Resonant scattering due to adatoms in graphene: top, bridge, and hollow position
AAPS/123-QED
Resonant scattering due to adatoms in graphene: top, bridge, and hollow position
Susanne Irmer, ∗ Denis Kochan, Jeongsu Lee, and Jaroslav Fabian
Institute for Theoretical Physics, University of Regensburg,93040 Regensburg, Germany (Dated: September 24, 2018)We present a theoretical study of resonance characteristics in graphene from adatoms with s or p z character binding in top, bridge, and hollow positions. The adatoms are described by two tight-binding parameters: on-site energy and hybridization strength. We explore a wide range of differentmagnitudes of these parameters by employing T -matrix calculations in the single adatom limit andby tight-binding supercell calculations for dilute adatom coverage. We calculate the density of statesand the momentum relaxation rate and extract the resonance level and resonance width. The topposition with a large hybridization strength or, equivalently, small on-site energy, induces resonancesclose to zero energy. The bridge position, compared to top, is more sensitive to variation in theorbital tight-binding parameters. Resonances within the experimentally relevant energy window arefound mainly for bridge adatoms with negative on-site energies. The effect of resonances from thetop and bridge positions on the density of states and momentum relaxation rate is comparable andboth positions give rise to a power-law decay of the resonant state in graphene. The hollow positionwith s orbital character is affected from destructive interference, which is seen from the very narrowresonance peaks in the density of states and momentum relaxation rate. The resonant state showsno clear tendency to a power-law decay around the impurity and its magnitude decreases stronglywith lowering the adatom content in the supercell calculations. This is in contrast to the top andbridge positions. We conclude our study with a comparison to models of pointlike vacancies andstrong midgap scatterers. The latter model gives rise to significantly higher momentum relaxationrates than caused by single adatoms. PACS numbers: 72.10-d, 72.15.Lh, 72.80.Vp, 81.05.ue
I. INTRODUCTION
In the past decade, graphene research has maderemarkable progress; from its first experimentalcharacterization , the way was paved towards high-quality graphene devices and proximitized graphene asingredients for electronic and spintronics applications .Graphene was the first realized two-dimensional crys-tal material with a linear dispersion at low energy. Asthe low-energy electrons can be described by an effec-tive Dirac equation for massless fermions, graphene wassuggested for studies of relativistic effects such as Kleintunneling or zitterbewegung . Apart from this funda-mental interest in the two-dimensional carbon allotrope,efforts were taken to tailor graphene properties for elec-tronic and spintronics devices.On the one hand, proximity effects in graphene wereexplored. It was found that exchange interaction canbe induced in graphene by placing it on a ferromagneticinsulator or, separated by a tunnel barrier, on a ferro-magnetic metal . Additionally, proximity-induced largespin-orbit coupling can cause topological effects ,giant spin lifetime anisotropy , and, together withproximity exchange, transport magnetoanisotropies .On the other hand, local adsorbates on graphene can beused to functionalize graphene. For example, graphene’sintrinsic spin-orbit coupling—of the order of 10 µ eV —was shown to be increased by more than a factor of100 by adatoms, such as hydrogen and copper mak-ing the spin-Hall effect accessible in graphene . Itwas also predicted that neutral adatoms with large spin- orbit coupling may stabilize the quantum spin Hallstate in graphene although experimental challengesstill remain . Furthermore, unconventional transportregimes were reported in theoretical investigations of spe-cific kinds of disorder .In experiments on orbital transport, long-rangeCoulomb scattering of charged adatoms canstrongly affect the measurements, whereas short-rangescattering off adatoms, on the other hand, highly in-fluences spin relaxation as the electrons feel theadsorbate induced local spin-orbit coupling ormagnetic moment. The local magnetic moments orig-inate, for example, from sp defects such as hydrogenadatoms , organic molecules , or vacancies .Vacancies furthermore give rise to zero-energy statesin graphene . Due to the small density of states atlow energy, graphene is especially sensitive to such in-duced states that affect strongly transport by resonantscattering . Another source for resonant states atlow energy can be substitutional impurities or ad-sorbates in graphene. The latter have been studied byexplicit tight-binding and density-functional theory cal-culations of specific adatoms . It was also re-alized by basic symmetry analysis that the adsorptionposition of an adatom plays an important role for theresonance scattering mechanism . For example, itwas established that the s orbital of an adatom in thehollow position is effectively decoupled from the states ofgraphene so that resonance scattering of such an orbitalis strongly suppressed.Here, we study resonant scattering off single adatoms a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b on graphene for the three stable adsorption positions,namely, top, bridge, and hollow, within the T -matrixformalism. We concentrate on adatoms with s or p z or-bital character and characterize them by hybridizationstrength and on-site energy in a minimal tight-bindingmodel. Our work extends and connects previous theoret-ical studies that are available in the relevant literature onthis topic. For example, Wehling et al.
59 studied withinthe T -matrix formalism single and double substitutionalimpurities. They showed the impact of selected orbitalparameters on the local density of states around the im-purity and addressed also the case of magnetic impurities.Here, we do not consider substitutional impurities butrather adsorbed elements which alter, for example, theenergy dependence of the local density of states. In con-trast, Robinson et al.
57 studied H + and OH − adsorbingin the top position. For small impurity concentrationsthey employed the T -matrix formalism and showed therise of an asymmetry in the conductivity due to the ad-sorbate in contrast to the symmetric contribution of lo-calized charged scatterers. We calculate resonance mapsthat scan a large portion of the orbital parameter spacefor resonance levels forming in the density of states andthus cover a broad variety of possible adsorbate realiza-tions on graphene. These maps show that adatom in-duced peaks in the density of states are in bridge andhollow positions much more sensitive to the variation oforbital parameters than in top position. Furthermore,the density of states and momentum relaxation rate showthat hollow adatoms are (almost) not hybridizing withthe π states of graphene. Considering the limit of diluteadatom concentration on graphene within supercell cal-culations, we investigate the localization of the resonantstates and find a clear power-law decay for top and bridgeadatoms in contrast to hollow adatoms. We complementour resonance analysis by a comparison of induced res-onances from general adatoms with vacancies and themodel of strong midgap scatterers in graphene.The work of Ferreira et al.
