Tunable Spin-Dependent Properties of Zigzag Silicene Nanoribbons

Nam B. Le,1,2 Tran Doan Huan,3 and Lilia M. Woods1,*1Department of Physics, University of South Florida, 4202 East Fowler Avenue,

Tampa, Florida 33620, USA2Institute of Engineering Physics, Hanoi University of Science and Technology,

1 Dai Co Viet, Hai Ba Trung, Hanoi 10000, Vietnam3Institute of Materials Science, University of Connecticut, 97 North Eagleville Road,

Unit 3136, Storrs, Connecticut 06269-3136, USA(Received 30 January 2014; revised manuscript received 10 April 2014; published 11 June 2014)

Silicene zigzag nanoribbons are studied using ab initio simulation methods. We find novel structure-property relations influenced by several factors, such as the magnitude of the width, spin polarization, spin-orbit coupling, and extended topological defects. It is obtained that while defect-free silicene nanoribbonsexperience antiferromagnetic-ferromagnetic transition as a function of the width, all defective nanoribbonsare ferromagnets. At the same time, the spin-orbit coupling role is significant as it leads to spin-dependentenergy gaps in the electronic structure. The origin of edged spin polarization is also studied in terms of thebalance between the exchange correlation and kinetic energy contributions. The uncovered unique spin-dependent properties may be useful for the application of silicene nanoribbons in spintronic applications.

DOI: 10.1103/PhysRevApplied.1.054002

I. INTRODUCTION

Two-dimensional graphene, a monolayer of honeycomblattice of C atoms [1,2], and its quasi-one-dimensionalderivatives, such as nanoribbons [3–5], have been studiedextensively due to their extraordinary properties. Eventhough graphene and its nanoribbons may have desiredcharacteristics for high speed electronics applications [6],their integration into Si-based electronic devices has beenchallenging. There has been considerable interest in seek-ing analogous materials, which possess the extraordinaryproperties of graphene, but they are more compatible withexisting electronic devices.Silicene, composed of Si atoms in a 2D hexagonal

lattice, is a great candidate, which may be better suitedfor practical electronic applications [7]. Other Si nano-structures have also generated much scientific interest[8–21]. Theoretical studies and density functional theory(DFT) calculations have shown that this Si-based grapheneanalog and silicene nanoribbons (SiNRs) possess not onlygraphenelike properties, but also other unique character-istics [8–14]. Recent experimental advances towards syn-thesis [15–21] have opened up new perspectives forapplications. Although energy bands in silicene and gra-phene have similar dispersions at characteristic points ofthe Brillouin zone, the massive nature of the Dirac carriershas made silicene suitable for realizing quantum Hall spineffects and topological insulator features [13,14,22].Although silicene has not been synthesized in a free-

standing form yet, it has been successfully grown on Ag

substrates by depositing Si atoms onto Ag(110) [15–17]and Ag(111) [18–20] surfaces. It can also be formedthrough surface segregation on ZrB2 thin films grown onSi wafers [21]. The honeycomb structure of silicene isidentified using scanning tunneling microscopy [15,16,18],in combination with low-energy electron diffraction [19] orangular-resolved photoemission spectroscopy [20].While C atoms are in a 2D configuration, characterized by

an sp2 orbital hybridization, Si atoms prefer a low-buckledhoneycomb structure [8,9,12,19]. As a result, some sp3

hybridization can be found in silicene. DFT calculationsreveal that the π and π� states around the Fermi level ofsilicene are graphenelike; namely, their energy bands arelinear at theDiracpoints of thehexagonalBrillouin zone [12].Silicene, however, has a large spin-orbit coupling (SOC)[14], strengthened by the buckled lattice. The large SOC canbe used to control the massive silicene Dirac electron, whichcan lead to prominent topological phase transitions [23].Despite the predictions for many exotic features in

silicene systems, their structure-property relationships arenot fully known and understood yet. The quasi-1D nature,edges, and strong SOC are of particular interest. In thispaper, we present computational studies using DFT meth-ods for zigzag SiNRs exploring how their characteristicsare influenced by the size of the width, magnetic orienta-tion, and the presence of extended topological defects. Itturns out that the SOC plays a critical role for the electronicstructure and that there is a width-dependent antiferromag-netic-ferromagnetic transition. In addition, extended topo-logical defects, containing pentagonal and octagonal ringsembedded in the hexagonal buckled structure, result innovel spin-polarized properties.*lmwoods@usf.edu

