Strain-Engineering of Band Gaps in Piezoelectric Boron NitrideNanoribbonsJingshan Qi,†,‡ Xiaofeng Qian,*,§ Liang Qi,§,∥ Ji Feng,⊥ Daning Shi,‡ and Ju Li*,§,∥

†College of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, China‡Department of Applied Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China§Department of Nuclear Science and Engineering and Department of Materials Science and Engineering, Massachusetts Institute ofTechnology, Cambridge, Massachusetts 02139, United States∥State Key Laboratory for Mechanical Behavior of Materials and Frontier Institute of Science and Technology, Xi’an JiaotongUniversity, Xi’an, 710049, China⊥International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China

*S Supporting Information

ABSTRACT: Two-dimensional atomic sheets such as gra-phene and boron nitride monolayers represent a new class ofnanostructured materials for a variety of applications.However, the intrinsic electronic structure of graphene andh-BN atomic sheets limits their direct application in electronicdevices. By first-principles density functional theory calcu-lations we demonstrate that band gap of zigzag BNnanoribbons can be significantly tuned under uniaxial tensilestrain. The unexpected sensitivity of band gap results fromreduced orbital hybridization upon elastic strain. Furthermore,sizable dipole moment and piezoelectric effect are found inthese ribbons owing to structural asymmetry and hydrogen passivation. This will offer new opportunities to optimize two-dimensional nanoribbons for applications such as electronic, piezoelectric, photovoltaic, and opto-electronic devices.

KEYWORDS: Elastic strain engineering, ESE, nanoribbon, piezoelectricity

Two-dimensional materials, such as graphene1−5 andhexagonal boron nitride (h-BN) monolayers,6 are

attractive for their fascinating physics and potential applicationsin next-generation electronics. Single atomic layer thicknessmakes it possible to fabricate devices with channels that areextremely thin. This will allow field-effect transistors (FETs) tobe scaled to shorter channel heights and higher speeds withoutencountering the adverse short-channel effects that restrict theperformance of existing devices.1 However, we must overcomea few fundamental difficulties. First, FETs with large-areagraphene channels cannot be switched off because it is asemimetal with zero band gap. To give a large on−off ratio inthe conductance, required for electronic logic, it is necessary forthem to possess a significant electronic energy gap Eg. The ball-park value for Eg should be around that of silicon, 1.1 eV. Forthis purpose, energy gaps were opened by cutting graphene intoultranarrow nanoribbons (GNRs).7,8 However, their band gapsdepend strongly on the types of the edges and widths7 of GNRsand therefore encounter serious issues with fabrication becauseof the small widths (less than 5 nm) required for getting a bigenough band gap. On the other hand, monolayer h-BN and h-BN nanoribbons (BNNRs) with edges passivated by hydrogenare all wide-gap semiconductors,9 making them unsuitable asFET channels at room temperature. Recently, another

strategy10 was experimentally demonstrated by hybridizing h-BN and graphene (h-BNC) domains. However, their randomdistribution may introduce difficulties in fabricating nanoscaleh-BNC films with uniform electronic properties. It would,therefore, be highly desirable to have capabilities tocontinuously control the band gap of two-dimensional materialsfor building future electronic and optoelectronic devices.One promising route toward the continuously tunable band

gap is elastic strain engineering (ESE),11 which allows one tocontrol the electronic properties of materials by simply applyingan elastic strain. Although large elastic strain rarely exists inbulk materials since they are easily relaxed by dislocationplasticity or fracture, plasticity and fracture are greatly delayedin nanostructures (“smaller is stronger”), allowing for a muchlarger dynamical range of applying elastic strain reversibly.11

For example, experimentally measured breaking strength ofgraphene monolayer was as high as 13% of ideal strength, whilethat of bulk graphite seldom reaches 0.1%.12,13 Unfortunately,uniaxial elastic strain will not open an energy gap for infinitegraphene within 20% strain.14−17 In the case of zigzag GNRs,

Received: October 11, 2011Revised: February 3, 2012Published: February 24, 2012

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uniaxial strain also has little influence on the bandstructure.18−22 Although the energy gaps of armchair GNRsare predicted to change as a function of the applied uniaxialstrain, the tunable window of the energy gap becomes narrowerwhen the width of ribbon increases.18−22