55 assumes a strong resonantscatterer sitting in the top position on graphene. In theirstudy of conductivity in single-layer and (biased) bilayergraphene they stress that the first Born approximationis not valid for strong resonant scatterers. Using partialwave analysis, they derive under certain approximationsan analytic formula for the influence of strong resonantscatterers on the conductivity. We comment later in themanuscript on the applicability of their assumptions andstress the consequences for quantitative analyses basedon this formula.The paper is organized as follows. We introduce inSec. II the framework of T -matrix formalism and inves-tigated adatom models. Sections III A, III B, and III Cpresent our resonance analysis for the top, bridge, andhollow adsorption position, respectively. The localiza-tion of the resonant states is discussed in Sec. III D, fol-lowed by a comparison between adatoms and vacanciesin Sec. III E, before we conclude in Sec. IV. II. METHODA. T -matrix formalism We study resonances from monovalent adatoms ongraphene in the single adatom limit within the non-perturbative T -matrix approach. Given a system de-scribed by the Hamiltonian H = H + V , with V be-ing the perturbation to the unperturbed system H , theretarded Green’s operator satisfies (cid:2) E + − H (cid:3) G ( E + ) = , (1)where E + = E + i δ and δ → G ( E + ) = G ( E + ) + G ( E + ) T G ( E + ) , (2)with T = V [ − G ( E + ) V ] − being the T -matrix. In thecase of a single adatom on graphene we use the standardnearest-neighbor tight-binding Hamiltonian of graphene H (hopping strength t = 2 . H = − t (cid:88) (cid:104) l,m (cid:105) | c l (cid:105) (cid:104) c m | , (3)whereas the presence of the adatom enters as a local(energy-dependent) perturbation V . We obtain this per-turbation by integrating out the adatom from the systemby the L¨owdin transformation (see Sections II B and IIIfor details).From the T -matrix and the Green’s functions of the un-perturbed graphene (see Appendix A), we directly ob-tain the perturbed density of states (DOS) per atom ν ( E ) = ν ( E ) + ∆ ν ( E ), where ν ( E ) ≈ | E | /W is theunperturbed DOS per atom with W = (cid:112) √ πt , and∆ ν ( E ) = ηπ Im Tr (cid:20) ∂∂E G ( E + ) T ( E + ) (cid:21) , (4)is the correction to the DOS of pristine graphene intro-duced by the impurity , where η = 1 / (2 N ) is the impu-rity concentration and N is the number of unit cells ingraphene.The itinerant electrons resonantly scattering off theadatom form the virtual bound state inducing a peakin the DOS . At the energy of the peak position, theresonance energy, the wave function is power-law local-ized around the impurity due to the hybridization ofthe impurity level with graphene’s low, though nonzero,DOS. The width (full width at half maximum) Γ of theresonance represents the resonance life time τ = (cid:126) / Γ.To characterize the resonances, we use the DOS, butwe also show the resonant behavior in the momentum re-laxation rate τ − m (see Appendix B for details). Note thatthe width of the resonance peaks in the momentum re-laxation rate is in general different from the peak widthsin the DOS. ωε (a) ωε (b) εω (c) FIG. 1. (Color online) Sketch of the orbital hopping Hamil-tonian for adatoms in (a) top, (b) bridge, and (c) hollow po-sition. The adatom is modeled by on-site energy ε and hy-bridization ω which connects the adatom to its nearest neigh-bors in graphene. B. Adatom models
We describe the adatom on graphene in the single-electron picture by an on-site energy ε of a single adatomorbital | X (cid:105) and its hybridization ω to the nearest car-bon neighbors in graphene. The orbital is assumed tobe invariant under C v point group, so it has s or p z or-bital character. This kind of model has already beenused successfully in tight-binding investigations basedon first-principles calculations for the top and bridgepositions .Three stable adsorption positions are typical foradatoms on graphene: top, bridge, and hollow. As de-picted in Fig. 1, the adsorption positions defer by thenumber of carbon hybridization partners for the adatom.While in the top position only one carbon atom con-tributes to the adsorption bond, bridge and hollow posi-tions offer two and six bonding partners, respectively, forthe adatom. The system is described by the Hamiltonian H = ε | X (cid:105) (cid:104) X | + ω (cid:88) (cid:104) X,l (cid:105) ( | X (cid:105) (cid:104) c l | + H . c . ) + H , (5)where l counts one, two, or six carbon sites, depending onthe adsorption position, and H is the pristine grapheneHamiltonian, Eq. (3).By down-folding H , we obtain the Hamiltonian H = H + V , including only graphene degrees of freedom, andan energy-dependent perturbation acting on the nearestcarbon neighbor(s) of the adatom, V ( E ) = | ω | E − ε P , (6)where E is the energy and P is the projection to the spaceof states formed by the p z orbitals of the hybridizationpartners of the adatom, P = (cid:88) (cid:104) X,l (cid:105) | c l (cid:105) (cid:88) (cid:104) X,m (cid:105) (cid:104) c m | . (7)The hybridization partners are therefore coupled amongthemselves by V ( E ). The non-vanishing block of the T -matrix can be written as˜ T = | ω | E − ε − ω A ( E ) P , (8) where A describes a combination of (retarded) Green’sfunctions that depend on the adsorption position, A = G ( E ) (top)2 (cid:2) G ( E ) + G AB12 ( E ) (cid:3) (bridge)6 (cid:8) G ( E ) + G AB14 ( E )+ 2[ G AB12 ( E ) + G NN13 ( E )] (cid:9) (hollow) (9)Here, G ( E ) is the (retarded) on-site Green’s function, G AB12 and G AB14 are the nearest and third-nearest neighborGreen’s functions of unperturbed graphene, respectively,which naturally couple opposite sublattices. Second-nearest neighbors on the same sublattice are coupled by G NN13 (see Appendix A for details).
