Confinement and edge effects on atomic collapse in graphene nanoribbons
Jing Wang, Robbe Van Pottelberge, Amber Jacobs, Ben Van Duppen, Francois M. Peeters
CConfinement and edge effects on atomic collapse in graphene nanoribbons
Jing Wang,
1, 2, 3, ∗ Robbe Van Pottelberge,
2, 3, † Amber Jacobs,
2, 3
Ben Van Duppen,
2, 3 and Francois M. Peeters
4, 2, 3, ‡ School of Electronics and Information, Hangzhou Dianzi University, Hangzhou, Zhejiang Province 310038, China Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium NANOlab Center of Excellence, University of Antwerp, Belgium School of Physics and Astronomy and Yunnan Key Laboratory forQuantum Information, Yunnan University, Kunming 650504, China
Atomic collapse in graphene nanoribbons behaves in a fundamentally different way as comparedto monolayer graphene, due to the presence of multiple energy bands and the effect of edges. Forarmchair nanoribbons we find that bound states gradually transform into atomic collapse stateswith increasing impurity charge. This is very different in zig-zag nanoribbons where multiple quasi-one-dimensional bound states are found that originates from the zero energy zig-zag edge states.They are a consequence of the flat band and the electron distribution of these bound states exhibitstwo peaks. The lowest energy edge state transforms from a bound state into an atomic collapseresonance and shows a distinct relocalization from the edge to the impurity position with increasingimpurity charge.
I. INTRODUCTION
Atomic collapse is a phenomenon where for sufficientlylarge charge of the nuclei bound states can enter the lowerpositron continuum and turn into quasi-bound states [1–4]. If the bound state is empty this process of entering thenegative continuum is accompanied with the productionof an electron-hole pair. Due to the very large nuclearcharge ( Z e ) needed in order to induce atomic collapseit was never conclusively detected in experiments [5, 6].However, the discovery of graphene enabled researchersto approach the atomic collapse problem in a differentway. It was shown that due to the enhanced Coulombinteraction in graphene, atomic collapse should occur atsignificantly smaller charge (i.e. Z ≈
1) as compared tothe original predicted one of relativistic atomic physics (i.e.
Z > ∗ [email protected] † [email protected] ‡ [email protected] den appearance of atomic collapse states, as in the gap-less case, in gapped graphene bound states gradually turninto atomic collapse states when entering the hole con-tinuum. Interestingly it was theoretized that atomic col-lapse should also occur in gapped 1D-Dirac systems [26].Such a strict 1D Hamiltonian is a rather crude approxi-mation for graphene nanoribbons demanding for a moredetailed study of the problem.In this paper we will consider how atomic collapsemanifests itself in finite width graphene nanoribbons.It is known that nanoribbons come in different forms.There are nanoribbons with armchair or zig-zag edgesand within these two types their can be either a gap orno gap depending on the number of atomic chains. Ontop of that the confinement in one of the spatial directionsleads to the appearance of multiple energy bands. Theseproperties are the reason that the atomic collapse of bulkgraphene will be different in graphene nanoribbons as wewill show in this paper. For example, we found thatthe Coulomb potential results in bound states at zig-zagedges. This is unexpected in view of the Klein paradoxthat electrons cannot be confined by electrostatic field inzero gap graphene. II. MODEL
Here we use the tight binding model which includes thegraphene lattice structure in contrast to the continuummodel used in e.g. Ref. [26]. For graphene we use thefollowing tight binding Hamiltonianˆ H = (cid:88) (cid:104) i,j (cid:105) ( t ij a † i b j + H.c. )+ (cid:88) i V ( −→ r i A ) a † i a i + (cid:88) i V ( −→ r i B ) b † i b i (1)The first term represents the tight-binding Hamilto-nian without any external fields. The hopping parame-ter is given by t ij and for graphene we take the generallyaccepted value -2.8 eV for nearest neighbour hopping.The operators a i ( a † i ) and b i ( b † i ) create (annihilate) an a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n electron on the i th site of sublattice A and B , respec-tively. The last two terms include an electrostatic po-tential which for our case is due to the presence of aCoulomb charge, Ze , which we model by a Coulomb po-tential V ( r ) = − β (cid:126) v F / (cid:112) r + r with β = Ze /κ (cid:126) v F thedimensionless coupling constant, v F the Fermi velocityand κ the effective dielectric constant. We took r = 0 . pybinding [28]. In all the calculations we use abroadening of 0 .