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II. METHOD OF CALCULATIONS

To investigate the SiNR properties, first-principles sim-ulations using the Vienna ab initio simulation package(VASP) [24] are performed. In this state-of-the-art code,Kohn-Sham equations are solved by the projector-augmented-wave (PAW) method [25] with a plane-wavebasis set and periodic boundary conditions. Exchange-correlation energy is calculated using the Perdew-Burke-Ernzerhof [26] functional. For all studied structures, anenergy cutoff of 500 eV and 1 × 1 × 11 automatic-meshk-point sampling of the Brillouin zone are used. Ionicrelaxation is obtained with force and total energy differencecriteria of 5 × 10−3 eV=Å and 10−6 eV, respectively.The SOC has a relatively large contribution to the

electronic structure due to the buckled structure and it isresponsible for the massive Dirac carriers at the K and K0points [14]. For the systems considered here, the SOC istaken into account via noncollinear magnetic calculations.The implementation in VASP [24] uses valence electronstaken into account through the second-variation methodand scalar relativistic eigenfunctions. Although this schemebrings a substantial additional computational cost, the SOCis necessary to properly describe the electronic structureproperties of the nanoribbons.

III. STRUCTURE OF SILICENE NANORIBBONS

SiNRs with widths in the 24–112 Å range (N ¼ 8–12zigzag lines), and hydrogen saturated zigzag edges arestudied. The hydrogen ends have a small effect on theintrinsic properties of the ribbons, which makes it useful tostudy the intrinsic properties of the silicene zigzag edge.However, other compounds can also be used to passivatethe highly reactive Si nanoribbon ends, as past research forgraphene nanoribbons has shown [27]. The supercells areconstructed to have 20 Å vacuum separation betweenribbons in neighboring cells. The width of each nanoribboncorresponds to a specific number of zigzag Si lines, asshown in Fig. 1(a). This nomenclature is similar to the oneused for graphene nanoribbons (GNRs) [5]. SiNRs withextended topological defects composed of alternatingpentagon-octagon pairs are also investigated. Such defectshave been realized experimentally and studied in grapheneand GNRs [5,28]. They can appear in different locationswith respect to the edges. Here we study N ¼ 8–30 SiNRswith a defect line located in the middle of the ribbons[Fig. 1(b)], and a SiNR with N ¼ 20 and with anasymmetrically positioned defect line specified by differentN1, N2 zigzag lines [Fig. 1(c)].The SiNRs have a buckled lattice. After relaxation, the

degree of buckling of 0.47 Å above the plane is found to bethe same for all studied nanoribbons. However, the Siatoms located on the defect line are protruded by 0.53 Åabove the plane. Comparing perfect and defective SiNRswith the same number of zigzag lines shows that the

topological defect results in an approximately 2 Å largerwidth due to the extra Si line at the defect. SiNRs withflatlike structure are also simulated, however, their totalenergies are higher as compared to the buckled ones. Forexample, we determine that the energy difference for theground state is approximately 0.39 eV for N ¼ 8 andapproximately 1.57 eV for N ¼ 32 ribbons in favor ofthe buckled lattice.For all structures we performed nonmagnetic (NM),

antiferromagnetic (AFM), and ferromagnetic (FM) calcu-lations. It turns out that all ribbons are magnetically orderedand the width is crucial for the particular magnetic state.For thinner ribbons (N < 23) the AFM state is the moststable one, while for thicker ribbons the ground state is FM.Figure 2(a) shows that similar to zigzag GNRs [5,29], theedge atoms bear the largest spin polarization and themagnitude of the atomic magnetic moments decreasesquickly towards the center of the ribbon. All SiNRs withextended topological defects are found to be ferromagnets

FIG. 1. Relaxed SiNR with N ¼ 20 zigzag lines with (a) nodefect present, (b) extended defect positioned in the center,(c) extended defect located asymmetrically. Characteristic Si-Sidistances and degrees of buckling are denoted. Atoms residing onthe defect line are shown in yellow. Zigzag edges are saturatedwith H atoms (shown in red). The infinite axial direction isdenoted as y.