III. RESULTS AND DISCUSSIONA. Top position
Following the procedure of Sec. II, we extract the res-onance energy E res and width Γ from the DOS undervariation of orbital parameters ω and ε and show DOSand τ − m for specific parameters, see Fig. 2. We restrictourselves to the experimentally relevant energy range of E res in [ − . , .
3] eV, which is equivalent to the variationof carrier density in the range [ − . , .
5] cm − .If we lower ω for a fixed ε , we gradually decouple theadatom from graphene. At ω = 0, the isolated adatomlevel induces a δ -peak on top of the DOS at E = ε .In the top position the resonance energy is mainly de-termined by the singularity in the denominator of the T -matrix. Since the real part of the Green’s function isan odd function of E (the imaginary part is even), theresonance energy changes sign, E res → − E res for fixed ω and ε → − ε . Apart from the sign in E res , the maps inFig. 2(a) therefore exhibit mirror symmetry with respectto the ε = 0 axis.Figure 2(b) presents DOS and momentum relaxationrate τ − m for selected parameter sets, indicated by thepath (1)-(2)-(3) in panel (a). Along path (1) to (2)the resonance level behaves as argumented above: twoadatoms with same ω = ω = ω but ε = − ε induceresonances on opposite sites of zero energy. Increasingthe hybridization strength ω for fixed ε , path (2) to (3),the resonance level shifts closer to zero with decreasingwidth. In the limit of ω → ∞ we have the effective po-tential on the adsorption site, Eq. 6), ω / ( E − ε ) → ∞ ,which enforces the wave function to vanish there. Thislimit simulates a vacancy in graphene, which induces azero-energy mode (see also Sec. III E). For a gen-eral impurity in top position we see that the larger theresonance energy E res is the larger is the resonance widthΓ. This is because with increased energy there are moregraphene states to which the impurity level can coupleto. (a) D O S [ ( e V a t o m ) − ] E [eV] -0.2 0 0.2 10 − − − τ − m [ p s − ] E [eV] (1)(2)(3) (1)(2)(3) (b) FIG. 2. (Color online) Resonance and momentum relaxationcharacteristics due to adatoms in top position. (a) Resonanceenergy E res and width Γ are shown as functions of ε and ω .(b) Snapshots of DOS and τ − m at three parameter sets (1) ω = 5 eV and ε = − ω = 5 eV and ε = 1 eV, and(3) ω = 7 eV and ε = 1 eV. The resonance levels are, re-spectively, at (1) E res = −
130 meV with Γ = 109 meV, (2) E res = 130 meV with Γ = 109 meV, and (3) E res = 64 meVwith Γ = 50 meV. DOS data are shown for adatom concen-tration of η = 10 ppm, for better resolution, and momentumrelaxation rates for realistic η = 1 ppm. The momentum relaxation rate shows characteristicpeaks at the resonance energies obtained from the DOScalculations with slightly different widths. The fartheraway from zero energy a resonance level forms the morenoticeable is the electron-hole asymmetry in the graphsof τ − m , Fig. 2(b). In experiments with dilute adatom cov-erage η (cid:39) −
100 ppm, for example fluorinated grapheneof Ref.75 and 76, the asymmetry is most probably maskedby the symmetric momentum relaxation rate profile ofcharged impurities in the substrate or additional strongmidgap scatterers contributing to transport. Asym-metric transport behavior was in contrast observed inhighly fluorinated graphene samples . B. Bridge position
The bridge position, Eq. (9), is affected by the inter-play of Green’s functions with different behavior under E → − E (see Appendix A). The symmetry arguments (a) D O S [ ( e V a t o m ) − ] E [eV] -0.15 0 0.15 10 − − − τ − m [ p s − ] E [eV] (1)(2)(3) (1)(2)(3) (b) FIG. 3. (Color online) Resonance and momentum relaxationcharacteristics due to adatoms in the bridge position. (a)Resonance energy E res and width Γ are shown as functions of ε and ω . (b) Snapshots of DOS and τ − m at three parameter sets(1) ω = 3 eV and ε = − . ω = 3 eV and ε = − . ω = 3 . ε = − . E res = −
27 meV with Γ = 17 meV,(2) E res = 14 meV and Γ = 9 meV, and (3) E res = 60 meV andΓ = 42 meV. DOS data is shown for adatom concentration of η = 100 ppm, for better resolution, and momentum relaxationrates for realistic η = 1 ppm. for resonance levels of the previous section are no longervalid. Figure 3(a) shows E res and Γ extracted from theDOS for the bridge position. We observe that the pa-rameter region leading to resonances, in the fixed energyinterval [ − . , .
3] eV, is dominated by negative on-siteenergy ε and constrained to a smaller region comparedto the top position. The resonance level position is muchmore sensitive to variation of parameters ω and ε whichwas also observed in Ref. 59 for substitutional doubleimpurities. Equating there the coupling strength U be-tween the substitutional impurities with their on-site po-tentials U leads to a local perturbation comparable toEq. (6) for the bridge position. However, the pertur-bation describing the substitutional double impurity isenergy independent, whereas for a bridge adatom it isenergy dependent due to the down-folding process.Similar as in the case of a single top adatom we canshift the resonance level for fixed ω from negative to posi-tive energies upon increasing ε along path (1) to (2). Thetransition from negative to positive resonance energy is (a) D O S [ ( e V a t o m ) − ] E [eV] -0.2 0 0.2 10 − − − − − τ − m [ µ s − ] E [eV] (1)(2)(3) (1)(2)(3) (b) FIG. 4. (Color online) Resonance and momentum relaxationcharacteristics due to adatoms in the hollow position. (a)Resonance energy E res is shown as a function of ε and ω .Widths Γ (not shown) are not resolved and are estimatedto be Γ ≤ τ − m at threeparameter sets (1) ω = 0 . ε = − .