In this Letter, we demonstrate by first-principles calculationsthat band gap of zigzag BNNRs (ZBNNRs) can be significantlytuned by applying a uniaxial strain. We find that the widerZBNNRs are, the more easily the band gaps can be tuned underthe same strain, demonstrating distinctly different physics fromGNRs. Moreover, ZBNNRs possess sizable dipole moment,and more importantly the magnitude and direction of thedipole moment change substantially under elastic strain,resulting in a polarization switch at finite strain. These findingsimply their potential applications in electronic, piezoelectric,photovoltaic, and opto-electronic devices. We will also elucidatebelow the underlying mechanism of strain- and width-dependent band gap and polarization.First-principles density-functional theory (DFT) calculations

were performed in a supercell configuration using the Viennaab initio simulation package (VASP).23 We employed projectoraugmented-wave (PAW) method,24 the local-density approx-imation (LDA)25 of exchange-correlation functionals, an energycutoff of 400 eV for the plane-wave basis, and 50 Monkhorst-Pack k-points along the nanoribbon axis. Geometry optimiza-tions were performed with a criterion of the maximum residualforce less than 0.02 eV/Å without any symmetry constraints.The supercell was adjusted to maintain a sufficiently largeseparation between adjacent nanoribbons (>15 Å from surfaceto surface). A representative atomic structure of 2 nm wideZBNNR is shown in Figure 1, which has two distinct edges

terminated by N and B atoms, called N-edge and B-edge,respectively, with all edge atoms further passivated by H atoms.As presented in Supporting Information, the band gap andpolarization using the generalized gradient approximation ofexchange-correlation functionals are almost the same as theLDA results except for a small constant shift. It is known thatDFT with LDA in general underestimates the band gaps ofmaterials and more accurate estimation of band gaps requires aquasiparticle approach.26 Nevertheless, the general trends of

width-dependent and strain-dependent band gap and piezo-electricity described here are unlikely to change.The width and strain dependence of energy gap are

presented in Figure 2 for monolayer h-BN and ZBNNRs

with uniaxial tensile strain applied along the zigzag direction.We can see that band gap of monolayer h-BN has slightchanges, while band gaps of ZBNNRs decrease significantlywith increased tensile strain. The wider the ribbon is, the morerapidly the gap decreases. For example, 10% elastic strainreduces the band gap of 6 nm wide ZBNNR from about 3.5 to1.0 eV. A previous study has indicated that the maximum elasticstrain which BNNRs can take is over 20%.27 Thus, ESE ofZBNNRs appears to be a promising approach to reduce thelarge band gap of h-BN down to the silicon range. The origin ofthe above width and strain dependence of the energy gaps canbe understood from electronic band structures and wavefunctions shown in Figure 3. Clearly, ZBNNRs are indirect

Figure 1. Atomic structure of 2 nm wide ZBNNR. Each B or N atomrow along nanoribbon width direction is labeled.

Figure 2. The variation in band gaps of monolayer h-BN andZBNNRs with different widths from 1 to 6 nm under the uniaxialtensile strain.

Figure 3. (a) Band structures of 2 nm wide ZBNNRs under differentuniaxial tensile strains. (b) Charge densities of HOMO and LUMOunder the different uniaxial strains.