(a) (b)

(c)

FIG. 2. Atomically resolved spin polarization for (a) perfectSiNR with N ¼ 8 zigzag lines, (b) SiNR with a defect line in thecenter, (c) SiNR with asymmetrically located defect line.

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with majority edge spins and minority spins at the zigzaglines surrounding the defect—Figs. 2(b) and 2(c). The totalmagnetization is M ≈ 0.78μB for all defect-free SiNR,and M ≈ 0.83μB for defective ribbons (μB is the Bohrmagneton).The origin of the spin polarization in zigzag nanoribbons

has been attributed to localized edge states and their effecton the electronic structure [30,31]. The existence of aflatband in some of the Brillouin zone and associatedelectronic correlations leads to a FM polarization of eachedge. It has been suggested that the flatband magnetism isnot specific to C atoms. It also occurs in edges with B and Natoms. Our calculations also show the existence of local-ized edge states in SiNRs, which is a testament that zigzagmagnetism can exist even in a staggered honeycomb latticewith significant SOC.The overall magnetic state of the ribbons can be

understood by investigating the energy difference ΔE ¼EAFM − EFM [32,33] and the contributions making upEAFM;FM. For the DFT calculations here, the kinetic andexchange-correlation contributions to the AFM and FMenergies are determining factors for ΔE.Let us first study ΔE as a function of the SiNR widths.

Figure 3(a) shows that jΔEj decreases nonlinearly as N isincreased and for N ¼ 23 the total AFM energy is N ≈1 meV lower than the total FM energy. Although thisdifference is small, the results did not change by increasingthe Brillouin zone sampling, energy cutoff, or otherprecision-related criteria. For the ribbon with N ¼ 24,ΔE ≈ 41 meV signaling the AFM-to-FM transition, withΔE being almost constant for all N > 24 ribbons. The insetof Fig. 3(a) shows that for N ≤ 23, the nonmagnetic state isthe least preferred as it has the highest total energy, while

for N ≥ 24, the AFM and nonmagnetic total energiesbecome practically the same.The energy difference ΔE shown in Fig. 3(a) is a key

component in understanding the spin polarization of thenanoribbons. It is directly related to the interedge inter-action, which determines the spin alignment at the edges[32]. The DFT calculations imply that ΔE ¼ ΔT þ ΔEXC,where ΔT is the kinetic (band) energy and ΔEXC isthe exchange-correlation contributions. The electrostaticenergies, which also contribute to EAFM;FM, are canceledin ΔE since they are identical. Both AFM and FMstates sacrifice kinetic energy to gain interaction energy.Figure 3(b) shows that both ΔT and ΔEXC are small andoscillatorylike functions centered at zero for 14 < N < 23.Nevertheless, the balance is such that the preferred stateis AFM. This ΔE vs N behavior is very similar to thatreported for graphene nanoribbons. ΔE decreasing as thewidth is increased but still favoring the AFM state has beenobtained by mean field Hubbard model theory and ab initiocalculations for graphene nanoribbons [32,33]. Comparingour results with the ones for the graphene systems showsthe decisive role of the exchange correlation for the AFM.As N is increased, the interedge interference decreases

and much more kinetic energy is sacrificed towardsexchange correlation for the FM state [Fig. 3(b)]. Themajority of this energy corresponds to the ka > 0.3π(a, lattice vector length) region in the Brillouin zone aswe further show. It is interesting to see that the oscillatorybehavior is still present for N > 24, but ΔT > 0 andΔEXC < 0. The competition is such that the FM alignmentis preferred with a rather large ΔE ≈ 40 meV indicatingthat the FM state is quite stable for wider SiNRs. Thedominance of exchange correlation for the FM orientationis unlike the case of graphene nanoribbons. For thesesystems no AFM-FM transition as a function of the width isfound since the exchange interaction dominates the AFMorientation for all ribbons.

