We present exact analytical and numerical results for the electronic spectra
and the Friedel oscillations around a substitutional impurity atom in a
graphene lattice. A chemical dopant in graphene introduces changes in the
on-site potential as well as in the hopping amplitude. We employ a T-matrix
formalism and find that disorder in the hopping introduces additional
interference terms around the impurity that can be understood in terms of
bound, semi-bound, and unbound processes for the Dirac electrons. These
interference effects can be detected by scanning tunneling microscopy.Comment: 4 pages, 7 figure

A. H. Castro Neto

Castro Neto A. H.

F. D. Klironomos

J. M. B. Lopes dos Santos

J. R. Santos

Li G. Andrei E. Y.

N. M. R. Peres

S.-W. Tsai

Wehling T. O. Balatsky A. V. Katsnelson M. I. Lichtenstein A. I. Scharnberg K. Wiesendanger R.

English

arXiv

Crossref

Electron waves in chemically substituted graphene

We present exact analytical and numerical results for the electronic spectra and the Friedel oscillations around a substitutional impurity atom in a graphene lattice. A chemical dopant in graphene introduces changes in the on-site potential as well as in the hopping amplitude. We employ a T-matrix formalism and find that disorder in the hopping introduces additional interference terms around the impurity that can be understood in terms of bound, semi-bound, and unbound processes for the Dirac electrons. These interference effects can be detected by scanning tunneling microscopy. © Europhysics Letters Association

