, note thatthe disposition of these interfacial atoms is distinct foreach structure. In fig 3 (a) we mark with black circlesthe re-hybridized sites where Usite must be modified. Weadjust the tigh-binding parameters to better fit with theenergy bands calculated with DFT.In fig 3 (b) and (c) we compare the DFT and TB energy

bands for the zz-1 interface, respectivelly. Despite somediscrepancies far from Ef , the general behavior is essen-tially the same for both DFT and TB calculations. Forthis defect, the adjusted parameters are t = 1.1eV , andUi = 0.25t (Ui = 0 for bulk atoms). The positive valueof Ui brings the defect states to the top of the occupiedenergy bands. Analyzing the Hamiltonian (see appendixA), it is possible to note that the effective coupling of thedefect atoms highlighted in Fig. 3 (a) to the rest of thesystem have a form of −t(1+e±ikz ), and are thus negligi-ble close to kz = π. The modulus square of the TB wavefunction at kz = π is depicted in Fig. 3 (f), and can beclearly seen a strictly localized wave function. For othervalues of kz, the states progressively become more dis-persive as a consequence of the increasing of the effectivecoupling of the defect sites to the rest of the system. Thewave function for kz = π − 0.2 is also depicted in Fig. 3(f), and a non-null contribution of the neighbouring sitesinduced by the increase of the effective coupling term ispresent.We also compare, in Fig. 3 (d) and (e), the DFT and

TB results for the arm-1 defect, respectively. Here, the

FIG. 3: (a) Black circles highlight the re-hybridized siteswhere Ui must be modified for the zz-1, zz-2, arm-1 and arm-2defects. Energy bands for zz-1 structure calculated with (a)DFT and (b) our tight-binding model; (c) and (d) show simi-lar calculations for the arm-1 structure. Tight-binding wave-functions for the highest-occupied energy band (kx = 0) forthe (e) zz-1; and (f) arm-1 structures. Here, the tight-bindingcalculations consider a total of 80 sites (N = 80).

adjusted parameters are t = 1.1eV , and Ui = 0.15t.Again, there is a great overall agreement between theresults calculated with both methods, and due to thepositive value of Ui, the defect states once again lie atthe top of the highest occupied band. However, contraryto the zigzag direction, in the armchair direction the ef-fective coupling between the defect sites and the rest of

the system never vanishes for any ~k point. We illustratethis behavior in Fig. 3 (g), where we show the modulussquare of the wave function for kz = π and kz = π/2.It can be clearly seen that, even though the states aremore localized at the defect sites, a contribution at sitesof distant neighbours are present. The numerical TB re-

4

sults also indicate that the states at the central regionof the valence band (around k = π/2) are slightly morelocalized than the states close to the Γ and Z.Now, let us focus again in the results obtained with

calculations based on DFT. The DOS are presented inFig. 2 (c) and (d) for the arm-1 and arm-2 interfaces,respectively. The behavior is quite similar for both arm-chair interfaces: i) there are no peaks associated withany localized states; ii) The DOS are very similar to thepristine case, except for small oscillations caused by theinteraction between the line defects and its complemen-tary images (see appendix B). Since our TB model in-dicates that the defect states are mainly localized at thecentral part of the valence band, we calculated the LDOSfor an energy range from E −Ef = −0.9 to −0.4 eV , forboth arm-1 and arm-2 cases. In both cases, it is pos-sible to see a slightly greater contribution of the defectsites, even considering that the defect levels are weaklylocalized, and some bulk states are been included in thisLDOS.An important remark to be made is that, even though

these interfaces create defect states at the top of the high-est occupied energy band, close to Ef they are always de-localized. As a result, the modifications in the DOS veryclose to the Fermi level in all cases are negligible. Forexample, the Dirac Cone characteristic V-shape is fullymaintained. Even with the inclusion of the spin-orbitcoupling in the calculations, the differences between thepristine and the defective systems are negligible aroundEf . Therefore, these defects will be quasi-invisible tomeasurements that depend solely on the Fermi-surface.