003 eV and the units of the Local Den-sity of States (LDOS) are [ eV.nm − ]. A 1000 nm longnanoribbon is used to simulate the infinite long nanorib-bon. For an armchair (zigzag) nanoribbon of width 4.8nm (5 nm) the system contains 1 . ∗ (1 . ∗ ) atoms. III. ATOMIC COLLAPSE IN GRAPHENE
For completeness and for comparison purposes we re-view atomic collapse in graphene. In graphene the atomiccollapse effect manifests itself in a different way as com-pared to relativistic physics. In relativistic physics boundstates inside the gap region (∆ = 2 m c with m theelectron rest mass and c the velocity of light) decrease inenergy with increasing value of the nuclear charge. How-ever, if the nuclear charge is sufficently large the boundstate(s) enters the positron continuum and hybridize withit. If this happens the bound state acquires a finite widthand turns into a quasibound state which is called atomiccollapse state. However, in graphene due to the gaplessnature the situation is very different. This is shown inFig. 1 where the LDOS at the impurity site is presented asfunction of the charge strength β and the energy. It canbe clearly seen that when β > . E ( e V ) FIG. 1. Density plot of the LDOS calculated at the impuritysite as function of energy and impurity strength β for bulkgraphene. L D O S (a) E (eV) L D O S (b) FIG. 2. (a) LDOS measured at the center of an armchairnanoribbon of width 4 . β = 0 inboth cases . IV. ATOMIC COLLAPSE IN ARMCHAIRNANORIBBONS
For armchair nanoribbons their are two major typesof ribbons depending on the width of the ribbon [29]:i) a gap in the spectrum is found when the number ofatomic chains is N = 3 p or N = 3 p + 1 with p a positiveinteger. This gap is proportional to the inverse of theribbon width. These nanoribbons are semiconducting; Beta0.60.40.20.00.20.40.6 E ( e V ) (a)0.0 0.5 1.0 1.5 2.00.60.40.20.00.20.40.6 E ( e V ) (b) 10 FIG. 3. LDOS measured at the impurity position as functionof the charge for a graphene nanoribbon of width 4 . N = 39 atomic chains. In(a) the LDOS is shown for a charge placed in the center ofthe nanoribbon and the LDOS is measured at the impuritysite while in (b) the LDOS is measured a distance 0.18 nmaway from the impurity in the direction perpendicular to thenanoribbon. ii) when N = 3 p + 2 the spectrum of the nanoribbon isgapless and these nanoribbons are metallic. However, itwas shown in experiments that all the armchair nanorib-bons are semiconducting [30]. This discrepancy can beexplained by including third nearest neighbour hopping.Since all armchair nanoribbons are gapped we will focuson nanoribbons that are gapped within a nearest neig-bour hopping tight binding model. Including third near-est neigbour hopping only leads to quantitative correc-tions making the metallic nanoribbons very similar to thesemiconducting ones. In Fig. 2(a) an example of a typicalLDOS of an armchair nanoribbon is shown. We calcu-lated the LDOS at the center of the armchair nanoribbonwith width 4.8 nm. The gap (∆ = 0 .
25 eV) is shown ingray. The cusps in the LDOS are typical for nanoribbonsand a consequence of the 1D nature of the spectrum andcorrespond to the onset of a new subband. In Fig. 2(b)the LDOS is calculated a distance 0 . nm from the cen-ter of the same nanoribbon in the direction perpendicularto the nanoribbon length. Figs. 2(a) and 2(b) illustrate that the electron probability corresponding to the differ-ent bands can be zero at some of the carbon rows.The gap region will allow for impurity bound states.This is in contrast with gapless pristine graphene whereonly quasi-bound states are possible. The effect of suchgap is clearly shown in Fig. 3(a) for -0.125 eV < E < β for an impurityplaced at the center of the nanoribbon. The width ofthe ribbon taken along x − direction is chosen to be 4.8nm, using the value of the inter carbon distance a cc =0 .
142 nm this gives N = 39 atomic chains. When thestrength of the charge is gradually increased a clear anddistinct state sinks into the gap region corresponding to abound state. The LDOS of this state increases when thecharge increases which is due to the fact that the stategets localized closer to the impurity. The lowest boundstate inside the gap keeps its bound state character untilthe charge reaches β ≈ .