Peres, NMR

Klironomos, FD

Tsai, S-W

Santos, JR

dos Santos, JMB Lopes

Neto, AH Castro

eScholarship - University of California

UC RiversideUC Riverside Previously Published WorksTitleElectron waves in chemically substituted graphenePermalinkhttps://escholarship.org/uc/item/67h9b5smJournalEPL (Europhysics Letters), 80(6)ISSN0295-5075AuthorsPeres, NMRKlironomos, FDTsai, S-Wet al.Publication Date2007-12-01DOI10.1209/0295-5075/80/67007 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaarXiv:0705.3040v1  [cond-mat.mtrl-sci]  21 May 2007Electron waves in chemically substituted grapheneN. M. R. Peres1, F. D. Klironomos2, S.-W. Tsai2, J. R. Santos1, J. M. B. Lopes dos Santos3, and A. H. Castro Neto41Center of Physics and Departamento de Fı́sica, Universidade do Minho, P-4710-057, Braga, Portugal2Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA3CFP and Departamento de Fı́sica, Faculdade de Ciências Univers dade de Porto, 4169-007 Porto, Portugal and4Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215,USA(Dated: August 26, 2018)We present exact analytical and numerical results for the electronic spectra and the Friedel oscillations arounda substitutional impurity atom in a graphene lattice. A chemical dopant in graphene introduces changes in theon-site potential as well as in the hopping amplitude. We employ aT -matrix formalism and find that disorderin the hopping introduces additional interference terms around the impurity that can be understood in terms ofbound, semi-bound, and unbound processes for the Dirac electrons. These interference effects can be detectedby scanning tunneling microscopy.PACS numbers: 73.20.Hb,81.05.Uw,73.20.-r, 73.23.-bIntroduction. Graphene [1], a one atom-thick layer ofgraphite, has been intensively studied [2, 3, 4, 5, 6, 7, 8, 9,10, 11, 12] due to its fascinating physical properties [8, 9,10].Graphene is a zero-gap semiconductor and its low-energyelectronic excitations are described in terms of a Dirac spec-trum. Because of this, disorder in the form of impurities[8, 11, 12, 13, 14], defects [15, 16, 17, 18] and surfaces[19, 20] can have a strong effect, especially when the chemi-cal potential crosses the Dirac point. In this work, we considerdilute chemical dopants in graphene incorporated as substi-tutional atoms, and calculate local single particle propertiessuch as the electronic energy spectra, local density of states,and electron density distribution, which shows Friedel oscil-lations as depicted in Fig. (1). The results presented herecan be measured by scanning tunneling microscopy (STM)[21, 22, 23].Atomic substitution in a carbon (C) honeycomb lattice ischemically possible for boron (B) and nitrogen (N) atoms.There have been several experimental studies of B and Nsubstitution in highly-oriented pyrolytic graphite (HOPG)[24, 25], graphitic structures [26], nanoribbons [27], carbonnanotubes [26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] andfullerenes [38]. When B or/and N atoms replace a carbonatom in a graphene sheet they have the following effects: (1)they act as impurity scattering centers; (2) they act as hole-or electron-dopants; (3) they introduce lattice distortions. Inthe case of B there is an increase of the absolute value of thehopping amplitude given its larger atomic radius (r ≃ 0.85Å)as compared to C (r ≃ 0.7Å). On the other hand, a N atomimpurity (r ≃ 0.65Å) can be modeled by a smaller hoppingintegral. For the change in the on-site energy, we assume theC on-site energy to be zero (our reference state); the smalleratomic number of B leads to a positive local energy, whereason a site occupied by N, the local energy should be smallerand therefore is modeled as a negative local energy. The ef-fect of a different on-site energy has been extensively studiedin the past [8, 11, 12, 13, 14], but the effect on the hopping hasbeen overlooked. In this work we study the combined effectof local changes in the atomic energy as well as the hoppingamplitudes. A vacancy has been modeled by an infinite, on-site energy potential, but it is also exactly represented bytheremotion of particular hopping processes from the Hamilto-nian. The opposite limit of very large hopping to the impuritysite corresponds to a four-atom molecule.FIG. 1: (color on line) Real-space electronic distributionand Friedeloscillations around an impurity with ionic radius smaller than carbon(such as N) (t0 = 0.5t).Hopping and potential disorder.The honeycomb lattice hasa unit cell represented in Fig. (2) by the vectorsa1 anda2,such that|a1| = |a2| = a (a =√3a0 ≃ 2.461 Å, wherea0is the carbon-carbon distance). In this basis any lattice vec-tor r is represented asr = na1 + ma2, with n,m integers.In Cartesian coordinates,a1 = a0(3,√3, 0)/2 and a2 =a0(3,−√3, 0)/2. The reciprocal lattice vectors are given by:b1 = 2π(1,√3, 0)/(3a0) andb2 = 2π(1,−√3, 0)/(3a0),and the vectors connecting anyA atom to its nearest neigh-bors are: δ1 = (a1 − 2a2)/3, δ2 = (a2 − 2a1)/3, andδ3 = (a1 + a2)/3. Using these definitions the Hamilto-nian can be written as:H = H0 + Vt + V0, whereH0 =−t∑r[b†(r)a(r)+b†(r−a2)a(r)+b†(r−a1)a(r)+h.c.],2an impurity atom213a 2a1t−t0t−t 0t−t 0δδδFIG. 