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band gap semiconductors with the highest occupied molecularorbital (HOMO) located at K point and the lowest unoccupiedmolecular orbital (LUMO) located at X point. Charge densitiesof HOMO and LUMO states under the different uniaxialstrains are shown in Figure 3b. We can see that HOMOcorresponds to an edge state decaying inward from the N-edgewith 2pz character on nitrogen atoms, while the LUMO is anedge state localized at the B-edge with 2pz character on boronatoms. These are in agreement with previous results.28−31 Asthe strain increases, HOMO becomes more localized towardthe N-edge, while LUMO does not change much. Theselocalized quasi-one-dimensional edge states fundamentallydiffer from the delocalized two-dimensional states in bulk h-BN. Thus, as indicated in Figure 2, the band gap of ZBNNRswill not converge to that of h-BN even if the width of ZBNNRsgoes to infinity. The width dependence of the band gap resultsfrom quasi-one-dimensional nature of the localized edge states,instead of quantum confinement of the bulk states (theconventional quantum kinetic energy argument). A detailedexplanation of the width-dependent band gap is included inSupporting Information. When uniaxial tensile strain is appliedalong the ribbon axis of ZBNNRs, the B−N bond along theribbon axis becomes longer, thus the hybridization of 2pzorbitals of N and B atoms on the edges reduces, resulting ina smaller band gap. Because of the spatial separation of the edgestates, electrons and holes are localized on the opposite edgesof the ZBNNRs, which may give rise to interesting transport,thermoelectric, and optical properties.The structural asymmetry of ZBNNRs not only leads to two

asymmetric edge states but also introduces spontaneouspolarization with a residual dipole moment along the widthaxis. Here we calculate the dipole moment density using thefollowing equation, P = (1/c)∫ supercell xρ(x,y,z)dx dy dz, whereρ(x,y,z) is the total charge density including negative valenceelectronic charge and positive ionic charge, w is the ribbonwidth, and c is the length of super cell (see Figure 1). As shownin Figure 4, the direction of dipole moment density is negative

under zero strain. With increasing strain, dipole momentdensity increases gradually, indicating electron transfer towardN-edge. The direction of the dipole moment flips when thestrain is above a critical value. For example, the critical strain of6 nm wide ZBNNR is about 4% only, which is a reasonablecritical elastic strain in the ultrastrength regime11 (but much

smaller than the ideal elastic strain limit of >20%27). Therefore,ZBNNRs act just like a giant polar molecule with a permanentdipole moment. The long aspect ratio of nanoribbon means wecan mechanically “grab” and stretch this molecule. Importantly,the magnitude and direction of dipole moment density of giantpolar molecules can be significantly tuned by stretching themalong the ribbon axis. We find that this polarization switch isattributed to the strain-induced electron transfer toward the N-edge, and more details are presented in SupportingInformation. ZBNNRs, therefore, combine the advantages ofsemiconducting properties and piezoelectric properties, whichhold promising potential in nanoelectro-mechanical devices,such as piezoelectric sensors and piezoelectric nanogenerators.However, the negative spontaneous polarization of ZBNNRs

under zero strain may sound counterintuitive. In fact, N has alarger Pauling electronegativity32,33 than B, χ(N) = 3.04 > χ(B)= 2.04, hence in principle it will take electron from B.Consequently, the N-edge and the B-edge seemingly should benegatively and positively charged, respectively, resulting in apositive dipole moment. Thus, the negative transversespontaneous polarization in Figure 4 must be related to thepresence of the hydrogen passivation at two edges. Indeed, ouradditional calculations yield positive dipole moments ifZBNNRs are “naked”, that is, without any hydrogenpassivation. However, when hydrogen is attached to the leftof the N-edge, it adds a strong negative contribution to P, asχ(H) = 2.20 < χ(N) = 3.04 and N would take electron from H,so the H−N bond would be strongly negatively polarized.Whereas, on the other side of the ribbon when a column of H isadded to the right of the B-edge, because χ(H) = 2.20 > χ(B) =2.04, H can actually take electron from B, and the B−H bondshould also be negatively polarized. Therefore, the two negativepolarization contributions overwhelm the originally positivepolarization of “naked” ZBNNR. In effect, the positive netcharge of the N-edge should be attributed to H of the H−Nbond, while the negative net charge of the B-edge should beattributed to the H of the B−H bond. This charge asymmetrybetween left and right is a consequence (albeit a bit subtle) ofthe structural asymmetry of ZBNNR, giving rise to the abovespontaneous polarization at zero elastic strain and a permanenttransverse dipole moment density, which is different from BNnanotubes.34