25. After this the bound stategets redistributed over the negative continuum states andaquires a finite width and turns into a resonant state.Note that for larger β more bound states appear in thegap region. All these bound states turn into resonanceswhen they enter the negative continuum region.In Fig. 3(b) we show the LDOS at a distance 0 .
18 nmaway from the center of the nanoribbon. Interestingly,more states and bands appear as compared to measuringthe LDOS at the center of the nanoribbon. This behaviorcan be explained from the fact that in armchair nanorib-bons some states show zero LDOS for certain rows ofatoms as discussed in Ref. [31]. The symmetric posi-tion of the charge in the middle of the ribbon impliesthat some states maintain zero LDOS at the center ofthe nanoribbon, and consequently do not show up whencalculating the LDOS at the center. This behavior isvery similar to the effect of defects studied in Ref. [31].In Fig. 3(b) we notice the appearance of an extra band(around E ≈ .
25 eV) with a diving series of states whichqualitatively behave very similar to the states inside thegap discussed earlier: they show a similar dependenceon β and turn into quasi-bound states when entering thelower continuum.So far we only studied the energy dependence of theLDOS at a single atomic position. In Fig. 4 we plot thespatial distribution of the LDOS for the first 5 electronicstates observed in Fig. 3(a) at β = 1. Figures 4(a)-4(d)correspond to states inside the gap while Fig. 4(e) corre-sponds to the first state outside the gap and belongs tothe next subband. In contrast to Schr¨ o dinger physicswith symmetric potentials, here we do not have evenand odd solutions and therefore no clear nodes in thewavefuntions (or LDOS) are found. The LDOS exhibitsrather a dumbell character which is made more clear inFigs. 5(a) - 5(d) where we show cuts through the LDOSof Figs. 4(a) - 4(d). The discrete nature of the LDOSreflects the discrete graphene lattice. These states havesome similarity to the ones found for a strict 1D-DiracHamiltonian [26]. y ( n m ) (a) 2 0 2x (nm)(b) 2 0 2x (nm)(c) 2 0 2x (nm)(d) 2 0 2x (nm)(e)0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 FIG. 4. Spatial LDOS for β = 1 of the first five states (seeFig. 3(a)) with energy: (a) E = − .
12 eV, (b) E = 0 .
037 eV,(c) E = 0 .
086 eV, (d) E = 0 .
105 eV, and (e) E = 0 .
272 eV.Red(blue) represents high(zero) LDOS while blue representszero LDOS. y (nm)0.00.10.20.3 L D O S (a) 0.000.050.100.150.20 L D O S (b)10 5 0 5 10y (nm)0.000.020.040.06 L D O S (c) 10 5 0 5 10y (nm)0.000.010.020.03 L D O S (d)10 5 0 5 10y (nm)0.0000.0250.0500.0750.1000.125 L D O S (e) 2 1 0 1 2x (nm)0.0250.0500.0750.1000.125 L D O S (f) FIG. 5. Cut through x = 0 in the y direction for the spatialLDOS calculations shown in Fig. 4. In (f) also a cut along x -direction for y = 0 is shown for the state corresponding to(e). In Fig. 3, next to the states located inside the gapregion quasi-bound states appear that are attached tohigher energy bands. Those states can be seen in theLDOS as peaks already at small charge right below theband edge. They hybridize almost immediately with theunderlying continuum acquiring a finite width. Such res-onances for positive energy are not atomic collapse states.
E (eV) L D O S (a) = 0.5 E (eV) L D O S (b) = 1.5 FIG. 6. Cut of the LDOS presented in Fig. 3(a). In (a) theLDOS is shown for β = 0 . β = 1 .
5. As in Fig.2 the gap is indicated by the gray region.
An atomic collapse state is a conduction band state re-distributed over valence band states that are located inthe negative continuum. The resonances observed forpositive energy are therefore resonant states but are notrelated to atomic collapse. These hybridized states wereinvestigated in Ref. [32] for small β within a continuummodel. Our tight-binding results show that these statesshould appear as a clear signature in LDOS measurmentswith e.g. a STM tip. Note that this is related to somestates that were recenly predicted for bilayer graphenewith a Coulomb impurity [33]. In Figs. 5(e) and 5(f) thespatial LDOS for the first quasi-bound state observed inthe positive continuum in Fig. 3(a) is shown. This stateshows no dumbell-like distribution but a more 1S likeatomic orbital shape which is confirmed by a cut of thespatial LDOS shown in Fig. 5(e) for x = 0. Also a cut for y = 0 is shown next to the latter which indicates that thestate is not confined along the nanoribbon. Note that inFig. 3(b) the states below the second energy band do nothybridize immediately with the underlying continuum.This behavior is a consequence of the symmetric place-ment of the charge. Further in the manuscript the effectof an assymetricly placed charge on these states will beinvestigated.In Fig. 6 a cut of the LDOS of Fig. 3(a) is shown fortwo values of β : (a) β = 0 . β = 1 .