2: (color on line) The honeycomb lattice with a carbon atomreplaced by an impurity atom, such a B or N. The local hoppingparameter changes fromt to t− t0.is the kinetic energy operator anda† (b†) are creation opera-tors in theA (B) sites. The operatorsVt = t0[b†(0)a(0) +b†(−a2)a(0) + b†(−a1)a(0) + h.c.] andV0 = ε0a†(0)a(0)are the impurity terms for hopping and potential disorder, r-spectively. In the particular caset0 = t, the scattering termVt represents a vacancy. In this work we consider the case ofzero chemical potential when the Fermi level crosses the Diracpoint. This is the case in which the system is most susceptiblto the presence of impurities. The equations of motion for theGreen’s functions can be readily written, and are given by:iωnGaa(ωn,k,p)=δk,p+∑q[λk,qGba(ωn,q,p)+ε0NcGaa(ωn,q,p)]iωnGba(ωn,k,p)=∑qλ∗q,kGaa(ωn, q,p)iωnGab(ωn,k,p)=∑q[λk,qGbb(ωn, q,p)+ε0NcGab(ωn,q,p)]iωnGbb(ωn,k,p)=δk,p+∑qλ∗q,kGab(ωn, q,p) ,whereλk,p = −tφp(δk,p − t0/Nct), φp = 1 + e−ip·a1 +e−ip·a2 , andNc is the total number of unit cells in the lattice.The sublattice symmetry is broken by the presence of the im-purity and this is manifested in the fact thatλpq 6= λqp and inthe asymmetric way in which theε0-term appears in the equa-tions above. The above set of equations can be solved exactly.The fact that the scattering termVt depends onφk leads to amore complex form for theT -matrix than usual. The exactsolution for the Green’s functions can be written in the form:Gaa(k,p) = δk,pG0k + g + h[G0k +G0p]+G0kTG0p ,(1)Gbb(k,p) = δk,pG0k +tφ∗kiωnG0kTG0ptφpiωn. (2)where all the terms also depend onω (omitted here forbrevity). They are defined as:g(ωn) = t20Ḡ0(ωn)/[NcD(ωn)], (3)h(ωn) = t0(t− t0)/[NcD(ωn)], (4)andT (ωn) = −iωnt0(2t− t0)− ε0t2NcD(ωn)(5)where the denominatorD(ωn) is given byD(ωn)=(t−t0)2+[iωnt0(2t−t0)−ε0t2]Ḡ0(ωn) (6)andḠ0(ωn) =∑k G0(ωn,k)/Nc with G0k = G0(ωn,k) =(iωn)/[(iωn)2 − t2|φk|2], the diagonal Green’s function forthe clean system.The significance of the termg(ωn) in (1) which onlyappears inGaa, is easily appreciated if we do the doubleFourier transform to real space. This term only contributestoGaa(0, 0), the return amplitude to the impurity site. The factor1/D(ωn) contains a sum over an infinite series of intermedi-ate scattering events, but the overall process is bounded andthet20 factor denotes hopping from the impurity to the nearestneighbor B-sites and back to the impurity site. A similar in-terpretation can be given to another term which only appearsin Gaa, namely,G0(ωn, k)h(ωn): this term contributes onlyto Gaa(r, 0) and describes an additional amplitude of propa-gation between the impurity site and anotherA site. No suchterms can, of course, appear inGbb when the inpurity is at anA site.The exact general solution for both diagonal components ofthe Green’s functions for a substitutional impurity with poten-tial and hopping disorder is contained in (1) and (2) . The lo-cal density of states (LDOS) can be obtained from the Green’sfunctions (1,2), after analytical continuationiωn → ω + i0+.For sites in theA andB sublattices, it is given by:ρν(r, ω) =−ImGνν(ω, r, r)/π , whereν = a, b andr is the position ofthe unit cell. The local number of electrons, for a half-filledband, is obtained by:na,b(r) =∫ 0−Wdωρa,b(r, ω) , (7)whereW = 3t is half of the total electronic bandwidth.Strong suppression of hopping and vacancies.A vacancycorresponds tot0 = t, leading to a simplified form forGaa:Gaa(k,p) = G0kδk,p +1Nciωn−G0k[Ḡ0]−1G0pNc, (8)whereGaa, G0 and Ḡ0 also depend onωn, and the on-siteimpurity potential was set to zero (ε0 = 0). Because the va-cancy creates a three-site zig-zag edge, one expects the ap-pearance of zero energy modes [8]. The1/iωn term in (8)leads to a contribution,ρa(0, ω) ∝ δ(ω) in the total DOS, asshown in Fig. (3). This contribution comes from the atomthat has been disconnected from the rest of the lattice. Itcorresponds to exactly one state and to properly represent amissing atom this contribution to the DOS should be ignored.At small energies (ω ≪ t) we obtain for the nearest neigh-bor B-sites thatGbb(0, 0, ω) ≈ −1/(9t2Ḡ0(ω)), in agreement3with previous studies [8, 39, 40] for vacancy modeled as in-finite on-site potential,ε0 = ∞. In this limit, ρb(0, ω) ≃(√3|ω|/6πt2){1+(36π2/27)[ω ln |√3ω2/6πt2|]−2} giving aresonance atω = 0. The full numerical calculation for all en-ergies is shown in Fig. (3) for the impurity site and in Fig. (4)for the nearest neighbor sites, with the spatial dependenceofthe amplitude of the low-energy states shown in the insets.FIG. 3: (Color online) LDOS fort0 = t andt0 = 0.9t at the impu-rity site. Main graph corresponds toρa(0, ω) for t0 = t (straight blueline) and fort0 = 0.9t (dashed red line). Left inset:ρa(r, ω = 0)for t0 = t. Right inset:ρa(r, ω = −0.075t) for t0 = 0.9t. Noticethe scale difference.FIG. 4: (Color online) LDOS fort0 = t and t0 = 0.9t at thenearest neighbor sites of the impurity. Main graph corresponds toρb(0, ω) for t0 = t (straight blue line) and fort0 = 0.9t (dashedred line). Left inset: ρb(r, ω = 0) for t0 = t. Right inset:ρb(r, ω = −0.075t) for t0 = 0.9t.For t0 slightly smaller thant, hopping to the impurity siteis strongly suppressed but not completely absent. Hybridiza-tion between the low energy states at the impurity site andat the nearest neighbors sites leads to splitting into two res -nant states, one with energy shifted to a negative value and theother to a positive value, as shown in Figs. (3-4).Nitrogen substitution.As discussed previously, a N atomhas a smaller atomic radius than a carbon atom, and this cor-responds to a smaller hopping amplitude between the N impu-rity and the neighboring carbon atoms. In addition, the largeratomic number of N atoms gives a negative on-site potentialε0 with respect to the carbon sites.−3 −2 −1 0 1 2 300.10.20.30.40.50.60.70.80.9ρ A(0,ω)ω−3 −2 −1 0 1 2 300.050.10.150.20.250.30.350.4ρ B(0,ω)ωFIG. 5: (color on line) Left: LDOS at the impurity site (above) andat its nearest neighbors (below), fort0 = 0.5t. Right: intensity plotsof ρa(r, ω = −0.525t) (top), andρb(r, ω = −0.975t) (bottom).We first study the effects of hopping disorder alone, andconsider the caset0 = 0.5t. Fig. (1) shows an intensity plotof the real-space distribution of the number of electrons, givenby (7). Interference effects give rise to Friedel oscillationsdisplaying the underlying six-fold symmetry. The main con-tributions in the sum over negative states (7) come from: (i)the resonance states created by the impurity; and (ii) the spec-tral weight under the van Hove singularity that is also affectedby the impurity. Fig. (5) shows these two contributions sepa-rately. In the left, we show the LDOS spectrum at the impu-rity site, and at its nearest-neighbors (B-sites). The spectrumshows a resonance peak atω0 = ±0.525t on the impurity,while for its nearest-neighbors it is mostly dominated by thevan Hove peaks atω ≈ ±t (ω0 = ±0.975t). The size of thesepeaks decreases away from the impurity and Fig. (5) showsthe intensity plots of these peaks in real-space, which can bedirectly measured using STM spectroscopy.When a negativeε0 is also added, the LDOS gets modi-fied as shown in Fig. (6) for the impurity and nearest neigh-bor sites. Any finite potential value at the impurity breaksparticle-hole symmetry and as a result the spectrum becomesasymmetric under reflection (ω → −ω).Boron substitution.In this case the impurity site has largerhopping amplitude and a positive on-site potential. Fig. (7)shows the LDOS at the impurity and the nearest neighbor sitesfor a case witht0 = −0.5t (so that the hopping amplitude tothe impurity,t − t0, is larger than the homogeneous system)andε0 = 0.525t. The peaks of the LDOS at certain ener-gies originate from impurity resonance states and van Hovesingularities. The real-space intensity plots are at resonance:ω0 = −2.935t for the sublattice-A LDOS, andω0 = −0.975t4−3 −2 −1 0 1 2 300.20.40.60.811.21.4ρ A(0,ω)ω−3 −2 −1 0 1 2 300.050.10.150.20.250.30.350.40.45ρ B(0,ω)ωFIG. 6: LDOS for the caset0 = 0.5t andε0 = −0.525t, at theimpurity site (left) and at the nearest neighbor sites (right).−3 −2 −1 0 1 2 300.020.040.060.080.10.120.140.160.18ρ A(0,ω)ω−3 −2 −1 0 1 2 300.050.10.150.20.250.30.35ρ B(0,ω)ωFIG. 7: (color on line) Electronic spectra for the caset0 = −0.5tandε0 = 0.525t, corresponding to an impurity such as boron. Left:LDOS at the impurity site (top) and nearest neighbors (bottom).Right: real-space intensity plot ofρa(r, ω = −2.935t) (top), andρb(r, ω = −0.975t) (bottom).