IV. EDGES AND EXTENDED DEFECTS IN SiNRs

We also calculate the energy band structure to furtherunderstand the magnetic edge states. Figure 3(c) shows thatthe energy gap between the highest valence and lowestconduction bands smoothly decreases as N is increased forthe AFM SiNRs (N ≤ 23). Looking at the energy bandstructure, displayed in Fig. 4, one finds that for N ¼ 8 andN ¼ 23 (AFM) Eg is located at ka ¼ 0.36π, betweendouble degenerate bands for each spin. The highest valenceand the lowest conduction bands are of π bonding and π�antibonding character, respectively. Flatbands appear inka ¼ 0.36π, but they are far away from EF because of thesignificant interedge interference due to the small width.For N ¼ 24, an AFM-FM transition occurs. The flat-

bands are brought closer to EF and much more kineticenergy is sacrificed for exchange correlation [Fig. 3(b)] forthe FM state. In contrast to the AFM state, the SOC is found

FIG. 3. (a) Total energy difference ΔE ¼ EAFM − EFM. Theinset shows EFM − ENM (green) and EAFM − ENM (blue),(b) kinetic ΔT ¼ TAFM − TFM and exchange correlationΔEXC ¼ EAFM

XC − EFMXC energy differences, (c) energy gap Eg,

(d) majority and minority energy gaps as a function of N fordefect-free SiNRs.

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to impact the energy bands significantly for N ≥ 24.Figure 4 displays the bands for the N ¼ 24 ribbon withand without SOC (SOC-off). If the spin-orbit interaction isnot taken into account, the π and π� bands cross at EF asthey are partly occupied and partly unoccupied. Theinclusion of the SOC results in opening of a small gap.This transforms the majority (minority) band into a fullyunoccupied (occupied) state. Similar results are found forall studied FM ribbons.The ferromagnetism and the SiNR width determine not

only the position of the minority and majority bands at EF,but also those of the next levels. The energy gap betweenthe majority highest valence and majority lowest conduc-tion bands is different as compared to the one for theminority highest valence and minority lowest conductionbands. Figure 3(d) shows that Emino

g ¼ 61.5 meV(Emajo

g ¼ 99.2 meV) is the highest (lowest) for N ¼ 30.Actually, for ribbons with N ≥ 29, these energy gapswithin each spin species become indirect as can be seenfrom Fig. 4.The SOC contribution to the total energy is calculated

via ΔESOC ¼ jEtotDFTþSOC − Etot

DFTj, where EtotDFTþSOC and

EtotDFT are the total energies with and without the SOC,

respectively. We find that ΔESOC is nearly unchangedfor the different ribbon. ΔESOC ¼ 0.17 meV=atom forthe defect-free structures. Once the extended defect isembedded, the strength of the SOC energy is ΔESOC ¼0.35 meV=atom. Doubling the SOC magnitude is relatedto the geometrical role of the extended line defect, whichessentially joins two nanoribbons with intact zigzagedges. If two defect lines are introduced in the ribbon,one obtains magnitude of the SOC for three ribbons.The energy band structure of the SiNRs exhibits further

interesting properties directly influenced by the magnitude

of the width, the edges, and the extended topologicaldefects. As discussed earlier, all defective ribbons areferromagnets regardless of the location of the defect.Several cases are shown in Fig. 4. We find that there isalways a majority band crossing the Fermi level, which isindicative of the metallic transport behavior of the majorityspins. At the same time, large flat portions of the minoritybands are found in the vicinity of EF. In some cases, theseflat levels lie exactly on EF (N1 ¼ N2 ¼ 4). In other casesthey are above and below the Fermi level (N1 ¼ N2 ¼ 10;N1 ¼ 6, N2 ¼ 14) with a finite indirect gap between thehighest valence and lowest conduction minority bands. Forexample, the calculated minority gap is 56.8 meV for N1 ¼N2 ¼ 10 and 54.0 meV for N1 ¼ 6, N2 ¼ 14, showing thatit is possible to obtain half-metallic characteristics in SiNRswith extended topological defects, where the majoritycarriers are conductors, while the minority carriers aresemiconducting.The nature of the electronic structure can further be