5. For β = 0 . β = 1 . β = 0 . ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● β W i d t h ( m e V ) FIG. 7. Width of first (blue) and second (red) bound stateturning into a resonant state shown in Fig. 3(a) as function ofthe impurity strength β . The width is defined as the energyrange over which the peak loses 30% of it’s intensity. of bound states appear due to the increased value of thecharge. For energy E > .
125 eV two resonant statesoriginating from the first energy band can be observed.In order to clearly show the bound state to quasi-boundstate transition seen in Fig. 3(a) we plotted the broad-ening of the first and second bound state as function ofthe strength of the charge in Fig. 7. For 0 < β < . ≈ .
003 eV which is the broadening used inthe calculation of the LDOS) which is representative fora bound state. However after β ≈ . β ≈ .
8. After the bound state has enteredthe continuum its width increases with decreasing energysimilar as in the case of bulk graphene shown in Fig. 1.The results in Fig. 7 thus show a distinct bound state toatomic collapse state transition.In all the above discussion, the charge was consideredto put at the center of the nanoribbon. The LDOS mea-sured at the impurity show less peaks than it was mea-sure 0.18 nm away from the impurity in Fig. 3. Thisphenomenon triggers us to study the effect of the posi-tion of the charge on the spectrum. The correspondingLDOS is shown in Fig. 8 for the same nanoribbon as theone in Fig. 3 but now for a charge placed 0.5 nm fromthe center of the nanoribbon in Fig. 8(a) and 2 nm fromthe center of the nanoribbon in Fig. 8(b). For a smallasymmetric placement of the charge the spectrum looksvery similar to the one shown in Fig. 3(b) for a symmetricplacement but where the LDOS is measured away fromthe charge position. However, the bound states origi-nating from the second energy band at E ≈ .
22 eV inFig. 8(b) start to show hybdridization with the underly-ing continuum acquiring a finite width. This confirms ourtheory that the special non-hybridizing behavior of thesestates discussed in connection with Fig. 3(b) is due to thesymmetric placement of the charge. Increasing the asym-metry even further as shown in Fig. 8(b) we notice thatthese states show an even stronger hybridization mak-
Beta0.60.40.20.00.20.40.6 E ( e V ) (a)0.0 0.5 1.0 1.5 2.00.60.40.20.00.20.40.6 E ( e V ) (b) 10 FIG. 8. LDOS measured at the impurity position as functionof the charge for a graphene nanoribbon of width 4.8 nm. In(a) the charge is placed a distance 0.5 nm from the center ofthe ribbon while in (b) the charge is placed a distance 2 nmaway from the center. ing our point even stronger. Qualitatively similar fea-tures in the spectrum are seen when placing the chargeasymmetrically. All the general features discussed for thesymmetric case remain: i) states inside the gap show atransition from bound to atomic collapse state with in-creasing charge β , ii) multiple energy bands appear, andiii) below these higher energy bands states appear thatalmost immediately hybridize with the underlying posi-tive continuum. This shows that the physics presented inthis manuscript should be robust in experiments almostindependent of the exact position of the charge, pavingthe way for the first experimental observation of a boundstate to atomic collapse state transition. V. ZIG-ZAG NANORIBBONS
In the case of zig-zag nanoribbons there is an extra ele-ment that is different from armchair nanoribbons, namelythe presence of zero energy edge states which have a flatband character. It is expected that these edge states willbe strongly influenced by the presence of the impurity.In Fig. 9 we show the LDOS calculated at the center of a -1.0 -0.5 0.0 0.5 1.0E (eV)0.00.10.20.30.40.50.6 L D O S / / -10-50510 E ( e V ) FIG. 9. LDOS measured at the center (dashed blue) andat 2 nm from the center (solid red) of a zig-zag nanoribbonof width 5 nm without an impurity. The insert is the bandstructure of the zig-zag ribbon. E ( e V ) FIG. 10. LDOS measured at the impurity site as function ofthe charge strength β and energy for a graphene nanoribbonwith zig-zag edges of width 5 nm. The impurity is located atthe center of the nanoribbon. The dashed line is the value ofthe Coulomb potential at the edge. zigzag nanoribbon (blue) of width 5 nm without an im-purity. The cusps signifying the multiple energy bandscan be clearly seen. Note that compared to the LDOS ofthe armchair nanoribbon discussed in the previous sec-tion the LDOS does not show a gap around E = 0 eV.When the LDOS is calculated 2 nm away from the centerof the nanoribbon (red) an extra peak at E = 0 emergesdue to the edge states.In Fig. 10 the