(at the van Hove singularity) for the sublattice-B LDOS. Justas for the case of N shown in Fig. (5) the van Hove singu-larities are more strongly affected at certain directions.Thevan Hove peaks are strongest on B-sites where the LDOScan be understood as being originated by three impurities (ththree nearest neighbor B-sites of the actual impurity site)andhence the star-shaped symmetry shown in the intensity mapsatω0 = −0.975t in Fig. (5) and Fig. (7). Near the band edge,the band structure of the clean system behave more like a con-ventional 2D system, so the resonance peak atω0 = −2.935thas a symmetric spatial distribution (Fig. (7), top).Conclusions. We find the exact electronic Green’s func-tions in the presence of an impurity that modifies both thelocal atomic energy as well as the hopping amplitude withneighboring C atoms. We have shown that the presence ofthe impurity leads to strong modifications of the local den-sity of states and also to the presence of unusual Friedel os-cillations, both quantities being accessible to measurementthrough STM spectroscopy. We have applied our theory tothe case of substitutional B and N and found that these twoatoms have very specific spectroscopic signatures. From theknowledge of these Green’s function one can also calculatethe transport properties of chemically substituted graphene.We thank the hospitality of the KITP at UCSB, (NSFgrant No. PHY05-51164), where part of this work was per-formed. We thank A. D. Klironomos and V. M. Pereira foruseful discussions. JMBLS and NMRP acknowledge financialsupport from POCI 2010 via project PTDC/FIS/64404/2006.A.H.C.N. was supported through NSF grant DMR-0343790.[1] K. S. Novoselov,et al., Science306, 666 (2004).[2] K. S. Novoselov,et al., Proc. Nat. Acad. Sc.102, 10451 (2005).[3] K. S. Novoselov,et al., Nature438, 197 (2005).[4] Y. Zhang,et al, Phys. Rev. Lett.94, 176803 (2005).[5] Y. Zhang,et al.Appl. Phys. Lett.86, 073104 (2005).[6] Y. Zhang,et al.Nature438, 201 (2005).[7] C. Berger,et al., J. Phys. Chem. B108, 19912 (2004)[8] N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev.B73, 125411 (2006).[9] A. H. Castro Neto, F. Guinea, and N. M. R. Peres, Phys. World19, 33 (2006).[10] A. K. Geim and K.S. Novoselov, Nature Mat.6, 183 (2007).[11] V. M. Pereira,et al., Phys. Rev. Lett.96, 036801 (2006).[12] Yu. V. Skrypnyk and V. M. Loktev, Phys. Rev. B73, 241402(2006).[13] V. V. Cheianov and V. I. Falko, Phys. Rev. Lett.97, 226801(2006).[14] T. O. Wehling,et al.cond-mat/0609503.[15] M. A. H. Vozmedianoet al., Phys. Rev. B72, 155121 (2005).[16] A. Cortijo and M. A. H. Vozmediano, Nucl.Phys.B763, 293(2007).[17] O. V. Yazyevet al., Phys. Rev. B75, 115418 (2007).[18] O. V. Yazyev and L. Helm, Phys. Rev. B75, 125408 (2007).[19] A. H. Castro Neto, F. Guinea, and N. M. R. Peres, Phys. RevB73, 205408 (2006).[20] N. M. R. Peres, A. H. Castro Neto, and F. Guinea, Phys. RevB73, 195411 (2006).[21] Y. Niimi, et al., Phys. Rev. Lett.97, 236804 (2006).[22] Y. Niimi, et al.Phys. Rev. B73, 085421 (2006).[23] G. Li and E. Y. Andrei, arXiv:0705.1185 (2007).[24] E. J. Mele and J. J. Ritsko, Phys. Rev. B24, 1000 (1981).[25] M. Endo,et al., J. Appl. Phys.90, 5670 (2001).[26] O. Stephan,et al., Science266, 1683 (1994).[27] T. B. Martinset al., Phys. Rev. Lett.98, 196803 (2007).[28] M. Yudasakaet al., Carbon35, 195 (1997).[29] R. Sen,et al., Chem. Phys. Lett.287, 671 (1998).[30] M. Terrones,et al., Appl. Phys. Lett.75, 3932 (1999).[31] D. Golberg,et al., Carbon38, 2017 (2000).[32] W.-Q. Han,et al.App. Phys. Lett.77, 1807 (2000).[33] R. Kurt,et al., Carbon39, 2163 (2001).[34] R. Kurt,et al., Thin Solid Films398-299, 193 (2001).[35] S. Trasobares,et al., J. Chem. Phys.116, 8966 (2002).[36] C. J. Lee,et al., Chem. Phys. Lett.359, 115 (2002).[37] M. Terrones,et al., App. Phys. A74, 355 (2002).[38] T. Guo, C. Jin, and R. E. Smalley, J. Phys. Chem.95, 4948(1991).[39] P. A. Lee, Phys. Rev. Lett.71, 1887 (1993).[40] A. V. Balatsky, M. I. Salkola, and A. Rosengren, Phys. Rev. B51, 15547 (1995).

We present exact analytical and numerical results for the electronic spectra and the Friedel oscillations around a substitutional impurity atom in a graphene lattice. A chemical dopant in graphene introduces changes in the on-site potential as well as in the hopping amplitude. We employ a T-matrix formalism and find that disorder in the hopping introduces additional interference terms around the impurity that can be understood in terms of bound, semi-bound, and unbound processes for the Dirac electrons. These interference effects can be detected by scanning tunneling microscopy

EDP Sciences OAI-PMH repository (1.2.0)

Electron waves in chemically substituted graphene

Abstract

Similar works

Full text

Available Versions

Crossref

Crossref

eScholarship - University of California

EDP Sciences OAI-PMH repository (1.2.0)