08 eV, (2) ω =0 . ε = 0 .
02 eV, and (3) ω = 0 . ε = 0 .
02 eV.The resonance levels are, respectively, at (1) E res = −
43 meV,(2) E res = 55 meV, and (3) E res = 99 meV. DOS data isshown for adatom concentration of η = 500 ppm, for betterresolution, and momentum relaxation rates for realistic η =1 ppm. also visible in the DOS and τ − m in Fig. 3(b). Though,fixing the on-site energy and increasing the hybridizationstrength shifts the resonance further away from zero en-ergy. This behavior is in contrast to the top position andoriginates from the effective coupling via the adatom be-tween the two carbon hybridization partners in graphene.Still we observe the natural broadening of peaks withincreasing resonance energy. Furthermore, the magni-tude of typical momentum-relaxation rate is comparableto the top position. C. Hollow position
An adatom in the hollow position that preserves the C v symmetry of the hexagonal graphene, as realized inthe model of Sec. II, is affected by destructive interfer-ence of electrons tunneling to the adatom. The effectivedecoupling of the adatom from a wide range of graphene D O S [ ( e V a t o m ) − ] E [eV] -0.2 0 0.2 0.4 10 − − − − τ − m [ p s − ] E [eV] tbhhp tbhhp FIG. 5. (Color online) Left and right panel show DOS andmomentum relaxation rate, respectively, for an adatom with ω = 0 .
54 eV and ε = 0 .
02 eV in different adsorption positions,namely top (t), bridge (b), and hollow (h). Correspondingresonant energies and FWHMs are (t) E res = 18 meV and Γ =3 meV, (b) E res = 82 meV and Γ = 10 meV, and (h) E res =261 meV and Γ ≤ E res = 286 meV, Γ = 114 meV. For bettervisibility, a concentration of η = 500 ppm is used in the DOS,the momentum relaxation rate data is shown for η = 1 ppm. continuum states in the Brillouin zone leads to distinc-tive features in several contexts such as local spin-orbitcoupling , scanning tunneling spectroscopy ,Kondo effect , Anderson localization , and graphenefor chemical sensing . We focus here on the dependenceof the resonance level on the orbital parameters describ-ing the hollow adatom in direct comparison to the topand bridge adsorption positions and show the results fora large parameter space, which was to our knowledge notaddressed before.Figure 4(a) displays the drastic reduction of the( ω, ε )-parameter space for resonance levels in energyrange [ − . , .
3] eV. Following the path (1)-(3) in param-eter space we see similar dependence of the position ofresonance levels on ω and ε as in bridge position. Wedo not display corresponding peak widths as we can notresolve them satisfactorily. The restriction to the resolu-tion of the peak widths comes from an energy broadening δ = 1 meV in the calculation of Green’s functions that wekeep due to numerical reasons (see AppendixA). We es-timate the width Γ ≤ τ − m calculations displayed atFig. 4(b). The adatom level presents itself as a very sharppeak in the DOS and is strongly sensitive to variation of ω and ε . This sensitivity is more pronounced than in thebridge case. Even at higher energies where one would ex-pect a stronger hybridization of the impurity state withthe graphene due to larger availability of graphene states,the peaks show no broadening but sit on top of the DOSof pristine graphene. Furthermore, we see that the mo-mentum relaxation rates for E (cid:54) = E res are much smallerthan for top or bridge for the same ω and ε . Figure 5shows a comparison of the DOS and τ − m for the threeadsorption positions. The hollow momentum relaxationrate only increases with larger peak energy. The hollowadatom, m z = 0, appears therefore as a weak scattererin graphene compared to top and bridge position. Thisresult is in accordance with the findings of Ref. 68 wherethe authors investigate the scattering cross section fur-ther.Overall, we find in our analysis of resonance levels,DOS and τ − m clear signatures for the decoupling of thehollow adatom from the graphene continuum states. Thedecoupled state is also visible in the tight-binding bandstructure of graphene supercells with hollow adatoms asan dispersionless energy band on top of graphene’s bandstructure (see Appendix C).We can break the destructive interference, for exam-ple, by considering a toy model describing an adatomorbital with magnetic quantum number m z = 1 in hol-low position (see Appendix C). As shown in Fig. 5, thetoy model momentum relaxation rate is now comparablein magnitude to the top and bridge adatom. Broadeningof peaks in the DOS as well as the enhanced momentumrelaxation rate indicate the lifting of the destructive in-terference seen for m z = 0 and restored effective couplingto the graphene states. D. Localization of resonant states
To investigate the localization of states around the im-purity at resonance energy, we calculate with the tri-angle method the local density of states (LDOS) forrepresentative adatoms from tight-binding supercells ofsize 40 ×
40. These supercells mimic a dilute adatom con-centration of about 312 ppm. We extract the LDOS atthe peak energy E tbres of the DOS which is found in allcases close to the predicted resonance energies E res forthe single adatom limit. Figure 6 displays the LDOS de-pendence on the distance from the adatom along selecteddirections in the case of a specific top, bridge, and hollowadatom.First, we note that the LDOS around top, bridge, andhollow positions shows symmetrical behavior in accor-dance with the local point group symmetries C v , C v ,and C v , respectively, of graphene around the corre-sponding adatom. This symmetry is also seen in theLDOS as calculated from the down-folded Hamiltonianin the T -matrix formalism (see Fig. 6). The tight-bindingsupercell calculations are used for the quantitative analy-sis. Due to the finite size of the supercells we can only in-vestigate the short-range behavior around the impurity—approaching the supercell border, the LDOS values sat-urate.The wave function profile for the top position was,for example, already investigated in Ref. 73 in the sin-gle adatom limit where it was also pointed out that thepower-law decay exponent depends on the direction of the − − − − − L D O S r [ a cc ]
30 B30 A (a) ◦ − − − L D O S r [ a cc ]
90 B330 A (b) ◦ ◦ − − − L D O S r [ a cc ]
30 B30 A (c) ◦ FIG. 6. Localization of resonant states in graphene around(a) top, (b) bridge, and (c) hollow adatom along selected di-rections. Insets show scaled LDOS (sphere radii indicate mag-nitude) calculated from the T -matrix formalism with down-folded Hamiltonian on the graphene lattice. The colors referto the two sublattices A (red) and B (blue). Top and bridgeposition show power-law decay in the dominant directions (seeparameters in the main text), whereas hollow shows no cleartendency. path taken away from the impurity and the sublattice ofthe investigated site. Furthermore, it is well known thatfor a strong scatterer in the top position the resonantstate is more localized on the opposite sublattice —inthe extreme case of a vacancy the resonant state popu-lates exclusively the opposite (intact) sublattice and theLDOS decays as r − . For top adatoms, the decayexponents depend on the orbital parameters of the chosenadatom and thus its resonance energy. For ω = 0 .