analyzed by considering the density of states (DOS). InFig. 5, we show the orbital and site-projected DOSwhen thedefect line is symmetrically located with respect to edges(N1 ¼ N2 ¼ 4). One notes that the majority DOS is muchlarger as compared to the minority DOS for the displayedregion. The contribution at EF for the majority DOS isattributed to the metallic dispersive band crossing the Fermilevel [Fig. 4(a)]. TheminorityDOS experiences a dip almosttouching the Fermi level, which is due to the small flatportion of the highest valence band close to theX point. Theflat region of the lowest conduction minority band for0 < ka < 0.19π is 8 meV contributing to the peakedDOS above EF.Figure 5(a) also indicates that the pz orbitals have a

significant contribution to the majority DOS at EF. The roleof the other orbitals is small, but not negligible. One findsthat while the pz contribution is large for E ∈ ð−3;−2Þ eV,

FIG. 5. Orbital-projected DOS and site-projected DOS for theFM defective SiNR with N1 ¼ N2 ¼ 4 symmetrically positionedextended defect.

FIG. 4. Energy band structure for several perfect and defectiveSiNRs. Energy bands for both spin orientations are shown.

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the px and py are more prominent in the deeper valenceregion of E ∈ ð−5;−1Þ eV. The s states are characteristicfor energies below −5 eV. All states contribute signifi-cantly in the conduction region for E > 1 eV. Wefurther note that the minority DOS is almost completelydetermined by the pz states and the sp3 hybridizationaround EF happens entirely for the majority carriers. Also,due to the much larger DOS around EF, EF the transport inthese systems will mainly be due to the majority carriers.Since the spin polarization is rather localized around the

edges and extended defect line, it is important to examinethe site-projected DOS to determine the contribution fromindividual atoms. Figure 5(b) shows the site-projected DOSfor those atoms that have the most contribution. It is foundthat the majority DOS is composed mainly of the zigzagedge atoms (1) and the atoms forming the defect line (8)and (9). One notes that although atoms (8) are in the middlethey have the same geometrical (zigzag) disposition asatoms (1). The minority DOS is much smaller and the peakat EF is attributed to atoms (8), while the edge atoms (1) areresponsible for the two peaks in the valence and conductionregions.

V. SUMMARY

In summary, comparing these results for zigzagSiNRs with previous studies for zigzag GNRs [5] showsthat besides the common properties, SiNRs exhibitunique structure-property relations. While all zigzagGNRs are found to be AFM with monotonously decreas-ing energy gap as a function of the increasing width [3,5],SiNRs experience AFM-FM transition, which has adramatic effect on the energy gap. The AFM state ofthe silicene ribbons is very similar to the one for GNRs,although the SiNR Eg decays faster as the ribbonsbecome wider. It turns out that the SOC does not affectthe electronic structure properties of the graphene rib-bons, however, this relativistic effect is much morepronounced for the staggered silicene lattice giving themassive nature of the Dirac electrons. The SOC isresponsible for a spin-dependent energy gap for theFM SiNRs. Differences and similarities in the propertiesare found when an extended topological defect isincluded in the GNR [5] and SiNR structures. Whilesuch a defect does not change the AFM ordering inGNRs, it makes all SiNRs ferromagnets. In some cases itis also possible to obtain half-metallic behavior depend-ing on the location of the defect. Furthermore, thecomposition of DOS for both types of ribbons is similarbut with more pronounced spin-dependent sp3 hybridi-zation for the SiNRs.The systematic ab initio studies of the structure-property

relations show that in order to understand and describesilicene nanoribbons one needs to take into account notonly the staggered hexagonal lattice, but also the spin-orbitcoupling and magnetic orientation. The unique properties

of SiNRs indicate that such nanoribbons may find appli-cations in spintronics or for topological insulator transi-tions, where SOC in Dirac electrons is necessary.

ACKNOWLEDGMENTS

Financial support from the U.S. Department ofEnergy under Contract No. DE-FG02-06ER46297 isacknowledged. We also acknowledge the use of theUniversity of South Florida Research Computing facilities.Discussions with Motohiko Ezawa and Rampi Ramprasadare acknowledged.

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