C. Simulations of scanning tunneling microscope

images

We show in Fig. 4 that it is possible to differenti-ate the various interfaces with scanning tunneling mi-croscopy (STM) images. We used the Tersoff-Ramanntheory23, with a bias voltage of 1.0 V , considering oc-cupied states and the constant current model. STM im-ages were commonly used to experimentally investigatesilicene1,3–8. Similarly to the experiments, our simula-tions show a clear signature of the buckling pattern atthe pristine region: the signal presents high spots over theup-shifted atoms, low spots over the down shifted atoms,and absence of signal at the center of the hexagons24. Forthe buckling interfaces, the out-of-plane displacementscreate distinct patterns depending on the interface sym-metry, as shown in detail in Fig. 4. In (a-1) we presentthe STM image for the zz-1 interface. The position ofthe interface atoms are indicated by black circles. Inthis picture, there is a clear line of low signals along thedimerised central interface atoms because they are down-shifted with respect to the STM tip. If we consider theother possible orientation of the STM tip, these sameatoms would now be up-shifted, and we would obtain aline of high signals, as shown in Fig. 4 (a-2). An inter-

FIG. 4: (Color Online) Images of the STM simulations forthe buckling phase interfaces. (a-1) and (a-2) show the zz-1interface; (b-1) and (b-2) show the zz-2 interface; (c) showsthe arm-1 interface; and (d) shows the arm-2 interface. Thefilled (non-filled) circles represent Si atoms with up (down)shifts. The lines are guides to the eye. The scale bar representthe tip height in A.

TABLE I: Formation Energy per unity length (Uform inmeV/A) for the line defects considered in this work.

zz-1 zz-2 arm-1 arm-2 θ = 19.11◦

Uform 17.5 17.9 20.9 20.3 22.8

esting feature of this image is the clear signal of π bondsbetween the central atoms of the defect. For the zz-2interface there are also two possible images, dependingon the tip orientation. These images are shown in Fig. 4(b-1) and (b-2). Here, the source of the differences is thefact that all first-neighbour atoms to the interface atomshave the same out-of-plane dislocation. Well defined πbonds can also be identified between interface atoms andits first neighbours via STM images.

For the interfaces in the armchair direction, differentlyfrom those in the zigzag direction, the STM images do notdepend on the tip orientation, as shown in Fig. 4 (c) and(d) for the arm-1 and arm-2 interfaces, respectively. Inboth cases, the out-of-plane rearrangement can be clearlyinferred from STM images, and the π bonds can also bedetected. Thus, we have shown that it is experimentallypossible to use STM images to identify and differentiateall these distinct interfaces.

5

D. Formation energies

The formation energies per unity length (Uform) for allthe defects are presented in table I. The interfaces alongthe zigzag direction have the lowest formation energies,because the π bonds are more clearly formed, leading toa slightly larger energy gain. Otherwise, the two zigzagand the two armchair interfaces have formation energiesquite similar among themselves. Besides the zigzag andarmchair directions, we also consider one intermediatecase, with an angle θ = 19.11◦ with the zigzag direction.The fully relaxed geometry is shown in Fig. 5, where canbe seen that the rearrangement of the out-of-plane dislo-cations are more complex, leading to a higher formationenergy due to the increase of the elastic contribution. Weexpect the same behavior for other chiral angles.

FIG. 5: (Color Online) Geometry for a chiral line defect. Theangle between the defect direction and the pure zigzag di-rection is 19.11◦. Here, the atoms are coloured according toits out-of-plane dislocation. The arrow represent the latticevector along the defect direction.

The values of Uform presented in table I indicate thatthe defects in the zigzag direction will be more easily cre-ated at low temperatures. Indeed, a recent work by LanChen et al. reports a spontaneous symmetry breakingphase transition in silicene over the Ag(111) substrate27.They have shown, at low temperatures, a spontaneouscreation of a 2D grid where the domains are triangleswith alternating buckling phases separated by interfacesin the zigzag direction. Although the authors did not dis-cuss the structural details of the triangles borders, it canbe inferred from their STM images that they are the zz-2interfaces described above. It is important to note thatthe particular triangular structure observed experimen-tally will most likely depend on the interaction betweenthe silicene and the Ag substrate. This indicates thaton the one hand the defects that we propose are verylikely to be observed, but on the other hand the par-ticular configuration that they will have on the silicenesheet will depend on experimental details such as sub-strate configuration and growth temperature.