54 eVand ε = 0 .
02 eV, corresponding E tbres = 19 meV, theLDOS is significantly smaller on sublattice A to which theadatom is adsorbed than on the opposite sublattice B, seeFig. 6(a). Along a selected line, starting at the adsorptionposition with 30 ◦ with respect to the x -axis, we extracta power-law decay on sublattice B, | ψ | ∝ r − p , p (cid:39) . | ψ | ∝ exp ( − qr ), q ≈ .
12, on sublattice A. The larger the resonant energythe broader the resonance peak gets due to stronger in-teraction with the graphene states, and the contributionson both sublattices will approach each other .The LDOS distribution for the bridge position aroundthe adatom is shown in Fig. 6(b). Its diamondlike shapereminds of two intertwined triangles with their centers onneighboring sites A and B. This appearance looks natu-ral if one imagines the bridge adatom as two neighbor-ing top adatoms or double substitutional impurities .The LDOS contribution on the hybridization partners ofthe bridge adatom is decreased due to the effective cou-pling between them which is mediated by the adatom[see Eq. (6)]. The overall resulting population of thetwo sublattices around the bridge adatom shows power-law decay: For a copper adatom on graphene in thebridge position , ω = 0 .
54 eV and ε = 0 .
02 eV, energy E tbres = 83 meV, we select two directions along 90 ◦ and330 ◦ shown in Fig. 6(b) with a clear power-law decaywith p ≈ .
09 and p ≈ .
45, respectively.As in the bridge case, the hollow adatom does not dis-tinguish sublattices which is clearly seen in the LDOSdistribution in Fig. 6(c). Extracting the LDOS on the lat-tice sites along the 30 ◦ direction for a hollow adatom with ω = 0 . ε = 0 .
02 eV and peak energy E tbres = 99 meVwe see tendencies to both power-law and exponential de-cay, p ≈ .
29 and d ≈ .
62 depending on the sublattice.Note that the resonance energy is comparable to the pre-vious bridge example. We know from the previous sectionthat the hollow position ( m z = 0) suffers from destruc-tive interference which seems to be related to the occur-rence of both power-law and exponential behavior on thesame order of magnitude. Indeed, we found that withlowering the adatom content in supercell calculations thehollow adatom ( m z = 0) loses LDOS contribution muchfaster than the top or bridge adatom. On the contrary,a clear power-law decay is seen for a toy calculation with m z = 1 for the same orbital parameters (see Appendix C,Fig. 9(c)). E. Vacancy vs. adatom
The top-position adatom is often compared to a va-cancy, the prototype of a strong resonant scatterer. Thisstructural defect in graphene induces a sharp midgapstate at zero energy. Leaving aside reconstruction of a single vacancy site in graphene, the vacancy canbe modelled either by removing the vacancy site fromthe graphene lattice or, equivalently, by assigning a lo-cal potential U to the corresponding site and taking U → ∞ . In our model description of the topadatom, this would mean to send the effective on-site D O S [ ( e V a t o m ) − ] E [eV] -0.2 0 0.2 10 − − − τ − m [ p s − ] E [eV] VacHCuF VacHCuF (a) − − − . . − . . − − R e ff [ Λ − ] E [eV] VacHCuF τ − m [ p s − ] E [eV] Vac (1)Vac (2)Vac (3) (b) (c)
FIG. 7. (Color online) Adatom in top position and vacancymodel. Panel (a) compares DOS (left) and momentum re-laxation rate (right), respectively, of a vacancy (Vac) withadatoms in the top position: hydrogen adatom (H), ω =7 . ε = 0 .
16 eV, copper adatom (Cu), ω = 0 .
81 eV and ε = 0 .