For device applications, effective mass of carriers is alsoimportant. It is defined as m* = ℏ2[d2E(k)/dk2]−1, where E(k)is the energy band, k is the coordinate vector in reciprocalspace, and ℏ is the reduced Planck constant. The effectivemasses of electron and hole are shown in Figure 5 in the unitsof me (me is the free electron mass). The effective electron andhole masses converge rapidly with increasing widths, indicatingweak width dependence for wide nanoribbons. However, forZBNNRs with the same width the effective hole mass increasessignificantly (by 50% from 0.85me to 1.30me with 12% strain)and the effective electron mass decreases substantially (by 25%from 1.6me to 1.2me with 12% strain). Over a certain largestrain (about 11%) the effective mass of the hole is larger thanthe effective mass of the electron. The large strain-dependentvariation of effective mass demonstrates the possibility oftuning carrier mobilities of doped ZBNNRs by applying ESE.Lastly, we remark on the physical significance of the band

gap. For photovoltaic applications, not only the band gap itselfneeds to be tuned (Shockley−Queisser analysis)35 forcontrolling the optimal light spectrum to absorb, but also theoscillator strength needs to be tuned for controlling the

Figure 4. The variation of the dipole moment density of ZBNNRswith uniaxial tensile strain.

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absorption coefficient. The direct transition between thespatially separated HOMO and LUMO is optically weak,because wave function overlap between edge-localized HOMOand LUMO decays exponentially with the ribbon width.However, optical absorption spectrum of ZBNNRs involves allpossible optical excitations by promoting electrons from everyvalence band to every conduction band. Our additionalcalculations confirm that there indeed exist high-lying valencestates spatially close to LUMO and low-lying conduction statesspatially close to HOMO, indicating the transition betweenthem are optically allowed. Furthermore, as shown in Figure 3a,the energy level between these optically allowed excitations alsoreduces upon elastic strain. This means the optical absorptionspectrum of ZBNNRs is also strain dependent, implying thatelastic strain could possibly be utilized to optimize the opticalabsorption spectra of ZBNNRs for photovoltaic applications.However, for FET applications the wave function overlap issueabove is not as important; the band gap should still be the mainconsideration because thermal excitation instead of opticalexcitation is utilized in controlling the carrier concentration.Indeed, electric field has been used in bilayer graphene36 togenerate a continuously tunable band gap up to 0.25 eV, andFET device based on this perpendicular electric field has beendemonstrated.37 The main difference with our case is that theelectric field there is externally applied, whereas it is internallygenerated here via the static dipole and piezoelectric effect.In conclusion, by first-principles calculation we show that the

band gaps of ZBNNRs can be tuned significantly by applying auniaxial tensile strain within the elastic range and this specialproperty roots in the reduction of orbital hybridization alongthe ribbon axis upon elastic strain. Furthermore, we find thatunder the same strain, the wider the nanoribbon, the smallerthe gap, which is a result of the localized edge states instead ofquantum confinement of the bulk states (the conventionalquantum kinetic energy argument). The above findings provideus a practical method to tune their electronic and opticalproperties with elastic strain engineering. In addition, due tostructural asymmetry and hydrogen passivation ZBNNRspossess a permanent transverse dipole moment, like a giantpolar molecule. The magnitude and direction of the dipolemoment can be tuned significantly by stretching this“molecule”, with a significant effect on the band gap andcarrier mobility. The phenomena predicted here should be

general and may open new avenues of optimizing two-dimensional semiconducting and insulating nanoribbons forelectronic, piezoelectric, photovoltaic, and opto-electronicapplications.

■ ASSOCIATED CONTENT*S Supporting InformationAdditional information and figures. This material is availablefree of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: (X.Q) qianxf@mit.edu; (J.L.) liju@mit.edu.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was supported by 973 Program of China(2010CB631003, 2012CB619402), NSF DMR-1120901 andAFOSR FA9550-08-1-0325. J.Q. also thanks the support byPAPD and NSFJS No. BK2010499.

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Figure 5. The effective masses (in units of the free electron mass me)of electron and hole for ZBNNRs with different widths under differentuniaxial tensile strains.