It is important to stress that the values presented intable I are one order of magnitude lower when comparedto either grain boundaries or dislocation linear defectsin graphene25,26. This is a consequence of to fact that

FIG. 6: (Color Online) Schematic representation of the moststable adsorption site of Au over the (a) zz-1, (b) zz-2, (c)arm-1, and (d) arm-2 interfaces.

grain boundaries in graphene always involve a chemicalbond reconstruction, and the hexagons are converted inother polygons, like pentagons, heptagons, etc. This kindof reconstruction highly increases the formation energyof the linear defects in graphene. On the other hand,as reported in this paper, the linear defects in silicenemay be generated by another mechanism: an out-of-planedislocation rearrangement, leading to lower values of theformation energy and, thus, a defect more likely to becreated.

E. Adsorption of Gold atoms

Another interesting point related to these line defectsis that they might be preferential adsorption sites. Toinvestigate this question, we performed proof-of-conceptsimulations with a single gold atom adsorbed over thesystem. In these simulations we used surpercells in whichthe Au atoms are separated by more than 18 A fromits periodic image. This is necessary to sufficiently de-crease the interaction between the Au atom and its im-ages. We choose Au atoms because there are previousexperiments of single atom adsorption over the Si(100)surface28, which bear some similarities with the line de-fects we propose due to the presence of dimers. Further-more, gold is frequently used in nano-devices.Considering a single gold atom over the pristine sil-

icene, we found a binding energy (Eb) of -3.50 eV whenadsorbed at the hollow site. This adsorption site is ener-getically favourable by 0.38 eV and 0.66 eV when com-pared to the top and bridge configurations, respectively.Corroborating previous works, this binding energy indi-

6

cates a much stronger bond in silicene when comparedto graphene16,29–32. To investigate the adsorption of Auover the buckling interfaces we considered several adsorp-tion sites, and the energetically most favourable ones foreach one of the four interfaces are shown in Fig. 6. Inthe same figure we also present Eb for each one of theseconfigurations. Confirming our expectations, the resultsshow that all these defects are more reactive than thepristine region, indicating that they will be preferentialabsorption sites. This opens up a variety of possibilitiesfor nano-engineering, where via adsorption of particularatoms or molecules at these sites on can taylor the prop-erties of 1D channels embedded on the silicene sheet.

IV. CONCLUSIONS

In conclusion, we have shown that due to the bucklingstructure of silicene, it is possible to have a new kindof low energy grain boundary associated with the rever-sal of the buckling phase, contrary to graphene, wherethe grain boundary defects have always high energy. Inparticular, we have investigated in detail these bucklingphase interfaces, showing that: i) modifications in theDensity of States will appear far from the Fermi energy;ii) The formation energy of these interfaces are very low,of the order of (kBT300K)/A. These values are approxi-mately one order of magnitude lower than grain bound-aries in graphene; iii) These structures can be experimen-tally identified by STM images; iv) These interfaces arepreferential adsorption sites when compared to the pris-tine region. Therefore, these interfaces are important andmost likely common defects in silicene, and may be thusused to control the adsorption of atoms and molecules,which could lead to many possibilities in molecular engi-neering.

V. ACKNOWLEDGEMENTS

We would like to thank the Conselho Nacional deDesenvolvimento Cientıfico e Tecnologico/Institutos Na-cionais de Ciencia e Tecnologia do Brasil (CNPq/INCT),the Coordenacao de Aperfeicoamento de Pessoal de NıvelSuperior (CAPES), and the Fundacao de Amparo aPesquisa do Estado de Sao Paulo (FAPESP).

Appendix A: Hamiltonian matrix elements for the

tight-binding model of the buckling phase interfaces.

In this appendix we explicitly show the matrix ele-ments for the effective model used to study the bucklingphase inversion in silicene. We first construct the Hamil-tonian for the pristine silicene with a large supercell, andwe then simulate the buckling inversion line defects bychanging the on-site energy only for specific sites.