08 eV, and fluorine (F), ω = 5 . ε = − . η = 10 ppm is usedin the DOS, the momentum relaxation rate data are shown for η = 1 ppm. Fluorine in the top position induces a broad reso-nance, Γ = 277 meV, at about E res = −
262 meV, leading to asmall shoulder in the DOS and momentum relaxation rate atnegative energies. Copper and hydrogen show stronger fea-tures at lower energies: copper induces a resonance level at E res = 68 meV with Γ = 9 meV. Hydrogen with E res = 7 meV,Γ = 5 meV, is comparable to a vacancy at zero energy, whichis fully symmetric with respect to negative and positive ener-gies. Panel (b) shows the dependence of the effective impurityradius R eff on the energy for a vacancy as well as hydrogen,copper, and fluorine adatoms. Panel (c) displays the momen-tum relaxation rate for a vacancy under different approxima-tions to Eq. (11). Neglecting the imaginary part of G , graph(2), increases τ − m slightly (by about 20% at E = 200 meV)compared to the exact result, graph (1). Further overestima-tion of τ − m (by a factor of 4 at E = 200 meV) originates inan artificially increased R eff = 4 . − , graph (3). potential ω / ( E − ε ) → ∞ in Eq. (8). We obtain the T -matrix for a vacancy , T vac = − G ( E ) | c (cid:105) (cid:104) c | . (10)The DOS and τ − m resulting from Eq. (10) are shownin Fig. 7(a). The vacancy introduces a sharp resonantpeak at zero energy, fully symmetric with respect to neg-ative and positive energies. This symmetric appearancedoes not hold for general top adatoms as presented inSection III A. DOS and τ − m for the adatoms fluorine ,copper , and hydrogen are included in Fig. 7(a). Theasymmetry is very small for a strong resonant scattererin the top position, such as hydrogen with a resonancelevel close to zero energy, which leads to the similarity ofa hydrogen adatom to a vacancy in graphene.Using the Boltzmann transport formalism, with thetransition rates from the T -matrix and the analytic resultfor the on-site Green’s function G (Appendix A), weobtain the conductivity for graphene in the presence ofan adatom in the top position, σ = e h πη E W (cid:20) ln (cid:18) | E | (cid:126) v F R eff (cid:19) + π (cid:21) , (11)where we have introduced R eff ( E ), R eff ( E ) = Λ − exp (cid:20) − W ω ( E − ε ) E (cid:21) . (12)with the momentum cut-off Λ (see Appendix A). Thequantity R eff ( E ) has the dimension of a length and canbe interpreted as an effective radius of a top positionedscatterer. For a vacancy we get R Vaceff = Λ − ≈ . R eff forthe top adatoms hydrogen, copper, and fluorine, in com-parison to a vacancy. The effective radius diverges atzero energy for the adsorbates.We can directly compare Eq. (11) for a top adatom tothe model of a strong midgap scatterer (SMS) or vacancyof Refs. 10, 54, and 55. There, the defect is modeled asa potential disk with finite (energy independent) radius R . The scattering cross section and conductivity are ob-tained from partial wave decomposition. The conductiv-ity reduces to σ SMS = e h k F π n i ln ( k F R )= e h π A uc n i E W ln (cid:18) | E | (cid:126) v F R (cid:19) , (13)where n i is the impurity concentration per unit area .Comparing the result of the SMS model to Eq. (11), weobtain σ = 2 · σ SMS if we set R eff = R and neglect theterm π / G . The quantities η and n i are related by η = 2 n i /A uc where A uc is the unit cell area. The additional factor oftwo was also found in the vacancy study of Ref. 63.Note that our T -matrix formulation uses a fixed mo-mentum cut-off that preserves the number of states. Inthe analysis of experimental data one uses σ SMS and fitsthe radius R together with n i , assuming that R is at theorder of a few angstroms . Figure 7(c) shows theeffect of the approximations to the momentum relaxationrate for a vacancy. Neglecting the imaginary part of G and using R = 4 . · Λ − ≈ . E = 200 meV for a vacancy is overestimatedby a factor of four. As the momentum relaxation rate is directly proportional to the impurity concentration, a si-multaneous fit of n i and R to experimental resistivitydata can lead to an underestimation of n i .Clearly, the model for strong midgap scatterers is notdesigned to reflect adatoms with resonance energies sig-nificantly different from zero. The previous sectionshave shown that the resonance level strongly dependson the orbital parameters and equally on the adsorp-tion position. Fluorine, an adatom also binding in thetop position , induces a dominant asymmetry in themomentum-relaxation rate, Fig. 7(a), due to a far-off (atabout -300 meV) and broad resonance. A single fluorineadatom or dilute concentration of non-interacting fluo-rine adatoms is not captured by a vacancy or SMS model.Note, that the SMS model yields generally higher τ − m than the T -matrix model for adsorbates, which can beunderstood as the consequence of several approximationsto G in the SMS approach. In experiments where theconductivity is well described by Eq. (13), also additionalsources of strong midgap scatterers, charged impurities ,or clusters can play a role so that the analysis of theexperimental data has to be done more carefully. IV. SUMMARY
In summary, we studied the effect of adsorption po-sition and orbital parameters (on-site energy ε and hy-bridization strength ω ) of single adatoms on graphene tothe formation of resonances in an energy range that isaccessible by experiments. Overall, we find significantdifferences between the three adsorption positions top,bridge, and hollow.In the top position, the resonance level lies closer tozero energy the larger ω , or the smaller ε , is. Especially,the resonance energy E res changes sign under ε → − ε .The resonance levels leave distinct features in DOS and τ − m . For resonance energies far away from zero energy, apronounced electron-hole asymmetry is predicted.For the bridge position, we find that the resonancelevel is more sensitive to changes in orbital parameters:the parameter range leading to resonances in the studiedenergy range is dominated by negative ε and increasing ω shifts resonance levels to higher energies. For same or-bital parameters, the resonance level for the bridge posi-tion lies at higher energy than for the top position. Rates τ − m are in magnitude comparable to the top position.A hollow adatom with s or p z orbital, on the contrary,acts as a weak scatterer in graphene as it is effectivelydecoupled from graphene due to destructive interferenceof electrons hopping on and off the adatom. Resonancelevels are seen within the studied energy range only for anarrow window of orbital parameters. Furthermore, theresonance peaks resulting in DOS and τ − m are very nar-row, and the rates τ − m are significantly smaller than inthe top or bridge positions. From the LDOS calculationof tight-binding supercells, we find that the resonancestate induced by a hollow adatom shows no clear ten- A B A B A B δ δ δ FIG. 8. (Color online) Site labeling convention inside thegraphene unit cell (red dashed diamond) as used for the realspace representation of the Green’s functions. The nearest-neighbor connection vectors δ j , j = 1 , , dency to power-law localization. On the contrary, topand bridge adatoms give rise to resonant states with aclear power-law decay.Finally, we compare our findings valid for monovalentadatoms on graphene with vacancies and the SMS modelof Refs. 10, 54, and 55. Both vacancy and SMS inducezero energy resonances resulting in electron-hole symmet-ric τ − m , which is also approximately the case for adatomswith a resonance level very close to zero energy, for exam-ple, hydrogen . However, variation of the effective de-fect radius R in the SMS model can enhance (and overes-timate) τ − m significantly compared to vacancies or singleadatoms from our T -matrix analysis. Therefore analyz-ing experimental data for adsorbates on graphene one hasto take carefully into account the particular limits of thedifferent models.Our results can help to understand experimentaltransport studies with dilute adatom concentrations ongraphene. Especially our comprehensive resonance maps,i.e., the dependence of scattering on different adatoms indifferent adsorption positions, can help to clarify the roleof specific adatoms in the limit of short-range scattering. ACKNOWLEDGMENTS
This work was supported by the European Union’sHorizon 2020 research and innovation programme undergrant agreement No. 696656, the DFG SFB Grants. No.689 and 1277 (A09), and GRK Grant No. 1570.