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1

Supporting Information for

Strain-Engineering of Band Gaps in Piezoelectric Boron Nitride Nanoribbons

Jingshan Qi1,2, Xiaofeng Qian3*, Liang Qi3,4, Ji Feng5, Daning Shi1 and Ju Li3,4*

1 College of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, China

2 Department of Applied Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

3 Department of Nuclear Science and Engineering and Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

4 State Key Laboratory for Mechanical Behavior of Materials and Frontier Institute of Science and Technology, Xi’an Jiaotong University, Xi’an, 710049, China

5 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China

Corresponding authors: qianxf@mit.edu, liju@mit.edu

In this supplementary document, we will compare the strain and width dependent band gap

and polarization of zigzag boron nitride nanoribbons (ZBNNRs) calculated using the

local-density approximation (LDA)1 and generalized gradient approximation (GGA)2 of

exchange-correlation functionals. We will also explain the underlying mechanism of the

piezoelectric phenomenon observed in ZBNNRs, i.e. strain-induced polarization swtich, by

investigating the effect of uniaxial tensile strain on atomic charges of ZBNNRs. Finally,

the width dependent band gap is explained in a simplified model.

1. Comparison of LDA and GGA band gap and polarization

We have used both LDA and GGA exchange-correlation functionals in the density-functional

theory (DFT) calculations of strain and width dependent band gap and polarization (dipole

moment density). As shown in Figure S1 and Figure S2, the DFT-GGA results are very similar to

the DFT-LDA results except for a small constant shift of band gap and dipole moment density.

The general trend of strain and width dependence for band gap and polarization is exactly the

same.

2

2. Mechanism of piezoelectric phenomenon in ZBNNRs

Based on Slater’s linear combination of atomic orbitals (LCAO) picture,3 localized ab initio

quasi-atomic orbitals4, 5 are unitarily transformed from extended Kohn-Sham eigen-states

calculated using the Quantum-ESPRESSO package6 while they are required to be maximally

similar to the Bloch subspace spanned by atomic orbitals. Then, electronic charge is

obtained by partitioning one-electron density matrix into the quasi-atomic orbital basis. Finally,

atomic charge is defined as total charge, i.e. the sum of electronic charge and

ionic charge. So positive (negative) atomic charge indicates electron loss (gain). Here

we take the 4 nm-wide ZBNNR as an illustrative example. For the sake of direct comparison, we

use the same atomic positions of B and N atoms of H-passivated ribbon for the “naked” ZBNNR,

and do not consider the effect of edge magnetization in the “naked” ribbon. The corresponding

polarization change with respect to the strain is shown in Figure S3a, which indeed shows the

placement of H is crucial, as noted above. Without H, the polarization is always positive and P

will not have the sign change with strain. Very interestingly, different from the H passivated

ZBNNRs, the polarization of “naked” ZBNNR is much less sensitive to the strain. This

remarkable difference roots in their fundamentally distinct electronic structure. Without H

passivation, ZBNNR is metallic due to the unpaired electrons on the N and B edges, while the H

passivated ZBNNR is insulating owing to saturated edges and exhibiting a large energy gap. The

direct consequence is the difference in the atomic charges under zero strain shown in

Figure S3b. Compared to the H-passivated ZBNNR, N and B atoms in the bulk of the “naked”

ZBNNR are strongly polarized, resulting in more electron transfer from B to N atoms. The

stronger electron transfer in the “naked” ribbon under zero strain leads to less bulk response

when it is under finite strain. Consequently, electron transfer in the “naked” one will be mostly

localized around its metallic edges. This is unambiguously demonstrated by their

atomic charge difference between 12% and 0% strain plotted in Figure S3c. In the case of

H-terminated insulating ZBNNR, larger polarization change occurs in the bulk. The changes of

both bulk and surface polarization due to finite strain are positive. They add up together and lead

to large positive change in total polarization. In addition, as shown in Figure S4, under the

increased strain the atomic charge transfer becomes stronger, resulting in larger

polarization increase. Overall, electrons transfer from B to N with the increased strain. Therefore,

hydrogen passivation plays two critical roles. On one hand, it largely determines the negative

spontaneous polarization of ZBNNR under zero strain, allowing the possibility of polarization

3

switch upon elastic strain. On the other hand, H atoms saturate the unpaired electrons of the

metallic “naked” ZBNNR, thus reducing the local charge transfer in the bulk of unstrained

ZBNNR and permitting the large polarization increase when the ribbon is stretched.