energy dependence of the LDOS is shownas function of the impurity strength for a zig-zag nanorib-bon of width 5 nm for an impurity placed at the centerof the nanoribbon. A number of states originating from E = 0 eV at β = 0 can be clearly observed. Thesestates are pulled towards lower energy with increasing E (eV) L D O S (a) = 1.45 E (eV) L D O S (b)= 2.25 FIG. 11. Cut of the spatial LDOS shown in Fig. 10 for (a) β = 1.45 and (b) β = 2.25. impurity charge. Since they originate from E = 0 eVthis band of discrete states can be attributed to the edgestates. They are bound states which we confirmed by thefact that their width increases linearly with the imposednumerical broadening and their position and width didnot change when we increase the length of the graphenenanoribbon. The lowest state shows interesting behav-ior with increasing impurity charge. It starts as a boundstate which can be seen from the narrow width of theLDOS. However, when the impurity charge increases thestate comes in contact with the lower continuum band(starting at E ≈ -0.50 eV) and gradually turns into anatomic collapse resonance. The width of this resonanceincreases with increasing charge. The reason that theband of edge states can support bound states lies in thefact that this band consists of a mixture of conductionand valence states. Consequently the conduction bandnature leads to the appearance of the previously dis-cussed bound states. The behavior of the states in thepositive energy range of Fig. 10 is similar to the onesshown for an armchair nanoribbon in Fig. 3 and there-fore will not need any further discussion. Just below E ≈ . E ≈ . β ≈ . β = 1 .
45 and β = 2 .
25 show-ing the situation before and after the first bound statethat started at E ≈ . y ( n m ) (a)202 y ( n m ) (b)6 4 2 0 2 4 6x (nm)202 y ( n m ) (c)0.0 0.1 0.2 0.3 0.4 0.5 0.6 FIG. 12. Contour plot of spatial LDOS of the lowest statein Fig. 10 for three values of the charge: (a) β = 0.781 ( E =-0.189 eV), (b) β = 1.607 ( E = -0.432 eV) and (c) β = 2.59( E = -0.871 eV). Fig. 11(a) the series of edge states can be clearly seen asa distinct number of peaks between E ≈ − . E ≈ − . E ≈ − . E ≈ . β = 2 .
25 as shown in Fig. 10 the low energybound state of Fig. 11(a) has turned into a resonant statewhich is being modulated by the edge states. This be-havior is seen in Fig. 11(b) around E ≈ -0.5 eV where theresonant peak shows subpeaks corresponding to the edgestates. The interesting behavior of the edge states mod-ulating the resonant states should be a clear signature tolook for in experiments.Now we look into the evolution of the edge states asfunction of the impurity charge β . In Fig. 12 we plot thespatial LDOS for the lowest edge state for three values ofthe impurity charge β . For small value of the impuritycharge (see Fig. 12(a)) the spatial LDOS is localised atthe edges confirming its edge state nature. Interestingly,the edge state turns into an impurity bulk state with in-creasing charge β (see Figs. 12(b) and 12(c)). This transi-tion from edge to impurity state explains the change in β -dependence (linear versus nonlinear) observed in Fig. 10and is a clear signature to look for in future experiments.In Fig. 13 the energy of the fan of bound edge statesas function of the impurity charge β , as derived fromthe LDOS, is shown for the first 8 impurity edge states.The lowest state which shows a clear edge to impuritystate transition discussed earlier is shown in red. Theenergy levels could be fitted (for the region 0 < β < E = − ( aβ + bβ ) with { a, b } respectively { } , { } , { } , { } , { } , { } , { } , and { E ( e V ) n a FIG. 13. First 8 states originating from the band of edgestates as function of the charge strength β . The lowest stateclearly visible in Fig. 10 is shown in red. The dashed line isthe values of the Coulomb potential at the edge. For small β the energy states behave as E = − aβ . In the inset thevalue of the fitting parameter a (in unit of eV) is shown forthe different curves. The straight line is given by a = 0 . − . n . y ( n m ) (a) r = 2.68 nm y ( n m ) (b) r = 2.85 nm y ( n m ) (c) r = 3 nm FIG. 14. Contour plot of spatial LDOS calculated for β = 0 . E = − . E = -0.182 eV and (c) E = -0.172 eV. The solid dotshows the position of the charged impurity and the dashedcircle indicates the radius(i) of the Coulomb potential at thisenergy. } eV. It is evident that for small β the energyis linear in β . This is in contrast with the states ofthe 2D hydrogen atom which exhibits a quadratic de-pendence E ≈ β [34] while the lowest atomic col-lapse states in bulk graphene (see Fig. 1) behaves as E = ( − β (cid:126) v F /r ) exp ( − π/ (cid:112) β − .