Appendix A: Green’s functions for graphene
Since we investigate resonant scattering with local im-purities, we focus on the real space representation of re-tarded Green’s functions. We start from the (full) k -dependent Hamiltonian for graphene whose eigenenergiesare given by ε nk = nt | f ( k ) | , (A1)where n = 1 for the conduction band and n = − f ( k ) = exp(i k . δ ) + exp(i k . δ ) + exp(i k . δ ). See Fig. 8 for our convention of the unitcell and the nearest-neighbor vectors δ j = (cos( j π − π ) , sin( j π − π )) a cc with j = 1 , , G ( E ) = lim δ → G ( E + ) = lim δ → VN (cid:90) d k (2 π ) E + E +2 − t | f ( k ) | , (A2)second, the Green’s function coupling opposite sublat-tices, G AB lm ( E ) = lim δ → G AB lm ( E + )= lim δ → VN (cid:90) d k (2 π ) − tf ( k )e i k . ( R l − R m ) E +2 − t | f ( k ) | , (A3)describing the propagation from site R m on sublattice B to site R l on sublattice A , and, third, the Green’sfunction coupling different sites at R m and R l on thesame sublattice N = A , B, G NN lm ( E ) = lim δ → G NN lm ( E + )= lim δ → VN (cid:90) d k (2 π ) E + e i k . ( R l − R m ) E +2 − t | f ( k ) | . (A4)Interchange of sublattices leads to complex conjugationof f ( k ) in Eq. (A3). Note that in this convention, vec-tors R m point to the actual positions of the lattice sites m and not to the unit cell hosting those sites. For thecalculation of the DOS, see main text Sec. II, we use thederivative of the Green’s operator, ∂∂E G , which we ob-tain in real-space from Eqs. (A2)-(A4) by differentiatingthe integrand with respect to energy E .There are several methods to calculate the Green’sfunctions for graphene. One, rather intuitive, approachrelies on the linear approximation of the spectrum aroundthe k -points K ± in the first Brillouin zone. The integra-tion is transformed to two integrals around K ± up to amomentum cutoff Λ, which ensures conservation of thenumber of states. An analytic result can then be directlyobtained for G , G ( E ) = EW ln (cid:12)(cid:12)(cid:12)(cid:12) E W − E (cid:12)(cid:12)(cid:12)(cid:12) − i π | E | W Θ( W − | E | ) , (A5)where W = (cid:126) v F Λ = (cid:112) √ πt is the cut-off energy andΘ( x ) the Heaviside step function.On the contrary, analytic formulas for G AB lm and G NN lm require the approximation Λ → ∞ . As alreadypointed out by Refs. 61 and 74, this approximationshould be taken with a grain of salt and the momentum-cutoff has to be chosen carefully in general .In fact, we found that applying Λ → ∞ in G AB lm and G NN lm has severe effects on the resonance energy calcu-lation in the bridge and hollow positions of Secs. III Band III C. Resonance levels in the bridge position shift0significantly: the resonance level of copper in the bridgeposition, as calculated in Ref. 23, changes from E res =128 meV to E res = 82 meV when going from the linearapproximation with Λ → ∞ to numeric integration overthe full Brillouin zone. Furthermore, peaks observed intight-binding supercell calculations for hollow adatomsare absent in the calculation with the linearized modeland Λ → ∞ .We therefore use full numerical integration over the2D Brillouin zone to obtain all Green’s functions thatwe need for the calculations discussed in the main text.Due to computational reasons we keep a finite imaginarypart of δ = 10 − eV in our calculations which inducesenergy broadening. We checked, by rescaling δ → δ/ δ → δ/
2. Very nar-row peaks Γ < δ = 1 meV.Therefore, in the hollow position, for m z = 0, all peakswidths appear to be twice the energy broadening. In thetop and bridge positions, we obtain, despite finite δ , thecorrect order of magnitude for Γ values.Symmetry considerations reduce the number ofGreen’s functions that are needed for the different ad-sorption positions. Green’s functions G AB lm show, apartfrom the natural translational symmetry, threefold ro-tational symmetry. Green’s functions G NN lm are invari-ant under sixfold rotations . Furthermore, it holds that G AB lm ( E ) = G BA ml ( E ). We thus end up with a maximumof four Green’s functions (and corresponding derivatives)that have to be evaluated for resonance energy calcula-tions in the hollow position, G , G AB12 = G AB16 = G AB32 , G NN13 = G NN15 = G NN35 and G AB14 = G AB52 = G AB36 . For ourconvention of the site labeling see Fig. 8.