However, we stress that the hydrogen passivation itself cannot give rise to the fascinating

phenomena above. In fact, beside the hydrogen passivation, both the asymmetric N and B edges

and the large electron negativity difference between N and B atoms play critical roles in the

elastic strain induced polarization switch in H passivated ZBNNRs.

3. The origin of width dependent band gap Based In general, band gap is defined as the energy difference between the lowest

unoccupied state (LUMO) and the highest occupied state (HOMO). In our ZBNNR case, HOMO and LUMO are two edge states with HOMO localized near the leftmost N-edge and LUMO localized near the rightmost B-edge. Let’s start with HOMO and LUMO states on a wide ZBNNR, shown in Figure S5. Since they are two eigen states of this wide ZBNNR, they satisfy

orthonormal condition, that is, , , and

. The corresponding energies are: , and

. Then, the energy gap between LUMO and HOMO of the wide

ZBNNR is . Now, let’s consider to move these two states onto the two edges of a

narrow ZBNNR. Due to the exponential localization of both states along the width direction, they will have more and more overlap when they become closer in space, which leads to an non-negligible interaction energy between HOMO and LUMO, that is,

. Then, we can obtain the new eigen energies of the HOMO

and LUMO states for the narrow ZBNNR by diagonalizing the following Hamiltonian,

. The solutions are: and

. Thus, the energy gap for the narrow ribbon

becomes: , which is clearly larger than the energy

4

gap for the wide ZBNNR, . Therefore, the energy gap between the localized

HOMO and LUMO edge states decreases when the ribbon width increases. Although the overlap matrix can be further taken into account, however, the trend and mechanism described here will not change.

In addition, the observed quick convergence of the energy gap with respect to the increase of ribbon width indeed comes from the fact that these HOMO and LUMO states are almost

exponentially localized along the width direction. The interaction energy will quickly

vanish as the width increase gradually, which means the off-diagonal term in the above Hamiltonian will go to zero. Thus, the energy gap will converge quickly as the width increases.

References:

1. Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, (10), 5048-5079. 2. Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, (18), 3865-3868. 3. Slater, J. C.; Koster, G. F. Phys. Rev. 1954, 94, (6), 1498. 4. Qian, X.; Li, J.; Qi, L.; Wang, C.-Z.; Chan, T.-L.; Yao, Y.-X.; Ho, K.-M.; Yip, S. Physical Review B 2008, 78, (24), 245112. 5. Qian, X.; Li, J.; Yip, S. Physical Review B 2010, 82, (19), 195442. 6. Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; Corso, A. D. J. Phys.: Condens. Matter 2009, 21, (39), 395502

5

Supplementary Figures

Figure S1. Uniaxial tensile strain and width dependent band gap of ZBNNRs using the LDA and

GGA exchange-correlation functionals.

Figure S2. Uniaxial tensile strain and width dependent dipole moment density of ZBNNRs using

the LDA and GGA exchange-correlation functionals

6

Figure S3. (a) The variations of the dipole moment density of 4 nm-wide ZBNNRs with and

without H passivation under uniaxial tensile strains. (b) atomic charge for 4 nm-wide

ZBNNR with and without H passivation under zero strain. Symbols filled with blue, orange, and

gray color stand for N, B, and H atoms, respectively. (c) atomic charge difference

between 12% and 0% strain for 4 nm-wide ZBNNR with and without H passivation in the upper

and bottom panel, respectively.

Figure S4. atomic charge difference between finite strain and zero strain for 4 nm-wide

ZBNNR. Negative (positive) value in blue (red) color indicates the electron gain (loss).

7

Figure S5. Schematic plot of the HOMO (red) and LUMO (blue) wavefunctions in (a) wide and (b) narrow ZBNNR. The HOMO and LUMO are initially the eigenstates of wide ribbon. After they move to the narrow ribbon, there are non-negligible overlap and interaction between HOMO and LUMO, which further split the energy difference between them.