25) [8]. Using firstorder perturbation theory with respect to the Coulombpotential explains the linear β -dependence of the boundedge states. The quadratic term gives a very small cor-rection to the linear behavior. The drop in b from thefirst to the second state is a consequence of the fact thatthe lowest state is turning into a bulk impurity state forlarge β .In the inset of Fig. 13 we show the fitting parameter a as function of the number of the edge state which showsa linear dependence a (eV) = 0.263 - 0.026 n for 1 7. This behavior can be understood from the factthat for small β the edge states remain confined at theedge, feeling a broader almost quadratic-like potentialfor small energy and distances. Consequently, these edgestates feel a softer potential, explaining the weaker β -dependence as function of the edge state number.In Fig. 14 we plot the spatial LDOS for β = 0 . β . We observethat with increasing energy the spatial LDOS localizationshifts further away from the center of the nanoribbon(where the Coulomb charge is placed). As a consequencethese states feel a weaker shift due to the decay of theCoulomb potential, explaining why they are higher inenergy. In Fig. 15 a cut of the spatial LDOS along one ofthe nanoribbon edges are plotted for the first four statesseen in Fig. 13. The cut is taken along the edge of thenanoribbon. The lowest state consists of one peak whilethe excited states consist of two peaks which move furtheraway from each other for higher energy states. Thesefigures are somewhat similar but not the same to theelectron probability of states found in Ref. [26] for theCoulomb problem in gapped Dirac materials. The twopeak structure in LDOS symmetric around x = 0 can beunderstood as follows. Lets consider the 1D edge statesand take the extreme limit of a flat band. The kineticenergy is quenched and the Dirac equation is reduced to V ( x, y ) ψ ( x, y ) = Eψ ( x, y ) (2)where y is the position of the edge. This equation hasas solution ψ ( x, y ) ≈ δ ( x − x i ) where x i is determinedby V ( x i , y ) = E as shown by the dashed circles inFig. 14. Because the Coulomb potential V ( x, y ) is sym-metric around x = 0 this gives two solutions x i = ±| x i | and thus the wave function becomes ψ ( x, y ) = c ( δ ( x − | x i | ) + δ ( x + | x i | ) (3)The separation between those δ -functions increase withenergy which agrees with Fig. 15. Those δ -peaks arebroadened in our numerical results because the edgestates exhibit some small dispersion and the edge statespenetrate into the bulk of the nanoribbon exponentialdecreasing away from the edge.From Fig. 10 and the previous discussion it is clearthat the states originating from the edge states are nar-row and consequently represent bound states. However, L D O S (a) 02468 (b)6 3 0 3 6x (nm)02468 L D O S (c) 6 3 0 3 6x (nm)02468 (d) FIG. 15. Cut of the spatial LDOS along the edge, calculatedfor β = 0 . E =-0.193 eV, (b) E = -0.182 eV, (c) E = -0.172 eV and (d) E =-0.160 eV. E ( e V ) n a FIG. 16. LDOS calculation for the same nanoribbon as inFig. 10 but now with a charge which is placed 2 nm fromthe center of the nanoribbon. The dashed line is the valueof the Coulomb potential at the closest edge. In the inset weshow the fitting parameter a (in units of eV) as function ofthe number of the edge state for the five lowest states. at first sight this seems strange because looking at Fig. 9reveals that the LDOS is not zero below these edge states.Consequently one may expect that these edge states willhybridize with the underlying continuum turning intoresonant states (similar to the upper band states pre-viously discussed for armchair nanoribbons). However,this seems not the case. From our previous results for y ( n m ) (a) r = 0.70 nm y ( n m ) (b) r = 1.15 nm y ( n m ) (c) r = 1.94 nm FIG. 17. Contour plot of spatial LDOS calculated for β =0 . 54 for three lowest states seen in Fig. 16: (a) E = − . E = − . 284 eV and (c) E = − . 