Appendix B: Momentum relaxation rates
We consider elastic scattering off adatoms on graphene,excluding multiple-scattering events which are sup-pressed in the dilute (single) adatom limit. The start-ing point is the transition rate between states | nk (cid:105) and | n (cid:48) k (cid:48) (cid:105) of same eigenenergy ε nk , expressed by the general-ized Fermi’s golden rule, W nk | n (cid:48) k (cid:48) = 2 π (cid:126) |(cid:104) n (cid:48) k (cid:48) | T ( ε nk ) | nk (cid:105)| δ ( ε nk − ε n (cid:48) k (cid:48) ) . (B1)Symbol n = ± k is the wave vector. The adatom orbital parametersand adsorption position enter the rate via the T -matrixterm.From the transition rate we obtain (for isotropic scat-tering in k space) the elastic transport scattering rate byweighting the rate with the transport factor (1 − cos φ kk (cid:48) ) and summing over final states, i.e., τ nk − = (cid:88) k (cid:48) ,n (cid:48) (1 − cos φ kk (cid:48) ) W nk | n (cid:48) k (cid:48) . (B2)Here, φ kk (cid:48) is the angle between k and k (cid:48) vectors. Aver-aging over the Fermi-contour gives the momentum relax-ation rate at given energy E , τ − ( E ) = (cid:80) k,n τ nk − δ ( E − ε nk ) (cid:80) k,n δ ( E − ε nk ) . (B3)As we are interested in the effects near the charge neu-trality point we use the linearized graphene spectrumaround the Dirac points.. The momentum relaxationrates are directly proportional to the adatom concentra-tion η = 1 / (2 N ). Appendix C: Hollow adatom - toy model
As seen in Sec. III C, the adatom with a m z = 0 orbitalin the hollow position suffers from destructive interfer-ence. This is because the adatom orbital couples equallyto its six carbon neighbors . Considering an adatom or-bital with non-zero magnetic quantum number, the hy-bridization coupling ω acquires a different phase for dif-ferent neighbors and hence the destructive interferencecan be lifted.As a toy model, we therefore take into account anatomic orbital with magnetic quantum number m z = 1coupled to all six neighboring carbon sites. Due to trans-formation of the orbital under the angular momentumoperator L z , the hybridization coupling ω gains a phasefactor of exp(i φ ) under rotation of angle φ . The modifiedHamiltonian H (cid:48) reads H (cid:48) = ε | X (cid:105) (cid:104) X | + (cid:88) n =1 ω n ( | X (cid:105) (cid:104) c n | + H . c . ) + H . (C1)where ω n = ω exp(i( n − φ ) with φ = π/
6. Time-reversalsymmetry determines whether the coupling ω is purelyreal or imaginary. For the present analysis this does notplay a role as the hybridization enters the perturbation,Eq. (6), as the square of the absolute value.The extension of the model changes significantly theresonances under variation of the orbital parameters ω and ε , see Fig. 9(a). Compared to the m z = 0 case,Sec. III C, a much wider parameter range leads to reso-nances in the energy interval [ − . , .
3] eV, which are alsosignificantly broadened. This is a fingerprint of a strongcoupling between adatom and graphene. Band structurecalculations for a 40 ×
40 supercell, for ω = 0 .
54 eV and ε = 0 .
02 eV, reveal in the case m z = 0 the decouplingof the impurity level from the graphene, see Fig. 10.The flat non-dispersive band at about E = 261 meVshows that the hybridization with graphene is strongly1 ω ω e i π ω e i π − ω − ω e i π − ω e i π (a) D O S [ ( e V a t o m ) − ] E [eV] -0.2 0 0.2 10 − − − τ − m [ p s − ] E [eV] (1)(2)(3) (1)(2)(3) (b) − − − L D O S r [ a cc ]
30 B30 A (c) ◦ FIG. 9. (Color online) Resonance and momentum relax-ation characteristics for a toy-adatom in the hollow position( m z = 1 orbital). (a) Graphical representation of the phase-dependent coupling between the m z = 1 orbital and its carbonhybridization partners as implemented in the model. Reso-nance energy E res and width Γ are shown as functions of ε and ω . (b) Snapshots of DOS and τ − m at three parametersets (1) ω = 1 . ε = − . ω = 1 . ε = − . ω = 1 . ε = − . E res = −
94 meV with Γ = 72 meV, (2) E res = 40 . E res = 106 . η = 500 ppm and momentum relaxation rates for η = 1 ppm. (c) LDOS around the impurity and along the 30 ◦ direction are shown for ω = 0 . ε = 0 .
02 eV, extracted at E res = 129 meV of a 40 ×
40 supercell calculation. Both A(red) and B (blue) sublattice contributions follow power-lawwith a similar exponent. -0.4-0.3-0.2-0.100.10.20.30.4 K Γ M K E [ e V ] K Γ M K m z = 0 m z = 1 FIG. 10. (Color online) Tight-binding band structuregraphene with adatoms in the hollow position with m z = 0(left) and m z = 1 (right) orbital for a 40 ×
40 supercell.A flat non-dispersive band (black-red dotted) is observed atabout E = 261 meV for m z = 0 with an almost undisturbedgraphene band structure. In the case of m z = 1 clear bandanti-crossings appear in the energy range [0 . , .
4] eV showinga strong hybridization of the impurity state with graphene. suppressed. On the contrary, the m z = 1 case withthe same orbital parameters shows strong coupling be-tween the impurity and graphene with occurrence of aband anti-crossing around the predicted resonance level E res = 285 meV (Γ = 114 meV).The momentum-relaxation rate, see Fig. 9(b), reachessame values as in the top or bridge positions, seeSecs. III A and III B. Figure 9(c) displays the localizationof the resonant state in a 40 ×
40 supercell for ω = 0 . ε = 0 .
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