For the pristine silicene, we construct a simple nearestneighbour tight-binding Hamiltonian with one effectiveorbital per site, with rectangular supercells having thesmallest possible size along the defect direction (z), andan arbitrary size along the lateral direction (x). For bothzigzag and armchair directions we used building blockswith 4 sites, as shown in Fig. 7 (b) and (c), respectively.Thus, the total pristine Hamiltonian can be written as:

H(kx, ky) =

HD VD 0 0 · · · VL

V †D HD VD 0 · · · 0

0 V †D HD VD 0

.... . .

V †L 0 · · · 0 V †

D HD

. (A1)

Here, HD is the building block Hamiltonian, VD is thecoupling between neighbouring building blocks, and VL

is the coupling between neighbouring supercells. A dia-grammatic representation of these interactions is shownin Fig. 7 (a). Since the building blocks have 4 sites, allthese are 4 × 4 matrices. Particularly, for the armchairdirection these terms are given by:

HD = −t

0 e−ikz 1 0

eikz 0 0 1

1 0 0 1

0 1 1 0

; (A2)

VD = −t

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

; (A3)

VL = −t

0 0 e−ikx 0

0 0 0 eikx

0 0 0 0

0 0 0 0

. (A4)

Where, t is the transfer integral, kz (kx) is the crystalline

FIG. 7: In (a) we present a diagrammatic representation ofthe Hamiltonian coupling terms. In (b) and (c) the buildingblocks for the zigzag and armchair structures are enclosedwith dashed lines.

7

momentum along (perpendicular to) the defect direction.For the zigzag direction, the Hamiltonian building blockterms are written as:

HD = −t

0 ck 0 0

c∗k 0 1 0

0 1 0 c∗k0 0 ck 0

; (A5)

VD = −t

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

; (A6)

VL = −t

0 0 0 e−ikx

0 0 0 0

0 0 0 0

0 0 0 0

. (A7)

Here, ck =(

1 + eikz

)

is an effective coupling term. Notethat ck → 0 when kz → π, as discussed in the sectionIII B.The re-hybridization of the defect sites leads to changes

in their on-site energy. Therefore, to simulate a singleline defect it is necessary to change the on-site energies(considered zero for the pristine sites) for a single buildingblock, as schematically shown in Fig. 7 (a).

Appendix B: Oscillations in the Density of States of

defects in the armchair direction

The Density of States (DOS) of these buckled line de-fects could present some oscillations when compared tothe silicene without any defect. This feature occurs withboth DFT and tight-binding calculations. This behav-ior is illustrated in Fig. 8, where we present the DOSfor the arm-2 defect. In this figure, we compare theDOS calculated with a small supercell having 40 sites(same defect-defect distance present in the ab initio cal-culations) with a DOS obtained with a system 10 timesbigger (N=400). In the latter system the line defects areseparated by 48nm from its periodic image, and in factthere is no interaction between them. As a consequence,the wave function presented in Fig. 8 (b) decays to zerobefore having any overlapping with the wave function ofthe neighbour line defect. In this case the DOS is almostindistinguishable when compared to the pristine case. Onthe other hand, for the smaller supercell (N=40), the dis-tance between the defect and its periodic image is 3.8nm.Thus, there is an overlap between the wave functions lo-calized at the defect and its periodic image, leading tothe appearance of several small spurious oscillations inthe DOS. It is important to stress that this interactiondoes not affect any of our main conclusions.

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FIG. 8: (Color Online) (a) Density of States (DOS) of silicenewith the arm-2 defect varying the distance between the linedefect and its lateral image. For N=40 the distance is 3.8nm,whereas for N=400 the distance is 38nm. We normalize theabsolute value or the DOS by the respective number of sites ofthe system. The inset shows a zoom of the DOS. Also consid-ering the arm-2 defect we show in (b) the modulus square ofthe wave function of the valence band at kx = 0, and kz = π.Here, the system has 400 sites (N=400). The defect lies inthe sites 201-204.

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