177 eV. The soliddot shows the position of the charged impurity and the dashedcircle indicates the radius(i) of the Coulomb potential at thisenergy. armchair nanoribbons one may expect that this behaviorcan be related to the symmetric placement of the charge.In order to confirm whether or not this suspicion is cor-rect we calculated the LDOS at the impurity for a chargeplaced 2 nm from the center of the nanoribbon in Fig. 16.We limited the figure to the edge states since they arehere of interest. Because the states are more clearly sep-arated it is now even more clear how these states originatefrom E = 0 eV. We checked that the width of those peaksin the LDOS scales with the numerical broadening con-firming that they are bound states. For β < β similar to the statesshown in Fig. 10. This time the states can be almostperfectly fitted to E = − aβ with { a } respectively { } , { } , { } , { } , { } , { } , { } , { } , and { } eV for the eight lowest states seen in Fig. 16. Inthe inset of Fig. 16 the fitting parameter a is shown asfunction of the number of the edge state. In contrast withthe results for a symmetrically placed charge shown in theinset of Fig. 13 no linear dependence is observed and thestates depend more strongly on the impurity charge β .The reason is that the Coulomb charge is placed muchcloser to the edge, consequently the edge states feel amuch deeper Coulomb potential instead of the broaderpotential felt by the edge states discussed earlier. There-fore the edge states are more strongly influenced by thecharge explaining their more profound dependence on thecharge β .In Fig. 17 the spatial LDOS is shown for β = 0 . 54 forthe three lowest edge states seen in Fig. 16. The behaviorof these states is very similar to the ones shown in Fig. 14 for the symmetrically placed charge with the differencethat the state is now entirely localized at only one edge,i.e. the edge closest to the potential center. VI. CONCLUSION We investigated how the finite width of graphenenanoribbons influences the atomic collapse phenomenonand found very different physical behaviours dependingon the type of edges.We showed with tight binding calculations that themanifestation of the atomic collapse in graphene nanorib-bons is fundamentally different from its manifestation inpristine graphene. In both armchair and zig-zag nanorib-bons bound states turn into atomic collapse states whenentering the lower continuum. This kind of behaviormimics closely the predicted atomic collapse in relativis-tic physics: bound states in the mass gap turn intoquasi-bound states when entering the negative contin-uum. Therefore, the experimental study of atomic col-lapse in graphene nanoribbons could pave the way tothe first observation of the true analog of the relativisticatomic collapse effect.We showed that in the case of zig-zag nanoribbons thewell known edge states lead to a modulation of the quasi-bound states when they cross the band of edge states.This modulation should be measurable in experimentswhen probing with an STM tip.Furthermore we showed that the atomic collapse ingraphene nanoribbons differs from the manifestation inpristine graphene in the following ways: i) instead ofthe sudden appearance of quasi-bound states in pristinegraphene a gradual bound state to quasi-bound statetransition is predicted in nanoribbons providing a veryclose analog of atomic collapse in relativistic physics. ii)The appearance of multiple energy bands leads to a richerspectrum as compared to pristine graphene with the ap-pearance of multiple quasi-bound electron states beloweach energy band. iii) In the case of zig-zag nanoribbonsan extra band of bound states appear, which are predom-inantly localized at the edge of the nanoribbon. The flatband character of those edge states are the origin of thesequasi 1D bound states and its LDOS consists of two iden-tical peaks whose separation increases with energy. Thebound character of these states are also a consequence ofthe fact that scattering on a zig-zag edge does not allowintervalley scattering.Last we would like to make a remark about the experi-mental feasibility of observing the effects predicted in thispaper. The production of different types of nanoribbonshave been realized over the last few years [35–39]. Re-cently [40], perfect edges in 2D materials was realized byusing a combination of top-down lithography with a nearanisotropic wet etching process. Placing the charges onthe nanoribbons should be possible using an STM tip asrealized in Ref. [9] for bulk graphene. Such an STM tipcan also be used to measure the LDOS. 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